This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
by a formula
consisting of n alternating unbounded quantifiers (beginning with an existential) A function i s Zn if i t s graph is.
followed by a limited formula.
The f i r s t type of parameter that we define measures the extent to which i s not a cardinal. that there i s a Z
p, p
The B -projecturn of injection of
p into y.
n
, i s the
Jensen shows
least ordinal
("31) that
a s the least y such that some Zn subset of y i s not p-finite. a Zibijection between
p
y such
this is the same
A s there is always
p and SB (see [i]), we can in fact inject S into p np via a
Z function. Our second set of parameters describes the extent to which The B -cofinality of n
p,
Z cf
p i s singular.
p, is the least y such that some 2n function with
domain y has range unbounded in
p.
same a s the least y such that some B
If n
p is
Bn-i-admissible, then this is the
function with domain y is not @-finite
(though this equivalence is not true for a l l p).
AN INTRODUCTION TO 6-RECURSION THEORY
117
In case n = 1, p p and Z cf j3 a r e alternatively written j3* and Rcf p, 1
1
respectively.
(Rcf abbreviates 'Recursive Cofinality".)
concerned with p*, Rcf p, p' and Z2cf (3. 2 is a regular p-r.e.
projectum and Z
1
We shall be mostly
Note that i f p i s admissible and A
set of degree 0'. then p p and Z cf 3 , a r e just the Z 2 2 1 cofinality of the relativized structure < L
B'
e, A>.
In case Rcf p 2 p* we say that p i s weakly admissible. many of the arguments from admissibility theory apply.
-
In this case,
The reason for this i s
that many priority arguments use p* to index a listing of requirements and the
*.
above assumption allows one to perform Z l inductions of length j3 Z -Uniformiaation is also easy in this case.
2
If
p i s admissible and
p 2 p! , then we say that j3 i s weakly C 2 -admissible. In this case, one can
Z cf
2
c a r r y out the construction of minimal p-degrees. minimal pairs of degrees and major subsets of
p-r.
p-r.
e.
e. sets.
If Rcf p < p* we say that p i s strongly inadmissible.
In this case, the
arguments of admissibility theory do not apply and new techniques a r e needed. This i s the difficult case of Z -Uniformization. 2
p < p!
Z cf
2
, then we
say that p is strongly Z2-inadmissible.
of minimal p-degrees, minimal pairs of
p-r.
If p i s admissible and
p-r.
The constructions
e. degrees and major subsets of
e. sets a r e all very difficult for such p and have only been accomplished in
v e r y special cases.
However, the techniques of p-Recursion Theory a r e now
beginning to apply themselves to this case (see Section 5).
Section 3 .
Weak Admissibility
As mentioned before, the methods of @-RecursionTheory apply in this case. prove:
In particular, the method of Shore blocking (see [17]) was used in [3] to
118
SY 0. FRIEDMAN Theorem 3.
If
p is weakly admissible, then there a r e regular t. r. e.
sets A,B such that A
WB
dwpA.
B, B
W. Maass (in [lo]) has found a technique for transferring many results f r o m a-Recursion Theory to a r b i t r a r y weakly admissible ordinals. ciates to each weakly admissible p an admissible structure
01-r. e.
degrees embed into the
p-r.
fi
about the admissible structure
He asso-
a such that
In this way, known results
e. degrees.
have consequences about the p-r.e.
degrees.
We now describe his construction in more detail. Let
K
= Zlcf B.
As
K
2 p*, there certainly i s a Ziinjection of
In fact, more i s true: there i s a 2 Let f:
B
+
be such a bijection.
K
< e, x. u> w
and T = f[T]. missible. (Zl over
Then TG
K,
Moreover, if A Define
).
5
< -0L
bijection of /3 onto K (see [ 3
F
c
a
then A is
x p-finite, x C_ A 4x c L
Then 5 -1'.
oc
and
I
B
- A +x
A
a-r.e.
Theorem 4 (Maass)
is p-immune if
K
K '
c LK ,
Maass shows that every
M-r. e. representative. This gives an em-
degrees 1-1 into the p-r.e.
( p weakly
degrees = the recursive degrees.
01 -r. e.
Then these two reducibilities
B'
C
= < L K , c r T > is ad-
e. if and only if A is
do agree on p-immune sets.
e. degree has a p-immune
bedding E of the
p-r.
analogously to 5
p-finite, x & K
p. 15).
weu
x c
p-recursive and
do not necessarily agree on subsets of K.
x
1,
K.
Let
T is K,
1
p into
admissible).
E (complete
An application of admissibility theory to
degrees.
The range of E = the t. r. e.
(n -r.e.
u(
set) = 0 1/2
.
([15] and [lb]) yields:
AN INTRODUCTION TO B-RECURSION THEORY
119
Any nonzero t. r. e degree is the join of two l e s s e r t. r. e.
Corollary.
If one t. r. e. degree i s below another, then there is a t. r. e. degree
degrees. in between.
Section 4.
Strong Inadmissibility
This is the most challenging case for p-Recursion Theory, for the lack of admissibility i s now so strong that many of the ideas from the admissible case become useless.
The alternative i s to employ deeper techniques from the Fine
Structure of
L a s developed initially by G'ddel [ 8 ] and more extensively by
Jensen [ 9 ] .
A l l of these techniques emanate from two basic lemmas due to
Godel:
-
Lemma. h: w X S
B
S
B
F o r each limit ordinal
which is Z over S 1
<SBl (>
Proof.
B
B,
there i s a partial function
such that f o r any Zlformula p(x, p),
3 x d x . P) +3 i e w(p(h(i,PI, P) .
Recall the canonical Zl well-ordering
ith E l formula is
3 y$(x.
S
B'
Then i f the
define h@(i, p) e least (in the sense of < )
p, y),
pair <x, y > such that $(x, p, y).
< of
Then h(i, p) = first component of hf(i, p).
The h above i s called the canonical Z Transitive Collapse Lemma.
1
If X < S 1
f
skolem function for S
B'
B
(i.e., X C_ S
B
and any Zl
formula with parameters from X and a solution in S has a solution in X ) then
B
< X , t > i s isomorphic to a unique
<SY,c
>.
Using these two lemmas, we can now illustrate in a simple example how Fine Structure technique can be used to generalize to arbitrary p a result whose "recursion-theoretic" proof only succeeds for admissible
B.
SY D. FRIEDMAN
120
Suppose A C y
Proposition (Jensen).
< p * and A i s p-r.e.
Then
A is p-finite.
p admissible.
Proof N u m b e r d ,
$ - r e c u r s i v e listing f: p
But then p* I s u p A 5 y Let p
G
S
B
< p*,
contradiction.
be a p a r a m e t e r defining A as a
F o r m X = Range h on O X (y u{p}), where h i s f r o m the
B'
4
Lemma.
Then X
j:X c r S
Let g = joh.
6 '
A.
p arbitrary.
Proof Number 2, s e t Zl over S
-
If A is not p-finite, then it h a s a 1-1
1
so apply the T r a n s i t i v e Collapse L e m m a t o get
S
B'
Now g is Zlo v e r S (simply t r a n s f e r the Zci definition f o r h o v e r X 6 to S 6 ) . f
Then so is g-'.
But i f f uniformizes
injects S6 into o X ( y u { p } ) ;
that 6
gWi, f Z o v e r S we s e e that 1 6* Since y C p*, we have proved
hence into y .
< p. But A is Z definable o v e r S 6 , so A 1
G
-/
S
B'
F u r t h e r ideas of Jensen, in p a r t i c u l a r a n effectivized v e r s i o n of h i s
0
principle, w e r e used in [4] to establish: T h e o r e m 5. then t h e r e a r e
p-r.
If
p*
is r e g u l a r with r e s p e c t t o p - r e c u r s i v e functions,
e. s e t s A , B 5
p* such that A
9
B,
WP
B
bwpA
.
This i s the b e s t solution to P o s t ' s P r o b l e m so f a r known i n the strongly inadmissible case.
This c o v e r s the c a s e where S
B
I=
"p* is a s u c c e s s o r
c a r dinal If. Open P r o b l e m . inadmissible
Does the conclusion of T h e o r e m 5 hold f o r a r b i t r a r y strongly
p?
F o r c i n g can be used to achieve a s t r o n g e r and m o r e model-theoretic inThe following r e s u l t will appear i n [5]:
comparability than that in T h e o r e m 5. T h e o r e m 6. and S
B
"p
Assume
p* is r e g u l a r with r e s p e c t to p - r e c u r s i v e functions
* i s the l a r g e s t cardinal.
'I
Then t h e r e a r e
p-r.
e. s e t s A , B
5 p*
121
AN INTRODUCTION TO B-RECURSION THEORY such that
A is not hi over < S [ B ] , c > ,
B
B is not A i
pth
(S [ A ] is the
B
>
over < S [A].€
B
.
level of the S[A]-hierarc
This
y.
ierarchy i s
fined exact .Y
a s the S-hierarchy except the function f(x) = A n x i s added to the schemes for the rudimentary functions. ) We conclude this section by sketching the proof of a theorem which illustrates the use of Skolem Hulls and Theorem 7.
There a r e
p-r.
0in p-Recursion Theory.
e. sets A , B such that A
$
f
s B,
B$
fB
A.
The proof of this theorem is not uniform in the sense that it divides into
p. Thus, the sets A , B will be defined relative
cases depending on the nature of
t o the choice of a parameter p c S
B'
Open Problem.
Can Theorem 7 be made uniform in that the sets A, B have
parameter-free Zidefinitions independent of We believe that the answer i s "yes.
p? In fact, we
There a r e integers m , n such that for all limit ordinals p,
Conjecture.
w,"
wft
over < S [w,B],c >
is not
B
is not A~ over < S
,
[w'I,~> ,
B
m
where W p = the nth parameter-free p-r.e. n
set.
Before giving our proof sketch of Theorem 7, we make some preliminary definitions and remarks.
In view of Theorem 3, it suffices to prove Theorem 7
-
i n the strongly inadmissible case. f .S 0' 8
i-i
_9
unbounded in
*
f3
and go: Rcf p
p. Let p'
parameter p i .
O
c S
B
Choose p-recursive functions
p such that go i s order-preserving, Range
be such that both f
Let po = < p i , p*>
.
0
g
0
and g a r e Zlover S in the 0 B
SY D. FRIEDMAN
122
< p*
Let h(i, p) be the canonical X i skolen function f o r S and for y
B
define H(y) = { h ( i , < y ' , p o > ) ( - ic w.y' < y } .
u
u
Thus H(y) is the
llZi
Skolem H u l l
.
H(y) = Sp In our construction, H(y) consists Y < B* (Thus the construction i s r e of those reduction procedures e of priority y. of y
Note that
{p,]":
dundant in that each reduction procedure is assigned a final segment of different Of special importance a r e those y
priorities.) Claim.
Let
K
< p* be a p-cardinal (i.e., Sp
the next p-cardinal. Proof.
Then' { y
= {y
,
put x into B and have y r e s t r a i n the m e m b e r s of y f r o m entering A . pairs
B
< R e , z > a r e handled similarly.
This ends the construction.
The idea, then, is that the m e m b e r s of D provide 8tguesses11at A
nHuh),
B
po defines the e n t i r e construction, j [ B n Hu(y)] E Dy i f y
6
a.
n H'(y).
Y
(via the bijection j: Hu(y)
implies that A!.
, one
j[A
H'(y)],
of the "guesses" is c o r r e c t .
Then t h e s e guesses are each used to s e a r c h f o r an x and y which attempt to satisfy
R:
( o r R B~ ) .
-
Of c o u r s e , since the p a r a m e t e r
oB"
So f o r y e
The
y)
124
D. FRIEDMAN
SY
A Now each pair < R e , z>, e e H(y), z c D
Y
is attacked at most once at
each stage of the construction; thus, any x put into A o r B by y and any y restrained from intersecting A or B by y must belong to H(yi-1). Lemma.
Otherwise, let y
Proof.
y have the same some
6
E
T
< u', y
H"(y't1)
-
Suppose y e H'(ytl)
6
-cardinality HT(6)
-
and so y
K
H
Let ,6,
Then y { H(y). Assume that y' and
be the least 6
A o
L e m m a 2. Proof. -
=
x
.
For A
t
< wL then C* 5 1
X,
i s @-finite, this shows that
5 w w , A and s o we a r e done by L e m m a 1.
-I
WB
A.
131
NEGATIVE SOLUTIONS TO POST'S PROBLEM.1 L
< S for sufficiently large A < w1 , then A i s X
If f ( h ) A
Lemma 4. p-finite. Proof. -
Suppose f (A) A
< SA for X LAi ? L o .
,
g(X) = pn[fA(h) < ] : S Then for some fixed n
for A
Define:
LA1.
0'
X = {A
I fA(X)
P(x,a))I,
-
d e n o t e s a sequence of v a r i a b l e s of t y p e s 0 and 1
and n L i, a r e c l o s e d under u n i v e r s a l q u a l i f i c a t i o n ( e f e e c t i v e l y i n a n index f o r P ) .
Proof;- T h i s f o l l o w s immediately from t h e f a c t t h a t t h e r e exist rec u r s i v e (indeed elementary) maps from Ass,,
8
(Observe t h a t f o r any a , Ca i s isomorphic t o CaxC,.
x
So
Cn i s isomorphic t o CnxCn which e a s i l y maps o n t o clxCn.
Assn.
for n t 1 The map so
c o n s t r u c t e d w i l l be o n t o a t t h e l e v e l of a s s o c i a t e s ) . p r o p o s i t i o n 3.2. Proof:-
1
For n t I, A s s r l i s a complete IIn-l
set.
The c a s e n = 2 , t h e b a s i s f o r an i n d u c t i o n , i s easy.
r e s u l t t r u e f o r n.
In
1
Then a n a r b i t r a r y
GI
( 3 a )@(;,a)
c ASS,,I,
f o r some r e c u r s i v e f u n c t i o n a l 8, i.e.
(3B)(B
set i s of t h e form
of t h e form
c Assll
&
(3 a ) ( B
= Q(2,a)))l-
Thus a n a r b i t r a t y IIA s e t A i s of t h e form
GI
( V a )(
so u s i n g (3.1) of t h e form
v B) (B
c ASS,
->
B # o ( 2 , a ) )3,
Suppose
COUNTABLE OR CONTINUOUS FUNCTIONALS Now d e f i n e Y by
(0 i f
Y(x') (u) =
Then
x'
(3vcu)T(e,
llOt
( l h (V),
V)
otherwise.
11
E
x'
141
is c l e a r l y I[ 1I1, w e have
Since
A i f f Y(x')s
completed t h e i n d u c t i o n s t e p . There a r e some immediate c o r o l l a r i e s of t h e above r e s u l t and L e t "0 be t h e everywhere z e r o f u n c t i o n a l of cyl'e
lxoof.
c o r o l l a r y 3.3.
For
Proof :- By t h e above argument, c o r o l l a r y 3.4. Proof:-
x'
~ k s-e t .~
i s a complete
z 2 , ASS("O)
11
E
A iff Y
(2)
E
11.
Ass('l0).
1
For n z 2 , ct-2-env(%)
= 11~~.
c o n s i s t s of a l l sets of t h e form
ct-Z-env("0)
{tl ( d a ) ( a
A S ~ ( " O )->
~ ( x ' , a ) )where ~ , P is
1.;
and i s c l o s e d under s u b s t i t u t i o n of
So c l e a r l y ct-2-env("O)
5
recursive functionals.
I t remains t o show t h a t ct-l-env("0)
i t a i n s a complete 11, s e t . argument f o r ( 3 . 2 ) .
@
to
Ass(%)
11;
con-
But t h i s f o l l o w s by t h e f i r s t p a r t of t h e
(The e x i s t e n c e of a recursive onto map from
x Ass(*'O)
i s much a s i n t h e proof of (3.1)
-
though
it l a c k s t h e s t r u c t u r a l m o t i v a t i o n of t h a t r e s u l t ) . The g e n e r a l i z a t i o n s of
(3.3) and (3.4) t o a r b i t r a r y F (of Lype
n z Z ) , i n v o l v e s u g l y coding problems; my p r o o f s r e l y on equivalences from Hyland [ & I so I do n o t g i v e them here, b u t simply s t a t e t h e results.
Suppose E' i s of t y p e n, n z % and l e t
$.
g i v e t h e value
of E' on some r e c u r s i v e dense sequence i n Cn-l. Theorem 3.5.
1
( a ) Ass(F) i s a complete JIn-l(hF) ( b ) ct-2-envtF)
set
= RA(hF).
Remarks 1) (3.5) ( a ) i s proved i n f u l l d e t a i l from a completely d i f f e r -
ent p i n t of view i n Norman [ill; (3.5) ( b ) c o u l d a l s o be o b t a i n e d u s i n g h i s methods. 2)
Ell],
(3.5) (b) should be c o n t r a s t e d w i t h t h e r e s u l t of Norman
t h a t i n t h e sense of Kleene (S1-SY)
lI;-:_,(hy)
(F of cype 3 o r more).
recursion,
L-env(F) =
142
J.M.E.
HYLAND
(3) The most s i g n i f i c a n t f e a t u r e of Norman [ i l l is h i s a b i l i t y
(see h i s Theorem 3 ) .
t o handle 1 - s e c t i o n s
A t t h e moment t h e r e is
nothing corresponding f o r c o u n t a b l e 1 - s e c t i o n s ( c f . Remark 3 of 52).
54.
D e f i n i t i o n s by r e c u r s i o n on t h e i n d u c t i v e d e f i n i t i o n of C2.
The o u t s t a n d i n g q u e s t i o n concerning t h e c o u n t a b l e f u n c t i o n a l s is whether one can o b t a i n t h e i n t r i n s i c r e c u r s i o n t h e o r y by applying t h e u s u a l i d e a s of g e n e r a l i z e d r e c u r s i o n theory. r a i s e d i n embryonic form by Kreisel
[io].
T h i s problem w a s
I t was c o n s i d e r e d i n
Hyland C71, b u t t h e t e n t a t i v e l y n e g a t i v e c o n c l u s i o n reached t h e r e
w a s based i n p a r t on c o n j e c t u r e s which have s i n c e been disproved. I t is d i s c u s s e d i n d e t a i l i n Feferman C43, where a p o s i t i v e answer t o t h e corresponding q u e s t i o n f o r t h e p a r t i a l continuous f u n c t i o n a l s A t f i r s t s i g h t t h e problem seems t o be one of fiiiding
is i n d i c a t e d .
t h e " r i g h t s t r u c t u r e " t o p u t on man C41).
However
be a structure. We w i l l assume i n the following t h a t M contains a copy N of u) and t h a t both N and the r e l a t i o n 5" which is the copy of the natural ordering
N,
2 with w,
w a r e among M, R1
5 in
... Rd.
As usual we shall i d e n t i f y
t+ following. I n general we shall use lower case l e t t e r s y, z as variables over M and c a p i t a l l e t t e r s A, B, C, X,Y, 2
..., x,
a, b, c,
E X)] = 8 (the Souslin quantifier).
2 f'unctionds.
A convenient reference for w h a t follows
i n t h i s and the next two sections
[K-MI, especially i n regards t o unexplained notation and terminology.
is
ALEXANDER S. KECHRIS
158
I f : f is a partial function prom M i n t o u d . A type 2 (partial, m s otone) functional on M is a p a r t i a l map cp : M"XPF(M)k+ u) which is monotone i.e.
Let PF(M) =
~ ( 2 f,l
co(t g1 Example. Here f ( 3
Let
6 be
2 a EM(f) =
R
f 0 abbrevlates
... fk ) = w A ... = w.
5 g,
fl
A
... A
fk
5%*
if 3x(f(x) = 0) if
Vx(f(x)
f 0).
"f(x) is defined and has value
a f i n i t e l i s t of type 2 finctionals on
V.
f
0".
Put
the 2-envelope of (p (on m) =def t h e collection of all second or2env(70,a der r e l a t i o n s on M which are semirecursive i n (p and t h e characteristic functions
of =, R1
... Rc.
These turn out t o be Spector 2-classes for c e r t a i n types of well-behaved (p called normal.
7.1.
Definition
called normal i n
(Moschovakis [Mos 41). A type 2 functional cp(f) on M is if there is a functional A (f,g) recursive i n such t h a t
T
cp
1 [x: g ( 3
i) cp(f
= 0)) 1 * Acp(f,g) = 0
-
{z: g ( 3 = 0) t_ d m ( f ) A r ::I g ( 3 = 01) t A'p(f,g) = I .
ii) g t o t a l A cp(f
Here I
I
means "is defined" and
general functionals
cp(z,?)).
"is undefined". (A similar definition applies t o
(p normal on ?4 i f each cpi
Call
t h e c h a r a c t e r i s t i c functions of =, R1 7.2.
Theorem
functionals on
V.
(Moschovakis [Mos 3,4]). Then 2env(m, E',;
such t h a t each functional i n Here a type 2 functional
5 is r
E
... RL.
3
(p is normal i n (p and
Let (p be a normal sequence of type 2
'4,
is the smallest Spector 2-class on 77f on r. Similarly for env(R 3.
...fk)
-
is r on r (where r is a 2-class a similar if for each 4 i(Fi, Fi, Ti), 1 5 i
yk))
r.
Again the key t o the proof of Theorem 7.2 is Theorem 3.4.
r
...
T h e condition t h a t (p is
guarantees that the operator defining a universal set i n 2env(q 2%,
say via KLeene's schemes, is
on
r.
3,
SPECTOR SECOND ORDER CLASSES AND REFLECTION
Remark.
Again, i f
m = 3 E#
159
can be dropped i n 7.2.
Let us consider now some examples. 2 1 ) Every type 2 object F 5 p(M) can be i d e n t i f i e d with the t o t a l function 2F : wM
+
u given by 2
{
F(f)
0, i f I,
if
f i s t o t a l A {x : f(x) = 0) E
2 F
f is t o t a l A (x : f ( x ) =
2 ~ .
01 k
2 2 F undefined on s t r i c t l y p a r t i a l f : M 4 (ti we can v i e w F i n a n a t u r a l way as a type 2 functional. Then it i s not hard t o check that 2EM, 2F i s normal BY making
for any f i n i t e l i s t
2
F of
type 2 objects.
Moreover
2env(R 2 ~ 23 , = 2env(q ,E:‘
= smallest Spector 2-class
2~M,
r such t h a t
so that we recover the example i n Section
2) (Hinman [Hi]). object 2F sdefQ i . e .
‘3 each object i n
6.
Let Q be a quantifier on M.
2F i s
i n A,
We attach t o Q t h e type 2
Q
2
0,
if
f i s t o t a l A Qx f ( x ) = 0
0,
if
Qx f(x) = 0
1,
if
&
t,
otherwise.
and the type 2 functional f(X) f 0
2 + It turns out again (see [K-MI) t h a t F i s n o h l Q and moreover by t h e various minimality characterizations mentioned before one can see t h a t
thus
2
F x =~ 2EM, ‘E$
=
2%
EM.
2 e n v ( ~,E‘: the smallest Spector 2-class 2env(?% ‘E,;
=
2 m ( ~ = ~ )
r on ?Ir closed under
both Q , 6 .
On t h e other hand
2FQ) = smallest Spector 2-class
on V such t h a t
Q E A = smallest Spector 2-class Q,< i.e. i f 8 , a ~ re i n
r
such that A i s uniformly closed under both then t h e r e a r e 8,s’ i n r such t h a t f o r each X,T:
This i s usually a l s o expressed by saying that Q and Q.
r is
closed under t h e “deterministic“
ALEXANER S. KECHRIS
160
‘4,
In general, 2env(vb example, i f ? =
JU,
2 ~ Q )< 2env(q ,:E‘
1
08.
For
is Q = 3 ) .
2 2 2 * and Q = 8 then 2env( E, 2Fs) = 2env(E1), while 2env( E, Fs) =
21ND(s) =21ND(zl) (by a theorem of G r i l l i o t ) . 21ND(x; )
- an exception
‘F;)
>> 2env(El )
.
It i s of course well-known that
The second order hierarchy.
The 2-classes
Ill(@are defined as usual:
-
el(@,
Ei (19 =
ni ( ~ =3
(3f
Q(X,i?,,jF)
: 8 (lightface!)
elementary on
(Q
1
r;+2(71) = { 3 f 8(X,T,,jF) : 8 E IIn+l (743 n;+2(171) = for any n
>- 0.
GI,
Various 2-classes i n t h i s hierarchy give r i s e t o Spector 2-classes i n certain cases as the next two theorems show. Theorem.
8.1.
z;
i) (Kleene [ K l 31 for ?I= Banrise-Gandy-Moschovakis [B-MG] i n genFor each countable l’((nl) E ~ I N D ( ~ ( ) .
eral).
ii)
1
(Moschovakis [Mosl]).For each countable ?(, Z2(V) is a Spector 2-
class. Theorem 8.1 holds also for certain uncountable q ’ s of “strong c o f i n a l i t y w“ e.g. where cof(k) = w (Chang-Moschovakis; see [Mos 11).
(V,,E),
8.2. If
n
i)
m is
>- 0,
{X 5 M:
Theorem. (Martin [Ma], Moschovakis [A-MI). Assume Projective Determinacy.
countable,
ii)
then
(Addison [Ad]).
are Spector 2-classes.
3% Ex,
i
4(@,
then
... vi
x ~ ( vis)
(xi,
1
all $n+l ( V ) , ZAn+2(@ for n
>- 0 are
Spector 2-classes.
Assume V = L. Then for any V, all &+2(R),
where
i s e s s e n t i a l l y uncountable i . e . X. 3= I f moreover ??I xi+l ) E X)
(5
t h e notion of wellfoundedness) is i n
a l s o a Spector z-class.
It is i n t e r e s t i n g t o note here t h a t i f one assumes AD, on top of ZF + Dc, then on h = f7 = structure of the reals, we have that C11 (R) is a Spector
the one hand for
AND
SPECTOR SECOND ORDER CLASSES
2-class (Martin, Solovay, Kechris, ...) while on the other i f
ll;(~o&)
i s a Spector 2-class (Kechris [Ke 1 I).
that Con(2FC) con(zFC)
I(
* Con(Zx
4(a)
5 9 . Recursion i n type A type 3 object on
Emple.
’EM = 14
3F and
2env(q
I),
= (al,
is a Spector 2-class).
3 objects.
M i s a collection 3F of subsets of p(M) i.e. 3F
We l e t again in
+
m = cul
Harrington has recently shown both
(z1)i s a Spector 2-class)
* Con(ZFC +
and also t h a t
161
REFLECTION
5 p(M)
3m
Edef
:X
C
p(p(M)).
f #].
all second order relntions which are semirecursive
the c h a r a c t e r i s t i c functions of =, R,
...RL.
L e t 3F be a f i n i t e l i s t of type 3 objects on M. Then i s a Spector 2-class. Moreover it i s closed under vp(M) the deterministic 3P(M) 3’.
9.1.
,env(Q
Theorem.
3EM,
%)
r is closed under r such t h a t f o r each x,P:
Here we say t h a t a 2-ChSS
in
r there
i s 63 E
We w i l l see later t h a t 2enV(!% 3EM,
’3
v’
and
t h e deterministic 8’ if for each Q,2
i s never closed under 3’.
The two c r u c i a l steps i n proving Theorem 9.1. are f i r s t the v e r i f i c a t i o n t h a t 3 2env(% EM, 33i s normed and t h i s i s due t o Moschovakis [Mos 51 and second the
verification that 2env(q 3EM, 3i9 i s closed under gM and t h i s is the G r i l l i o t Harrington-MacQueen Theorem (see Harrington-MacQueen [ H-Ma])
.
O u r next goal i s
t o provide again a minimality characterization of these envelopes. 3
3F.
For t h a t it w i l l be convenient t o separate the r o l e of EM i n the l i s t 3EM,
9.2. Definition. A Spector 2-class on F7 which i s closed under V’ and the deterministic 8’ w i l l be called an E-Spector P-class. Recall a l s o t h a t i f
r
i s a 2-ClaSs and 3F a type 3 object then we say that on A i f for each 8,2 i n r there i s 6) E r such t h a t for each ?,y:
If
B(o(%T,z)0 ?L(z,P,Z)),
then
63(;,3
0
{Z : f?(z,Y,Z))
E
3F.
3
F is
C a l l also
ALEXANDER S. KECHRIS
162 3F -A
i f both
3
F,
7
3
3
Thus for example EM i s A on A i f f
F are A on A. 1
closed under the deterministic 3
, v' .
r
is
W e are now ready t o state 9.3.
Theorem (Moschovakis [Mos 31.
31be
Let
a l i s t of type 3 objects on L on ?$ such that each ob-
Then 2env(n(, 'EM, '3 is t h e smallest E-Spector 2-class j e c t i n 3~ is A on A.
The key t o the proof is t h e f u l l F i r s t Recursion Theorem for Spector 2-classes
-
a basic closure property of such classes under appropriate inductive definitions One of course needs t o consider such
operating on p(M) instead of M this time.
inductive definitions since universal sets i n ,env(m,
3q'3
are defined by op-
erators which a c t e s s e n t i a l l y on p(M) as opposed t o those used i n Sections 4-7 which operate on M and have the m e m b e r s of p(M) carried through as parameters. Let
*(;,?,a)
be an operative t h i r d order r e l a t i o n i . e . S varies over second order
r e l a t i o n s of the appropriate signature so t h a t S(x,?) makes sense. w e l e t again
c'(;;,q
42 ’.
we obtain a proof that v)
v) l e t %(x,Y)
- &(Y)
A
w
be universal. E
- &(Y).
9
I).
Then
~(X,Y)(W = 3tY A
A 3 e E w(We =
X)
A
LX”>
kx).
-1
r
ALEXANDER S. KECHRIS
168
Our next goal i s t o bring f o r t h some aspects of the theory of r i g i d Spector 2classes r e l a t e d t o the comparison between monotone and nonmonotone inductive definability. I f 3 is a 2-class, by
l"on we
which are monotone i n S.
V
= w,t 21ND(E:)
denote t h e collection of operative cp(ii,jf,S) i n 3
A well-known r e s u l t of G r i l l i o t a s s e r t s that for
=,IND(Z; jmon).
The next r e s u l t p r w i d e s the proper content that It was proved originally for most interesting r i g i d r's as a straightforward combination of a r e s u l t of Harrington (14.1. below) and a reexplains this theorem.
sult. of Harrington-Moschovakis (14.2. below).
The general version given next
( a n d a different proof) i s due t o Harrington-Kechris [H-K 11. 13.3.
Theorem.
Let
r
be a Spector 2-class on 7R.
If
F is rigid, then
,IIVD(P) '21ND((?)mn).
It i s an open problem i f tne converse holds as W e l l . A corresponding r e s u l t holds f o r inductions i n
m.
r
Theorem (Harrington-Kechris [H-K 1 I ) .
13.4.
I f ?f3 E A and
as w e l l ( f o r most Pa). Let
r
be
a Spector 2-class on
i s rigid, then 21ND(r) =,1ND(I'mon).
The byyothesis ?f 3 E A i s needed as the counterexample
m shows.
=,IND(m) on a countable
It seems also relevant t o mention here the following 13.5. Theorem (Aanderaa [ A a ] ) . I f 3 is typical, nonmonotone on i s normed and closed under EM then ,IND(Z) < ,IND($. I n particular, for m y Spector 2-class 014.
r,
,Im(r)
o r t = 1,...,l.
Let; = 0, choosing to go one route say to
191
&, or another say to 2, accordingly; so we
have S5.1 (where "cs" is for "case"). S4.0-54.3
allow the result of a computation by
computation by$ ("composition" of computations).
x
to be fed into another
By 54.1, indeed not just a
computed number but a computed function ~5 0 x(0;6 0, I k ) (one whose value for each 0 6 is computed by
x)
is made a type-1 function argument for the further
computation by $; and similarly by S4.2 and S 4 . 3 for type-2 and 3 function arguments.
(In 1959, we did not boldly put the schemata S4.i for 1 > 0 in at the
beginning, but only special cases S8.2, S8.3 of S4.1, S 4 . 2 ; and we depended there on a theorem, XXIII p. 21, rather laboriously proved under certain caveats, in
other cases.) Finally, 57.147.3 give us the Turing oracle-principle by which we can ask for, and receive, the value of a function argument ai (J = 1,2,3) for an argument and SO similarly gives us the value of an assumed function B t (one of 0) for -,
L.
arguments
What else could one need for computation?
& I postpone for a bit describing exactly how computation with these schemata is to be conducted.
It will suffice for the nonce
to
know that the
expressions on the two sides of each schema will receive the same value, if either receives a value, for given values of the variables
and of the functions 0.
First, we must consider how the schemata are used in concert. A functional AeWL $@;@I) will be called partial recursive, or the function X 0 1 $ ( 0 ; d ) partial
recursive in 0 (so if 0 is empty,
An$(a), or
simply 0,. is partial recursive),
iff $ ( 0 ; U Z ) is introduced by a succession of applications of the schemata. Say the applications introduce successively functionals (or functions) $
0 is E SO,
ls-*s@E where Thus, for each 1 @ = 1, ...,E), 0, may be introduced outright by one of
$.
-
In the cases of the other schemata, the $I
S1.0, S1.l, S2.0, S3. S5.1, S7.1.
on the right (S6.1, Sll), or the $ and
$l,...,$i-l.
-
x
on the right (S4.1), must come from among
...,2 z for the schema applications
The list of indices c1,
introducing successively $l,...,$
E
contains complete in itself all the details of
the definition of $ from 0. Indeed, just 2 = z does!
72
192
S . C . KLEENE
RECURSIVE FUNCTIONALS AND QUANTIFIERS OF F I N I T E TYPES REVISITED I
193
so
Thence, with
the result is easily seen. In this example, $1,...,$15 are introduced as functions of number variables partial recursive in
n,e, where
q,e
can each be any assumed partial function of
two number variables. But by application of S11, $16 is partial recursive just in 8;
and then $17 and $18 are also.
.4,
..,$,
Thus, if we had before us just the derivation
and didn't know of the coming application of S11, we would be asking
for values of two assumed functions r1,8 in our computations. But when $l,...,$15 is built into the derivation of $18, the
more explicitly with A& not $,.,$, shall feed ,$,
comes to be identified with $16 (or
$16(e;y,k). It is $18 we are really interested in (and At the step introducing $9, which is where
per se).
back in rather than using a value of
TI
enters, we
n as an independent assumed
function. We could have emphasized this by writing $16 in place of
q
in
$l,...,$15; but that could be puzzling to a person reading the derivation forward
for the first time. We shall usually be interested, ultimately, in what functions $ are partial recursive in a fixed list 0 , either of fixed assumed functions or possibly of variable assumed functions (function variables). computation relative to completed derivations $,,...,$
So
we shall want to define
E
of functions $ = 6
E
from 13. This can require, as we have just seen, looking forward as well as backward in handling a 0, with
-
). then Oi(A1, ...,A - - 5
0-expression E3.
(i=
..
1,. ,E).
If A is a 0-expression, and
0
u = 0,1,2,3).
...,B
..,D%3
are 1-expressions, 31 are 3-expressions (where (z+)~
-
=
,C1,.. .,C ,D1,.. .,D 1 is 31 %2 3 3
,B1,.. .,B
& is a
type-j variable, then Aaj A is a
j+l-expression (j = 0,1,2), called a type-j+l A-functor. E4.
If A is a j+l-expression, and B is a i-expression, then IAl(B)
abbreviated A(B))
is a 0-expression (j = 0,l.Z).
E5.
0 is a 0-expression.
E6.
If A is a 0-expression, so are (A)'
AA1).
(often
and (A121 (often abbreviated A' and
If A, B and C are 0-expressions, so is cs(A,B,C).
RECURSIVE FUNCTIONALS AND QUANTIFIERS OF F I N I T E TYPES R E V I S I T E D I
In computing, after fixing z (and thence $l,...,$E,
etc.).
199
we take any
0-expression E (under the resulting definition in 2.2) and an assignment 0 (as above), and seek the numeral W (i.e. 0'"''
with
w 2 0 accents)
as expression for
the answer "f(if it exists) to the question "What is the value of E under n?". (Incidentally, we may abbreviate 0',O",
... as 1.2,. .. .)
We shall represent a computation as in the form of a tree with E at the initial (leftmost) vertex. We draw our trees lying on their sides (as in 1959 and as some in Madison after the ice storm of March 4 , 1976).
The trees branch to the right.
The principal branch, ending (if the computation can be completed) in the numeral W expressing the value initial vertex.
w of E under R,runs
horizontally rightward from the
(We didn't show it quite so in 1959 p. 22.)
When a numeral is
recognized as the value of an expression, whose computation is therewith completed, we shall celebrate by flagging it there with a "t".
A flagged numeral gives the
value of the expression at each vertex along the branch running leftward from it as far as the branch runs horizontally, under the assignment in force at the vertex. We shall define completed computation trees inductively. If E is 0 , we flag E immediately. Otherwise, we graft onto the vertex carrying E subtrees issuing from next vertices, of which there are either one along the horizontal (principal) branch, or also others below it. Yes, I have been a successful grafter, with apples and plums. The ripe fruit at the ends of all the branches of a completed computation tree will be flagged numerals. In the simplest case, when we need a new variable, we will choose the next unused one in the list of variables of its type (assumed in 2.2).
The computation
of an expression E will be based on having an assignment 0 of values to the free variables of E and of appropriate functions to all the function symbols in the list 0. As we define computation trees moving leftward (the computations proceeding rightward), the list of the free variables of the expression E at the vertex we have before us will vary.
It will be simplest now to think of n as an
assignment to exactly the free variables of the E of the moment, but to all the
S . C . KLEENE
200
function symbols in 0 (which list remains fixed, after fixing 5 and thus fixing the class of j-expressions for j= 0,1,2,3). In special circumstances, we may wish to reserve a finite list!
of variables
that will not be introduced as new free variables in the computation and which may have values assigned to them throughout a discussion. Now we are ready to give the inductive definition of being the computation tree for E under Q,
by cases corresponding to the cases in the
definition of 0-expression in 2.2.
E is a type-0 variable 5. The tree is completed by adding one vertex
El:
horizontally rightward bearing the numeral R for the value
of 5 in the
assignment Q, flagged:
a-
Rt
) as under E2. We pass from E horizontally rightward -i3 to a next vertex bearing the result F of the same substitution for the variables
E is +i(A1,...,Dn
E2:
-
on the right of the schema (determined by
zi) - that introduced +,- as gives E on the
left. There is indeed one detail in the schemata (after having fixed the symbols
...,"$211,"81",. ..,"e -1
") which isn't determined by the index
"+1",
namely what variable is the
fd-'of an application of S 4 . 1
(1=
of
+
=
1,2,3).
+P We pick
it here to be the first variable in the list of its type (other than a variable in a reserved list )!
which does not occur free in E.
result in a "collision". =
lh((g)ii-l)
-
=
lh((ps(z))i-l)
Thereby, this step will not
Should +i be introduced by SO. then if t 2 gi = lh(gi)
-
-
-
-
(end 1.4), then the "Ot" we use here for the right
-
side of the schema application is to be the symbol for the function 0 the "et" +,....,+E fed back in by S11; if t > q, -
%i
is 8
t-q
from the list
from
el...,
Y
Clearly, F will again be a 0-expresslon. (Likewise, in ail the other steps.) Anticipating ultimate "success" if we are describing computation (proceeding rightward), or having "success" by the hyp. ind. if we are giving the inductive definition of a completed computation tree (proceeding leftward), the subtree we graft onto the vertex bearing E is say "F step is:
... Wt", and the whole result of the
201
RECURSIVE FUNCTIONALS AND QUANTIFIERS OF F I N I T E TYPES R E V I S I T E D I E4.h:
E is a 0-expression under E4 of the form {AaL A}(B)
1
$
pass rightward to
(i=
0,1,2).
We
Al, the result of substituting B for the free occurrences of
aL in A.
If this would result in a "collision", i.e. if free variables of some k substituted parts B would become bound by A-prefixes Xg- in A, we first change
k k such bound variables 8- in respective parts Af3- C of A to avoid this happening, using the first eligible variables not occurring free in E from the appropriate lists, doing this in order on the offending h S t s { h a 1 AI(B)
E4.1:
-9
a1
AI
from left to right in A.
...
wt. 1
E is a 0-expression under E4 of the form a (B).
vertices.
There will be two next
For the tree to be completed, the lower next vertex must begin a
subtree leading to a value I of B represented by the flagged numeral Rt; and then the upper next vertex (horizontally rightward) receives the flagged numeral Nt for the value
of a
1
&)
under 0, i.e. for the value 2, for I as argument, of the
function assigned by a to the function variable a
1
.
1
a (B) -Nt.
...
\B E4.2:
Rt
2
E is a (B). a
2
Mt
.
B(c)
...
... 2 evaluated by ...
Nrt
... c evaluated by 1 ... ... 2 evaluated by 0 ...
NOt
...
B(c)
'B(2) The assignments
-
lowest upward) 0.1,
-
for the lower next vertices include respectively (from the
...J,...
as the values of the new variable 5 (to be picked as
the first in the list of type-0 variables not occurring free in B and not in a reserved list
x). B may or may not contain
lack a value for a function a'
2
.
a2
free; if not, the assignments n
-
If all the lower subtrees can be completed, a type-1
is determined as {cO,n >,c1,zl>.. -0
..,,...)
are the numbers expressed by the numerals NO,N1 value of B ) .
where n
-0
,=1,...,n7,...
,...,Nr-,... (in effect, 'a
is the
Then we can complete the computation as shown, where M is the
202
S.C.
KLEENE
numeral for the value of a2 (a1) under the assignment in E4.3:
(s
c
E is a'(B).
$2
to a2
.
Similarly, using some well-ordering a0,al,...,a3,...
< 5) of all the type-1 objects. a 3 (B),-St.
...
B(y)
v ( y )
B(y)
... Y evaluated by a3 ... M3 t ... ... y evaluated by al ... Mlt ... y evaluated by .a ... Mot
If all the lower subtrees can be completed, S is the numeral for a3 (a2 ) , where a
2
= {
I
5
0, using the A-operator); i.e. we have
variables
L h of
210
S.C.
KLEENE
OI left over. For example, into $(~,a,F,oZ) substitute ~ ~ ( 0 2for ) 2, F to get
for a , A 6 ~~(6.61)for
$ ( m )= J,(Xl(m),xb In a
h a x2(b,fi)
full substitution 1959 p.
X2(l?,n),x8 X3(8,02),LT).
6 , the final OZ is not present.
(Here, unlike 1959,
we are putting the bound variables first.) (VIII)
in 0
(Standard substitution.)
The class of the functions partial recursive
is closed under standard substitution.
-,
like 1959 V p. 6 except we have h extra.
express xl(d). x2@,&)
x,(h,F,GZ).
as x,(a,F,d),
By (IV) and (VI), we first Then by applications
successively of S 4 . 0 , S 4 . 1 and S 4 . 2 , +l(a.F.Ot)
= J,(x,(a.F,a),a,F.~)
J,(xl(PZ).a.F,
00,
JI1(Xbx,(k,F,d),F,W $(xl(O-O.xb x,@,M),F,n). + =@ $,(A8 x , ( B , &I ) , @)= J,(x,(bC).xk x2@.&),W x3(6,m).fZ). $2(F,0t) =
Of course, we have made similar assumptions to those for (VII), as we shall do again below. (Full substitution.)
(IX)
The class of the functions partial recursive in 0
is closed under full substitution. (Cf. 1959 V p. 6.)
similar to
(VIII) and illustrated by 1959.
(Explicit definition.)
(X)
Let E be
any 0-expression containing no function
where $l,...,
symbols except 01,...,t31,$1,...,+g,',fl,cs. in 0 = -
(el,
...,e-1
-
are partial recursive
*I
), and containing free at most the (distinct) variables
there is a function
+
partial recursive in 0 such that $(Ol)
E.
&. Then
(Cf. 1959 VII
P. 6 . )
-
Proof. For clarity, let us understand that each J, (k = 1, ...,g) is introduced -k (qk = $ (k) -0 ) from 0, and by a different partial recursive derivation ...,d 2E the formation and computation rules are those of the q-fold treatment in 2.5. In
dF),
FiL
building up the $ to express E, we add other derivations, extending the formation and computation rules accordingly.5 We use induction on the number of the symbols in E.
-
Case El:
sl,...,%-0 .
E is a number variable.
Say the number variables in & are, in order,
If E is s1,take $(UZ) =
slby
S3.
If E is a with & > 1, take 1_
to
RECURSIVE FUNCTIONALS AND QUANTIFIERS OF F I N I T E TYPES REVISITED I
be & with a moved to the front, let $l(&)
a
I I
=
Case E2: E is$i(A1,...,A
.B1,...,B
-
_E
%
and use (VI) to get
?L'
+l(ib.
,C1,...,Cn
9
-2
,D1,...,D
23
).
are all of them variables but maybe with repetitions, then we get
+ ( n=) $i(A1 -
variables.
,...,Dn ) by -3
Now suppose some of A1
(VII) with (IV).
with (IV) as already remarked, we get +l(id',OL)
are the new variables. By the hyp. ind., each of A1,. variable can be expressed as a x(0L);
As B,
6 gives +(GO
Case E4:
_ E i
...,C
= +,(
E is A(B).
= ai(al-')
$o(ai,&-l,@C)
each of Bl,
can be expressed as a
and similarly with C1, for
If Al,
,...,D
...,Dg 3
=3
are not
Replace each which is not a variable by a distinct new variable to get
$,(a'). By (VII)
say it is
211
9
and D1,
g3
where
...,B9which is not a variable, ind. applied to B;
Now a standard substitution (VIII)
...,GZ) = Qi(A1 ,...,Dn 1. -
-3 We can write this as $(&)
by S7.i.
-
..,A9which is not a
Arx(~,h)), by the hyp.
...,D .
= $i(UZ')
= +o(A,B,Dt) where
Then we can operate with +O(A,B,OC) as under
Case E2, noting that for the treatment there the hyp. ind. will be needed now only for C if A (not a variable) is '-&A and if
1>
C, and if B is m t a variable, for B if
1 for D where B is hai-'
Case E5 is immediate by 5 2 . 0 .
(XI)
1,
D. Each of these has fewer symbols than A(B). E6 is reduced to Case E2 in the same manner
_=
as Case E4, and
1=
a E7 likewise after introducing +o($,m)= 0L( 5 ) by SO.
(Definition by cases.)
in 8 ,
Ilf X~("),X~(~),...,X,(P~) -
1
xl(m ir xotoz)
+(me
...
xn-,trn) if xo(m x,(uz) if Xo(W
are partial recursive
= 0,
= q-2,
2 2-1,
where +(a) l a defined exactly if x ,(h)is defined, say it fits the i-th case, and -
xi(fl)
-
is defined (irrespective of whether
&&, e.g. for q
= 4.
Using (X), let
in 8, so
(Primitive recursion.) is the function
+(a,
(GI) is defined for 1 # A ) .
1
(a), cs (x, (0t)'Z 'X3 ( 0 ; )'X4 ( m ) )) ) ) . g $(t.) and x(a,c,Z are ) partial recursive
4 ( 0 )= cs(xo (UZ)'X1 (02)I (cs(xo(a)'1,x2 (XII)
x
6 ) defined by
2 12
KLEENE
S.C.
Proof.
{
@(O,L) = *&I, @ ( 5 1 , 6= ) Xk,@@,L),S,.
The usual proof by induction on a that the
(Cf. IM Example 3 p. 350.)
pair of recursion equations has a unique solution recursive derivation 01,
@
works here. We get a partial
...,@ from 0 of that function Q P -
use (11) and (111) to reconstrue
$,x
= @
P -
as follows.
First,
as functions partial recursive in 17.0 (but,
as in 2 . 2 we won't show the 0 explicitly, nor here the n since Jl,X don't actually depend on n).
Next, using (X) for Pn((a,L)
as its "O",
q,0
= csk,*(A
let
,x(5zl,n(5%L)
,A)).
Finally, using S11, let
+
$(2,6)=
P$k,&.
To elaborate a bit, by saying that the recursion equations have a unique
solution @ we mean that, when we treat "@"
in them as a symbol for a variable
assumed function, call it n, and use the computation rules as from n,O, then (for a given interpretation of the function symbols 0) there is exactly one interpretation @ of a,&.
n that makes both equations hold for all choices of values of
It is a fundamental property of our computation rules 2 . 4 (to be explored
11k.R) will be carried along
in Part 11) that, in computing X@,n(a,L),&,
intact (perhaps being substituted, and the results being again substituted, etc., for variables in other expressions, but never having anything substituted for its variables a,L), until and unless (as may happen more than once, on different branches) the evaluation by E7 of n(zt,& depends on n only via to X(5~l,q(~~l,&),
that is, the value of this will depend on
.
applied to evaluate nkzl,L) is defined (and
In brief, x@,nk,d),A)
as one could reasonably expect. The like applies
q(5,&),
&);
is called for.
n only via E7
For a given type-0 value of 5, the expression 521
kl will have given values of the appropriate types, the same as
the subordinate computations for them under E7 will give). result will be obtained for n(&+l,L)
by E7 with
using Sll to feed in a computation of @(&l,&) derivation Ql,...,QP
-
of
Consequently, the same
n interpreted by our 9 as by
via the partial recursive
@.Thus, the phenomenon exhibited in Example 1
(following (V)) will not arise here.
V
RECURSIVE FUNCTIONALS AND QUANTIFIERS OF FINITE TYPES REVISITED I
213
A function is primitive recursive in total functions 0, iff it is definable
from 0 using the schemata we listed in 1.3 except that (XII) (called 55 in 1959 p. 3) used as a "postulated" schema replaces S11, whereupon Sl.1 and S5.1 become redundant; primitive recursive then, if 0 is empty.
Primitive recursive
functions are total, as are functions primitive recursive in total functions 0. The present definition of "primitive recursive" functions is equivalent for 0 empty to that in 1959 p. 3 (for types 0,1,2,3) using 1959 Remark 1 p. 6, and for 0 non-empty by 1959 1.8 p. 7.
For variables of types 0 and 1 only, it agrees with
ORT, by 1959 1.7 pp. 6-7. Since we didn't use S11 in establishing (I)-(XI), recursiveness (with
6=$
they hold for primitive
in (V)).
(XIII) (Least-number operator.)
If $(y,ot) is partial recumive in 6 ,
$(d) = uy[*(y,~)=Ol, where py[$(y,a)=O] is the least y (if it exists) such that $ ( O , U Z ) ,
...,$(y,&)
are all defined and $(y,D) = 0, and is undefined if such a y does not exist.
-
Essentially as in IM Example 4 p. 350. Or apply (V) to the example
Proof.
$19,,,,$18 in 1.4, thus. Example 2.
To adapt the derivation L$~,...,$,~ as given in 1.4 to the present
context, first replace its
"L"by
"Pt" and its ''0" by "JI".
uy[$(y,&)=O] is partial recursive in (of
+
= 018),
a.
Thereby,
$(a)
From the (adapted) derivation $ l,.
..,$ 18
it is not hard to see that in the computation of $ ( h ) from $ fot a
given assignment to UZ (and a given $), E7 will be called upon only to evaluate
JI(A,n)for the original 0 2 and a succession of A ' s which can be evaluated as
.
0,1,2,...
Therefore we get no fewer values with JI an assumed function than when,
in applying (V), we replace the introduction by SO of +,(y,bL,d,e)
as the assumed
function $(y,fl) by the identification of $,(y,GZ,d,g) with the last function in a partial recursive derivation from 0 of the same function J I ( y , U Z ) expanded by (IV) to JI(y,a,g,g). That is, the phenomenon of Example 1 does not arise, so the ~
~~
presence of Sll in 1.3 made the schema of primitive recursion replaceable by introductions of some particular functions, just as did the p-schema in u-recursiveness (cf. IM p. 320, Kleene 1936a).
S.C. KLEENE
214
function given by the new derivation from 0 constructed by (V) is py[$(y,fi)=O], not a proper extension of it.
(XIV)
(Totally undefined function.)
$ ( m ) such
n, $(&)
that, for each
There is a partial recursive function
is undefined.
FirstDroof. As in IM Example 1 p. 350; i.e. $(r1;07) 1 TI(&)by SO, = $ ( $ ; P I ) by S11.
$(&)
$(m)
Second uroof.
2:
py[$(y,OL)=O] by (XIII), where
$(y,oZ) = y' by S1.0.
&& Our next objective is an enumeration (or parametrization) theorem and its consequences, still taking the ranges of the variables to be types 0, 1, 2 , 3.
In
1959, while we had to work hard to get a restricted result on substitution of
A-functionals, the enumeration theorem was almost the starting point (S9 p. 13, XI1 p. 15).
It won't be
so
hard though to get the enumeration theorem here, if
we refrain from laboring over the sort of detail with a GBdel numbering which by now has become routine. We establish a GBdel numbering of the symbols and expressions, as we described them in 2 . 2 , which will apply to any
el,
...,e-2 and $,,...,$
E
0.1.2.3;
...,
'D2i3
where
'Ir
''I
Thus $i(A1
...,c > where 7 -
,...,D
) has 3 3 indicates Gbdel numbers, and similarly with
-
Ad-1
); A(B) has ; A has ; A' has -t3
As in 1959 2.1 p. 7 , for and
..,ai> = Ad--' -
0 let
-
in these are primitive
1;6
recursive functions of their variables, and, for 0
... - - = 4. -
(ao
s...,
3
a )
-
if o&e)
&
@) 0=8
E is { A 8 4 A)(B). Similarly: 3 * 0 3 = $ (z,*E4*A(z,e),a a ) if ?j(Z,C) +*(z,e,aO a 1 E4.1: E is Bi(B). Then _1. = ( c ) ~and , ~rB7 = ( E ) ~ . E4.A:
A
,...,
,...,
-
+*(z,e,aO ,...,a3) E4.2:
= (al(+*(z, (e),.aO
2
E is Bi(B).
-
Say that 2
=
&
0
ao = with 0
(e)1='
(e)0=8
(d1=.
,...,a3)))
(e)1,2 if ",e) & (=)o=8
0
Bk, where
-
k
=
$E4.2(=)
&
@)l=.
with $E4.2 primitive
recursive, is the first number variable not occurring free in B. 0
&
Then if
217
RECURSIVE FUNCTIONALS AND QUANTIFIERS OF F I N I T E TYPES REVISITED I Using a totally undefined function qo (cf. (XIV)):
Otherwise:
$*k,e,a 0
-
,...,a3) = J-, ~ ( E , = , ~,..., O a3) if gk,g).
Now all the case specifications, with $* replaced by definition by cases fitting (XI).
q,
For, we can define the
x0 primitive
recursively
,...,a3) so that ~~(&,=,a~,..., a ) = 0, ... , "-1 ... , E7.1 or Otherwise applies; and the
0
(in fact independently of qrO,a
can be assembled into a
3
according as the case hypothesis of El,
xl, ...,xn
-
we get by use of (X) for n,0 as its " 0 " .
,...,a
J,n(z,g,ao
3
,...,a3),
), or m r e explicitly +qs0(2,e,a0
defined by combining the cases (with $* replaced by
,...,a 3 )
$*(z,=,ao
Thus there is
= J,$* (=,=,ao
q).
representing what is
So using Sll, we can get
,...,a3) to satisfy all the specefications.
Now it can be proved under the assumption s(g.g), by induction over any 0 1 2 3 completed computation tree for E under the assignment R extracted from a ,a ,a ,a 3 as in the theorem (with given values of 0). that then $*k,e, ,'a a ) = w for
...,
the value of E under
a.
Conversely, still assuming 0
completed computation tree for $*(z,g,a
,...,a3
0
of ~ , = , a
,...,a
3
g(=,=),by
) with result
induction over a under given values
and 8 . E Z x under the assignment for E extracted from a
And of course, g(2,g) gives that $*(g,e,a
0
w
0
,...,a3) is undefined.
,...,a3 .
In these inductions, we will have that the use of S11 to give $ (z e a0,...,a
=
$~~,(z,r,aG ,...,a
given 0 and as
q
3
), where
+l-l(&,z,a
0
E -'-'
,...,a3) has been defined from n,Q,
z,e,a0,a1,a2,a3 11
for the
in hand.
So. for
ranging over our types 0,0,0,1,2,3, $* defined via Sll is a 0 3 0 of the recursion q(z,e,a a ) = $:-l(z.,e,a a3), just as in
,...,
,...,
the (simpler) case of a primitive recursion (for (XII)).
0
~,e,a,a1,a2,a3, values of
..
That is, in the
3 ,a ) for given type-O,O,O, 1.2.3
computations of +;-l(=,=,ao,. q
values of
will be called for under E7 only for argument
expressions which will all of them be defined in our types (so the phenomenon of Example 1 will not arise).
In particular, for values of =,=which, via the
for the application of (XI), give us e.g. Case 4.2 (read s n(. ..) appearing, the part a2 (As q(...)) will with "q" replacing "$*") with A 2 eventually be evaluated via a new application of E4.2 with a as the 'a of E4.2 evaluations of
)
a variable assumed function, will treat "previous values" of $* just
n is treated as an assumed function by E7 for the arguments
solution for
3
CS(...)
2 18
S.C. KLEENE
(as will be seen in Part 11). having the values 0,1,2,
Thereby (A= TI(...)}(=) will start subtrees with
..., and after using E4.X
=
we will come to evaluate by E7 TI
applied to argument expressions with the new variable 8 in one of them.
Similarly
with E4.3 and with E7 when the Ot has some arguments of types > 0. All the
-
arguments of TI that will arise in the computation will be primitive recursive functions of -z,e,a0,a1,a2,a3 _ and new variables like =,u,u
2
introduced into
subcomputations for E4.2, E4.3 and E7. V We write ( X V ) also as (XV.3.3)
to express by the first "3" that $l,...,$E
may
have variables of each type 5 3 , and by the second "3" that variables of each type =
1 does not increase the class of functions of variables of types 5 1 partial recursive in functions of variables of types 5 1. Similarly, we have (XV.j.&)
for any 0 (k 51.5 3.
(1959 I pp. 3, 15 doesn't hold in the present theory.) (XV.3.1)
For each fixed
0
(possibly empty) which consists of functions of
variables of only types & 1, there i s a function
+* (of four variables)
partial
recursive in 0 in the theory with only types 2 1 (thus omitting 54.2, S4.3, S6.2, 56.3, S7.2, S7.3) such that:
For 5,s as in (XV) from "If" to "then" (in the
theory with types 0,1,2,3) but with E containing free only variables of types 5 1, +*(Z,e,a
0 1 ,a )
E
0,l and each i) each variable Bi1. occurring free in E 0 1assigned the value (aL),. Otherwise, +*(z,e,a ,a ) i s undefined.
when (for each
1. =
-
m. To adapt the proof of
(XV.3.3),
we observe that, when the E in its
hypothesis contains free only variables of those types,* none of the possible computation steps will introduce a higher-type variable.
Indeed, E4.2 and E4.3,
81f furthermore, the functions 0 have only variables of type 0, then only finitely many branches can issue from any vertex, so any completed computation tree i s finite, and our partial recursive functions (functions partial recursive
RECURSIVE FUNCTIONALS AND QUANTIFIERS OF F I N I T E TYPES REVISITED I
219
and E7 with arguments of types > 1, will be inapplicable. So we set out to define a
$ * k , c , u 0,u 1)
instead of
0 $*k,e,a ,...,a3).
The case hypothesis and the
specifications for $* will then be entirely within the theory for types 5 1, in which the constructions used from earlier are all good (as we remarked in 1.1). (XVI)
(Enumeration theorem.)
-
0,1,2,3; & g
m = -
&q,gl,g2,g3variables of types
& e J
; and similarly let
as in 1.3.
-
There is a function 0
{ z ) ( n ) partial ), recursive in 0
(for 0 -,
0
be characterized by ( z , U Z ) , also written
m,g-
{z)'(&)
such that, if $(d)is partial
recursive in 0 with index 5,
{zI0'(a) = $(a). M. There is no loss of generality in taking variables of the respective lists (preceding ( X V ) ) . recursive function 9,
to be the first n ,g ,g ,g - 0 1 2 3 There is a primitive
such that: if Ix(z) & @)ly& @).)=g, then
-*-
GHdel number e of the 0-expression 0 derivation determined by
z;
E
and
(PI),
n(z)= -s-
$ J ~
where $l,...,$ 0 otherwise.
E
(z) is the g,g is the canonical I)
,
Now put
n(E,a) = {=10(bl) = -9-
0
,(g),ss~50s...,5: -l>). -1 -2 -3 1, there is a primitive recursive
-l>, J c e i v e s a t t e n t i o n a t any s t a g e
R
T
.
0, We
before
A1
x
E
W:
A similar argument shows t h a t
, A2
B
i s a-recursive.
The theorem now follows e a s i l y from Lemma 3.3 SECTION 4:
SIMPLE SETS
Let g(x) for a l l with
M
T,Q
E
x < w
.
L Y1-M
.
But t h e n
If g were not a c o f i n a l i t y f u n c t i o n , choose
L
K1-M
As
The theorem w i l l follow once we show
must b e fi:-unbounded,
Axf(x,r) = Axf(x,p)
The admissible o r d i n a l s f o r which m a x i m a l u-r.e.
, an
there are
u
T > p >
@
impossibility.
s e t s e x i s t have been c l a s s i f i e d
by Lerman [13]. P a r t i a l r e s u l t s were p r e v i o u s l y obtained by K r e i s e l and Sacks [51, Sacks [ 211 and Lerman and Simpson THEOREM 4.2.
Maximal a-r.e.
[163 .
s e t s e x i s t i f and only i f t h e S -projecturn of
is
u
3
w .
Maximal u-r.e.
s e t s w i t h a-bounded complements have been s t u d i e d by K r e i s e l and
Sacks [51, Owings [ l a ] and Leggett
[a].
Leggett has c l a s s i f i e d t h e admissible
o r d i n a l s f o r which such s e t s e x i s t . THEOREM 4.3.
Maximal a-r.e.
s e t s w i t h a-bounded complements e x i s t i f and o n l y i f
( x * = w .
An u-r.e.
set
H
i s s a i d t o b e hyperhypersimple ( h h s ) i f
i n &(a)
a
-H
i s not a*-
f i n i t e and t h e l a t t i c e of s u p e r s e t s of
H
forms a boolean a l g e b r a .
Post 1191 o r i g i n a l l y defined hhs w-r.e.
s e t s d i f f e r e n t l y , and Lachlan [61 discov-
ered t h e above d e f i n i t i o n and showed it e q u i v a l e n t t o P o s t ' s d e f i n i t i o n for
a = w
.
M a x i m a l s e t s a r e hhs, so hhs w-r.e.
sets exist.
For
u = w
, Lachlan
[61
has c l a s s i f i e d t h e boolean a l g e b r a s which can occur a s l a t t i c e s of s u p e r s e t s of
hhs s e t s . w
.
From lheorem 4.2, hhs a-r.e.
For such
supersets. THEOREM
is w
4.4.
, or
u
, Cooper
s e t s e x i s t i f t h e S -projecturn of
3
a
(unpublished) has c o n s t r u c t e d hhs s e t s w i t h no maximal
F u r t h e r r e s u l t s o b t a i n e d by Chong and Lerman [l] a r e now summarized. s e t s e x i s t if e i t h e r t h e S -projecturn of
Hyperhypersimple u-r.e. i f t h e S -cofin€dity
2
of
< u*
a
3
u
i s l e s s t h a n t h e tame S2-proJectum of
I n t h e l a t t e r case, t h e hhs s e t s a r e e x a c t l y t h o s e s e t s whose complements have order-type
is
w i t h a f i n a l s e w e n t of order-type l e s s t h a n t h e tame S2-
u
.
2 32
MANUEL LERMAN
projectum of of
.
a
Hyperhypersimple a-r.e.
i s greater than
a
An a-r.e.
set
R
w
s e t s f a i l to exist i f the S -cofinality
3
.
i s s a i d t o be r-maximal if
every a-recursive s e t
either
W
W ('1 ( a
-
a
R)
-
i s not a*-finite and f o r
H
or
(a
- W)
0 (a
I t i s easy t o see t h a t any maximal s e t i s r - m a x i m a l .
finite.
Lachlan [61 have studied r-maximal w-r.e.
sets.
-
.
a
Lerman and Simpson [16] use t h e
r-maximal a-r.e.
f a i l to exist i f
If
A G B
, we
every a - r . e . finite.
s e t s e x i s t i f the S -projecturn of
3
i s not a l i m i t of a-cardinals and the
a*
.
a*
s e t s f o r cer-
The known r e s u l t s a r e summarized i n t h e following theorem.
THEOREM 4.5.
is
a*-
Robinson [20] and
construction of Theorem 4 . 1 t o r u l e out t h e existence of r-maximal tain
is
R)
call
set
W
a major subset of
A
, if
a
-
(W
if
Lachlan [ 6 ] has shown t h a t every w-r.e.
has a major subset, and t h a t i f ma1 if and only i f
B
-
A
set
i s r-maximal and
B
i s f i n i t e or
way, he obtains r-maximal w-r.e.
B
-
A
A
B
w
is
.
They
X - c o f i n a l i t y of 3
a
i s not a*-finite and f o r
i s a*-finite then
B)
(J
B
a
a
-
(W
i s a*-
0 A)
which i s not a-recursive
A c B
then
i s a major subset of
s e t s which a r e not maximal.
i s r-maxi-
A
B
.
In t h i s
The following theo-
rem of Leggett and Shore [ 9 ] sunmarizes t h e known r e s u l t s and subsumes e a r l i e r r e s u l t s of Lerman [141.
THEOREM 4.6. every a-r.e.
I f t h e Z2-projectum of
a
equals t h e X2-cofinality of
a
, then
s e t which i s not a-recursive has a major subset.
A natural question t o ask a t t h i s point i s whether, f o r all
a
, there
i s a form-
ula of t h e language of l a t t i c e theory s a t i s f i e d by some, but not all, simple ar.e. sets. THEOREM
4.7.
An affirmative answer i s given by For all
a
, there
i s a formula of t h e language of l a t t i c e theory
with one f r e e v a r i a b l e s a t i s f i e d by some, but not all, simple a-r.e.
PROOF. If the S -c&ina;Lity of 2
a
sets.
i s l e s s than the tame S2-projectum o f
a
,
LATTICES
OF a-RECURSIVELY ENUMERABLE SETS
233
then Chong and Lerman [l] show t h a t t h e formula " S i s hyperhypersimple" d i f f e r e n t i a t e s between simple s e t s .
Otherwise, Leggett and Shore [ 9 ] show t h a t the
formula "S i s a major subset of some a - r . e .
s e t " d i f f e r e n t i a t e s between simple
(This l a t t e r formula w a s previously used by Lerman 1141 t o o b t a i n the re-
sets.
sult i n t h e s p e c i a l case when a
i s a regular c a r d i n a l of
D
L .)
An important open question o f recursion theory i s t o determine t h e degree of uns o l v a b i l i t y of t h e elementary theory of
.
Lachlan [ 7 ] showed t h a t & ( w )
E*(w) a r e equidecidable, a r e s u l t generalized by Lerman [12] t o all
and Let
&(a)
8
a
.
be t h e language of t h e pure p r e d i c a t e calculus with e q u a l i t y , binary r e l -
a t i o n symbols t o be i n t e r p r e t e d as union and i n t e r s e c t i o n , a unary function symbol t o be i n t e r p r e t e d as complementation, and a unary r e l a t i o n symbol t o be i n t e r preted as distinguishing t h e a - r . e .
sets.
n e language $ applied t o the boolean
algebra generated by t h e w . e . s e t s i s equivalent t o t h e usual language of l a t t i c e theory applied t o t h e r . e . s e t s , and i s useful
for studping d e c i d a b i l i t y
questions. THEOREM
4.8.
The
1-3
theory of
e i t h e r t h e S - c o f i n a l i t y of 2 a
, or
a
i f the S - c o f i n a l i t y of
3
2
i n t h e language
.&a)
i s decidable i f
and t h e tame S -projecturn o f
a
2
a
and t h e Z -projectum of
a
3
are
w
a r e both
and a
a* =
and
t h e r e is a g r e a t e s t a-cardinal. The d e c i d a b i l i t y for
a = w
was obtained by Lachlan
"'(1.
were obtained by Lerman [ll] and include t h e case where
of
L
The remaining cases a
i s a regular cardinal
.
The types of simple s e t s considered i n t h i s section play an important r o l e i n the A complete c l a s s i f i c a t i o n of those
a
e x i s t would be valuable f o r extending Theorem 4.8 t o a l l
a
decision procedures.
for which such s e t s
.
Progress has been
made r e c e n t l y by Lerman and Soare towards obtaining a decision procedure f o r t h e
y-3
theory of
g ( u ) i n the language
id
obtained from
by adjoining a
unary r e l a t i o n symbol t o be i n t e r p r e t e d as distinguishing t h e m a x i m a l s e t s .
2 34
MANUEL LERMAN
SECTION 5:
DEFINABILITY AND AUTOMORPHISMS
The f i r s t q u e s t i o n which we cofisider i n t h i s s e c t i o n i s t h e c h a r a c t e r i z a t i o n of a l l definable i d e a l s , f i l t e r s , and congruence r e l a t i o n s of & ( a )
.
One o b s t a c l e
towards o b t a i n i n g such a c h a r a c t e r i z a t i o n i s t h e determination of whether "a-bounded" o r e q u i v a l e n t l y " a - f i n i t e " i s d e f i n a b l e . "a-bounded" able.
and "a*-finite"
"a-finite''
i s a-recursive &
5 - M2 &
&(a)
, and
i s known t o b e d e f i n a b l e i n o t h e r c a s e s , e.g.,
Owings [l8] shows t h a t A
a r e all equivalent over
If a* = a
5
A
-
A)
i s maximal &
5 ) ) .A
for
w = a*
M2 i s maximal i n M1
cannot b e s p l i t i n t o two p i e c e s , each non-a*-finite,
(B)(M2 # B
so a l l a r e defin-
,
a
(3%)(3M2)(M2t_M1C_A
i s a - f i n i t e i f and only i f
0 (a
then "a-finite",
&
(i.e.,
by any a-r.e.
set)
summary of t h e c a s e s where a - f i n i t e i s known t o b e de-
f i n a b l e can b e found i n Lerman [lo]. The d e f i n a b l e i d e a l s of &(a) f i n i t e s e t s i s one such i d e a l .
a r e c h a r a c t e r i z e d i n Lerman 1121.
The i d e a l of a*-
There i s a t most one a d d i t i o n a l d e f i n a b l e i d e a l ,
t h e i d e a l of a-bounded s e t s , b u t t h i s i d e a l i s d e f i n a b l e only when "a-bounded"
is
definable. Several d e f i n a b l e f i l t e r s a r e known t o e x i s t , b u t some become t r i v i a l ( e q u a l t o t h e f i l t e r of s e t s with a * - f i n i t e complements) for v a r i o u s choices of
a
.
The
f i l t e r of simple s e t s i s always d e f i n a b l e , and t h e f i l t e r of s e t s simple f o r &,(a)
( t h e q u o t i e n t of &(a)
obtained upon f a c t o r i n g by t h e i d e a l of bounded
s e t s ) i s d e f i n a b l e e x a c t l y when "a-bounded''
One of t h e s e f i l t e r s
i s definable.
w i l l always b e t h e l a r g e s t d e f i n a b l e f i l t e r (Lerman [lo]).
Other d e f i n a b l e f i l -
t e r s which a r e sometimes n o n - t r i v i a l a r e t h e f i l t e r of hhs s e t s and s e t s w i t h a*f i n i t e complements, t h e f i l t e r of s e t s with no maximal s u p e r s e t s , t h e f i l t e r of s e t s w i t h no r-maximal s u p e r s e t s , t h e f i l t e r o f s e t s w i t h no hhs s u p e r s e t s , and t h e f i l t e r of s e t s with no r - m a x i m a l
o r hhs s u p e r s e t s .
n o n - t r i v i a l and d i f f e r e n t over & ( w )
.
For example, t o o b t a i n a s e t w i t h no r-
maximal or hhs s u p e r s e t , we s t a r t w i t h a maximal s e t s u b s e t of
M
.
Let
f
All t h e s e f i l t e r s a r e
M
and l e t
A b e a major
be a one-one w-recursive f u n c t i o n enumerating
M
, and
let
LATTICES OF cbRECURSIVELY ENUMERABLE SETS
.
B = f-l(A)
i s the desired set.
B
Since
i s r-maximal,
A
2 35 it follows t h a t
has no hhs s u p e r s e t , e l s e by Lachlan [6], t h e r e would be a r e c u r s i v e s e t that
A IJ R = M
c o n t r a d i c t i n g t h e r-maximality of
rem [is] implies t h a t
A
.
such
Owings' s p l i t t i n g theo-
It is unknown whether t h e r e
has no r-maximal s u p e r s e t .
B
R
B
a r e i n f i n i t e l y many f i l t e r s which a r e d e f i n a b l e over &(a)
.
Various d e f i n a b l e congruence r e l a t i o n s n o t corresponding t o f i l t e r s o r i d e a l s have been i d e n t i f i e d . c o n t a i n s no a-r.e.
One such i s :
i s simple w i t h
A
s e t which i s not a * - f i n i t e .
congruence r e l a t i o n of € ( a )
B
if
&b ( a )
if
(A
-
u n l e s s "a-bounded"
B ) IJ (B
-
A)
- B)
IJ (B
i s definable.
c o n t a i n s no a - r . e .
I n the l a t t e r A
i s simple with
if
a
-
B
i f for all a-r.e.
(W IJB )
i s a*-finite.
sets
,
W
a
-
B
s e t which i s not a-bounded.
Another d e f i n a b l e congruence r e l a t i o n which i s sometimes n o n - t r i v i a l i s : major w i t h
- A)
This w i l l b e t h e l a r g e s t d e f i n a b l e
c a s e , t h e l a r g e s t d e f i n a b l e congruence r e l a t i o n i s given by: for
(A
(W I J A )
i s a*-finite
A
is
i f and only
It i s unknown whether t h e r e - a r e i n f i n i t e l y many
d e f i n a b l e congruence r e l a t i o n s i n &(a)
.
One way t o t r y t o c o n s t r u c t i n f i n i t e l y
many might be t o i t e r a t e mixtures of t h e above congruence r e l a t i o n s and f i l t e r s t o successive q u o t i e n t s of &(a)
and t o show t h a t t h e procedure does not terminate.
A d e t a i l e d d i s c u s s i o n of t h e s i t u a t i o n can b e found i n Lerman [lo].
The o t h e r t o p i c which we c o n s i d e r i n t h i s s e c t i o n d e a l s w i t h automorphisms of
&(a) and
&(a)
.
a # w
L i t t l e i s known f o r
.
A d e t a i l e d summary f o r
a = w
can b e found i n Soare [26]. It i s easy t o s e e t h a t every automorphism of €(a)
i s determined by a permutation of
, but
a
g i v e s r i s e t o an automorphism of &(a) determines a n automorphism of there are
,"O
8(u)
automorphisms of
g(a) f o r
arbitrary
automorphism of
.
automorphisms of &(a)
s m e automorphism of are
z(a)
&*(w)
a
.
t h a t n o t every permutation of
Furthermore, every automorphism of &(a)
Using maximal s e t s , Kent
.
a
[41 showed t h a t
A l l t h e s e automorphisms g i v e r i s e t o t h e
. h c h l a n used a d i f f e r e n t method t o show t h a t t h e r e r"(u) . The number of automorphisms of &(a) and
remains t o be determined.
Soare [26] shows t h a t every
comes from some automorphism of & ( w )
, so
t h e determina-
MANUEL LERMAN
236 t i o n of t h e automorphisms of
€(a)
E*(w) a r e c l o s e l y r e l a t e d .
and of
i s t e n c e of such a r e l a t i o n s h i p for a r b i t r a r y A subset
.of
&(a)
a
i s a n a - o r b i t i f f o r any
an automorphism o f &(a)
carrying
c(
{B : B i s a-recursive and n e i t h e r
as
iS
to
A
B
Bl nor
. a
{ B : B i s a - f i n i t e and has a - c a r d i n a l i t y
{B : B i s maximal}
mal s e t s ]
and
has y e t t o b e determined. A c
18 , J = {B
which have been c l a s s i f i e d .
n
.
c &(a) : t h e r e i s
It i s easy t o s e e t h a t f o r a l l
-B K)
i s a-finite}
.
i s an a - o r b i t
Soare [26] has shown t h a t
{B : B i s t h e i n t e r s e c t i o n of e x a c t l y
a r e w-orbits f o r each
The ex-
n
d i s t i n c t maxi-
These, e s s e n t i a l l y , a r e t h e only w-orbits
Maximal s e t s do n o t form an a - o r b i t f o r c e r t a i n
a
a s was demonstrated by Leggett [ 8 ] by producing two maximal s e t s whose complements have d i f f e r e n t order-types which do n o t allow an automorphism. i n t e r e s t i n g t o determine whether t h e a - o r b i t of a maximal s e t
M
It would b e
i s determined
j u s t by p r o p e r t i e s of t h e order-type and t h e boundedness of t h e complement of
M
.
I t seemed n a t u r a l following S o a r e ' s c l a s s i f i c a t i o n of t h e w-orbit of a m a x i m a l s e t t o conjecture t h a t t h e c l a s s of hhs s e t s w i t h no maximal s u p e r s e t s a l s o forms an w-orbit.
Unfortunately, t h i s i s n o t s o , a s was r e c e n t l y shown by Lerman, Shore,
and Soare [15]. Another i t i t e r e s t i n g c l a s s of problems d e a l s w i t h b a s e s f o r automorphisms Of &(a) and e x t e n d a b i l i t y of automorphisms from s u b l a t t i c e s of
&(a) t o a l l Of
€(a)
.
This question has been s t u d i e d by Shore and Soare, and i s summarized i n Shore [231. Such problems l e d Shore t o t h e discovery of a new d e f i n a b l e c l a s s i n € ( a )
, the
nowhere simple s e t s [ 2 4 ] .
References [l]
C.T. Chong and M. Lerman: Hyperhypersimple a-r.e.
s e t s , Ann. of Math. Logic 9
(1976) 1-48. Friedberg: Three theorems on r e c u r s i v e enumeration, J. Symbolic Logic
[2]
R.M.
[3]
K . W d e l : Conrnstency proof f o r t h e g e n e r a l continuum h y p o t h e s i s , Proc. Nat.
23 (1958) 309-316.
237
LATTICES OF a-RECURSIVELY ENUMERABLE SETS Acad. S c i . U . S . A . 25 (1935) 220-224.
C.F. Kent: Constructive analogues of t h e group of permutations of t h e natura l numbers, Trans. Amer. Math. SOC. 104 (1962) 347-362. Sacks: Metarecursive s e t s , J. Symbolic Logic 31 (1966)
G. K r e i s e l and G.E. 1-21.
A.H.
Lachlan: On t h e l a t t i c e of r e c u r s i v e l y enumerable s e t s , Trans. Amer.
Math. SOC. 130 (1968) 1-37. : The elementary theory of r e c u r s i v e l y enumerable s e t s , Duke
Math. 3. 35 (1968) 123-146. A . Leggett: Maximal a - r . e .
s e t s and t h e i r complements,
hiiii.
of Math. Logic
6 (1974) 293-357. A. Leggett and R.A.
Shore: Types of simple a-recursively enumerable s e t s , J.
Symbolic Logic 41 (1976) 681-694. M. Lerman: Congruence r e l a t i o n s , f i l t e r s , i d e a l s and d e f i n a b i l i t y i n l a t -
t i c e s of a-recursively enumerable s e t s , J. Symbolic Logic 41 (1976) 405-
418. : On elementary t h e o r i e s of some l a t t i c e s of a-recursively enu-
merable s e t s , t o appear. : I d e a l s of generalized f i n i t e s e t s i n l a t t i c e s of u-recursive-
l y enumerable s e t s , t o appear. : Maximal a-r.e.
s e t s , Trans. Amer. Math. SOC. 1 8 8 (1974) 341-
386. : Types of simple a-recursively enumerable s e t s , J . Symbolic
Logic 41 (1976) 419-426.
, R.A.
Shore and R.I. Soare: R-maximal major subsets, i n pre-
paration. and S.G.
Simpson: Maximal s e t s i n a-recursion theory, I s r a e l J.
Math. 1 4 (1973) 236-247.
M. Machtey: Admissible o r d i n a l s and l a t t i c e s of a-r.e. Logic 2 (1971) 379-417.
s e t s , Ann. of Math.
238
MANUEL LERMAN J.C.
Owings: Recursion, metarecursion, and i n c l u s i o n , J . Symbolic Logic
32
(1967) 173-179. E.L. Pqst: Recursively enumerable s e t s of p o s i t i v e i n t e g e r s and t h e i r decis i o n problems, B u l l . Amer. Math. SOC. 50 (3.944) 284-316.
R.W. Robinson: Two theorems o n hyperhypersimple s e t s , Trans. Amer. Math. SOC. 128 (1967) 531-538. G.E.
Sacks: P o s t ' s problem, admissible o r d i n a l s , and r e g u l a r i t y , Trans.
h e r . Math. SOC. 124 (1966) 1-23.
G.E. Sacks and S.G. Simpson: The a - f i n i t e i n j u r y method, Ann. of Math. Logic 4 (1972) 343-367. R.A.
Shore: Determining automorphisms of t h e r e c u r s i v e l y enumerable s e t s , t o
appear. : Nowhere simple s e t s and t h e l a t t i c e of r e c u r s i v e l y enumerable
s e t s , t o appear. S.G.
Simpson: Recursion theory over admissible s t r u c t u r e s , R-series,
Springer-Verlag, Heidelberg, i n p r e p a r a t i o n . R.I.
Soare: Automorphisms of t h e l a t t i c e of r e c u r s i v e l y enumerable s e t s
P a r t I : Maximal s e t s , A n n . of Math. 100 (1974) 80-120.
J.E. Fenstad. R.O. Gandy, G.E. Sacks (Eds.) GENERALIZED RECURSION THEORY I I North-Holland Publishing Company (1978)
Q
HIGH a-RECURSIVELY ENUMERABLE DEGREES Wolfgang Maass Msthematisches Institut der Universitat Miinchen A degree g
is said to be high if &' = 0"
the jump of 2 and
where &'
is
0 is the degree of the empty set. Thus 0'
is a high degree but in ordinary recursion theory (ORT) there exist as well high recursively enumerable (r.e.)
degrees below
0'
according to a theorem of Sacks 1121. The proof of this result is a very nice application of the infinite injury priority method. It follows from the theorem of Sacks that the notion high is not trivial. Further results show that the notions high and low (
is low if
a' =
0' ) are in fact important for the study of
the fine structure of the r.e. degrees in ORT. The intuitive meaning is that 2 is high if 2 is near to if
2 is near to
0
0'
and 2 is low
in,the upper semilattice of the r.e.
degrees.
Therefore these notions are useful for the study of non-uniformity effects in this structure where one looks for theorems which hold in some regions of this semilattice but not everywhere (see e.g. Lachlan L43). In addition high degrees are interesting for technical reasons. Some results have been proved for high degrees and it is not yet known whether they are true for all r.e. degrees (see e.g. Cooper
tll).
Finally high degrees are a link between the structure of r.e. degrees and the structure of r.e. Martin
(see[l5]):
sets according to a theorem of
A degree contains a maximal r.e.
set if and
only if it is a high r.e. degree. In a-recursion theory for admissible ordinals 2 39
OL
the deeper
WOLFGANG MAASS
240
properties of r.e. degrees and r.e. sets are explored in a general setting and one tries to find out which assumptions are really needed in order to do certain constructions. We refer the reader to the survey papers by Lerman and Shore in this volume for more information. It turned out that in fact several priority arguments can be transferred to u-recursion theory (see e.g. Sacks-Simpson 1141, Shore 1163, Shore 1181). Other results of ORT have been proved for many admissible
u but 'it is still open whether they hold for all
admissible a( e.g. the existence of minimal pairs of a-r.e. degrees 161 ,[21]
and the existence of minimal a-degrees [171,[7]).
Lerman [5] closed the gap between provable existence and provable non-existence in the case of maximal cc-r.e. sets.
For some time one thought that the existence of high tx-r.e. degrees below 0 '
was as well completely settled by Shore c 2 0 3 ,
but an error was found in the proof of Theorem 2.3.
in [ 2 0 ]
* . The
problem was then open again except for z2-admissible u where the existence proof from ORT works and for
d
such that
0'
is the
only non-hyperregular a-r.e. degree where every a-r.e. degree below 0'
is low according to [201 (these are the types ( 1 ) and
( 4 ) in our characterization in $ 3 ).
We close the gap in this paper by proving that high a-r.e. degrees below 0'
exist if and only if u2cf u a w2p u
. This re-
sult was not expected and is different from the result in 1201. We think that the new result is a lucky circumstance for a-recursion theory since it was thought in C201 that the situation is somewhat trivial (every non-hyperregular cL-r.e. degree is high). Now it turns out that inadmissibility (in form of non-hyperregularity) influences the behaviour of the jump of an a-r.e. degree but is *I would like
thank R . A . Shore for informing me about this.
HIGH a-RECURSIVELY ENUMERABLE DEGREES
241
not s o s t r o n g t h a t it overruns everything ( t h i s w i l l become even c l e a r e r i n our forthcoming paper C113 ). The plan o f t h i s paper i s as f o l l o w s : contains some basic d e f i n i t i o n s and f a c t s . In
we construct high ci-r.e.
> o2cfa 8 v2pa
degrees below
0'
f o r t h e case
. We give some motivation f o r t h e construction
s o t h a t t h i s chapter should be readable f o r anyone who has seen bef o r e an i n f i n i t e i n j u r y p r i o r i t y argument i n ORT (e.g.C23]).
construction r e f l e c t s s e v e r a l t y p i c a l f e a t u r e s of
The
a-recursion
theory and uses s t r a t e g i e s which would not work i n ORT. In
$&
0'
i n t h e case o 2 c f o
we prove t h a t t h e r e e x i s t no high w-r.e. 4
w2pa
degrees below
by using some basic p r o p e r t i e s o f
s t r o n g l y inadmissible s t r u c t u r e s . Along t h e way some first r e s u l t s a r e proved about a distinguished degree between which we w r i t e
O3I2.
A summary i s given i n
. Four types o f
0,'
and
0''
admissible o r d i n a l s have
t o be distinguished as f a r as t h e behaviour o f t h e jump o f r.e. grees i s concerned.
for
de-
242
WOLFGANG W S S
$0. P r e l i m i n a r i e s
p
Lowcase greek l e t t e r s a r e always o r d i n a l s , always l i m i t o r d i n a l s and
a i s always admissible i n t h i s paper.
W e consider only s t r u c t u r e s ';G = < Lp,B> r e g u l a r over D
s LR i s
Lp
, i.e.
zn% i f
' d x < (3 ( L r D
may c o n t a i n elements of
For
2,s
t h a t some
w r i t e s rrnp'p
p
ancfa
A set
D
ancflqa
An o r d i n a l
and
D
E.
$6-r.e.
{ K a Lo
I)-finite i f
& s U,"3
is
D
I K
C
D
.
zns
C
sets
i s E n % - i f and only i f
e
E (3 )
U,"
if
we w r i t e
for
(i.e.
D =
zn'&
which are given by some
n = 1
of
for
Wes
A ,D c L o
A srLD )
one says t h a t
i s %-reducible t o
A
i f t h e r e i s some index
D
e e (i such t h a t f o r
K c Lg
all
K
sn&i f t h e s e t i s xn$v . A s e t
i s called a (regular) 0-cardinal
for some
is
D
if
tame-
D3
.
.
anpLa,
~r3
For s e t s (written
.
such
zn$function
($-recursive)
K a Lo
d e f i n i t i o n . I n t h e s p e c i a l case (xi <e,x? s
(which
and one
instead of
W e f i x f o r t h e following u n i v e r s a l every s e t
dr
such t h a t some
anpa
b [ b is a (regular) cardinal]
L,.
En formula
d cofinally into
W e say t h a t a s e t
i s called
say t h a t a s e t
f o r the l e a s t
d s (3
s Lo i s c a l l e d
" p o s i t i v e neighborhoods" K C L,,
. We
6 ( i . e . p maps (3 1-1 i n t o b ). We w r i t e
into
I,% ( A ,$).
Lp )
is
B
L p as parameters) over t h e s t r u c t u r e %
f u n c t i o n maps
i n s t e a d of
L
B G LR and
i s d e f i n a b l e by some
for the least
projects
p
where
B
r\
one w r i t e s a n c f d X
1 c (3
A are
and
A
t)
3 H1 H2 8 L B ( E We
d
A
H1 si D
A
H 2 s Lo
-D
)
and KcLp-Ae3H,
H ~ r L ~ ( ( K , 1 , H l , H 2 > 6 WH~1*s D * H 2 C L R - D ) .
H I GH
The index
e
~ 1 -RECURS I VELY
can be communicated by w r i t i n g
One f u r t h e r defines t h a t
[XI
(written
A SwSD
L,,
6
says t h a t a degree
D
r e s t r i c t e d t o single-
i s defined by
A =$D
A asD
and t h e equivalence c l a s s e s a r e c a l l e d 6-deRrees
D **A
c
K
to
).
An equivalence r e l a t i o n
A E
.
A LZD
i s weakly $-reducible
A
i n t h e same way but with t h e s e t s tons
243
ENUMERABLE DEGRE ES
A
. One
has c e r t a i n p r o p e r t i e s i f t h e r e e x i s t a s e t
which has a l l t h e s e p r o p e r t i e s .
We study i n t h i s paper t h e
a-jump
operator
(see Shore c201
f o r a discussion o f t h e d e f i n i t i o n ) : A'
:= C<e,x* 13
H1 H2
i s t h e jump o f a s e t write
instead o f
We
L, (<x,H1,H2> c
E
we^
). Since we have
A
4e
H2 c I.,-
A
i n a - r e c u r s i o n theory
A S L ,
WeL&
H1 5 A
A)
(we always
D + A'
ss D f
t h i s d e f i n i t i o n gives r i s e t o t h e d e f i n i t i o n of t h e u-jump
c w ~ 'f o r
operator
We w r i t e s t e a d of
(0')'
bility)
U2La =~
"1
for the
0
6
At
w-degree o f t h e empty s e t and
A
in-
0"
-
. Observe t h a t UlLU 0' and ( u s i n g the a d m i s s i 0" . Furthermore we have f o r r e g u l a r s e t s A t h a t . E
One says t h a t an u-r.e. otherwise
.
2
at-degrees
set
A
i s complete i f
A
0
;
0'
i s c a l l e d incomplete.
We o f t e n use without f u r t h e r mentioning t h e r e g u l a r s e t theorem
of Sacks which says t h a t every a - r . e .
regular
a-r.e.
For a s e t
set A
(see [13],
c L,
[22],[81
one w r i t e s
such t h a t a c o f i n a l function
f :
a-reducible t o
A
rcf A = u
A
. The s e t
, otherwise
A
rcf A
degree contains a
f o r proofs).
f o r the least
b + o e x i s t s which
6
ot
i s weakly
i s c a l l e d hyperregular i f
i s c a l l e d non-hyperremlar.
d
2 44
WOLFGANG MAASS
Observe t h a t we have f o r r e g u l a r ticular
r c f A = o l c f ' L a t A ' a , i n par-
A
i s non-hyperregular i f f
A
Hyperregularity i s
i s inadmissible.
a p r o p e r t y of de-
-contrary t o regularity-
g r e e s r a t h e r t h a n of s i n g l e r e p r e s e n t a t i v s : gree then
i s t h e same f o r every
rcf A
& i s an a-de-
if
A E &
.
Simpson proved i n h i s t h e s i s t 2 2 1 t h a t f o r any
r = rcf
have t h . a t
u - c a r d i n a l and
A
f o r some eC-r.e.
v2cfLu8
= c2cf u
.
A
yc
Q
we
is a regular
iff
The f o l l o w i n g Lemma combines of Shore c19]
i n b ) Simpson's r e s u l t w i t h Theorem 2.1.
. The proofs
z2 p r o j e c t i o n
of a ) and c ) a r e s t r a i g h t f o r w a r d ( c o n s i d e r a
from x
i n t o a2por f o r c ) ). Lemma 1 :
a)
0'
b)
There e x i s t s an incomplete non-hyperregular
either
i s a non-hyperregular
c)
such t h a t
*
o2cfLOLw = u2cfcr
w2pa < x
and
(we w r i t e
w2cfu
zn'&s e t
i n s t e a d of
Jn,p
d-r.e.
OL
. degree
w2cfa
I,
L,
v3cfa < e3p u
set. According We w i l l con-
.
2 48
WOLFGANG MAASS
s t r u c t i n L111 a n
ci
where
cr 3 cf u
a3pu
that this set
A
w
. We will show in the
has the properties we want. The
construction for the case a2cfu =
o
is rather close to the con-
struction in OR1 and will be discussed briefly afterwards.
250
WOLFGANG MAASS
W e f i x f o r t h e following r e g u l a r a - r . e . that
and
C 4, D
(Dw)w
& B : ~ 4( ( ( 3 = 0 h y e D ) v ( ( 3 > 0
LaI= Vy’
and f i x a
0”
6
S tr L a b 3 y v x Y((3,y,x)
(3
B s
S
A
y 3 x 1WP,gt,x)))
Then we have f o r (3 > 0 : f3
E
+ I~yIa B l =pd( L a k Vx’f’(p,J,x))
S
lfi6 s
4
Iyl
25 1
ENUMERABLE DEGREES
.A
P
> r2cfa 8 02pu
if
H H
A
5
-
L,
.
As)
c2cfa >
and
for
"C Qe A"
.
(3
The next d e f i n i t i o n is t h e f i x p o i n t device which was mentioned i n point 5 ) of t h e motivation.
A i s an
e-fixpoint at s t a g e
f o r every
7
there i s a
, uo(
(1)
g(x)J
(2)
V limits A
and t h a t
f*
"x
x
w i l l not be
either.
Qo
w a s not put i n t o
and
A
and not
a t stage
U
since there e x i s t s
U'GQ
( z l + l ) n dom g f f c ( z ' + l ) fi dom gQ'
.
:
HIGH a-RECURSIVELY ENUMERABLE DEGREES
Since
( z t + l ) n dom g"
at s t a g e
s ( z ' + l ) n dom guO
3
x i s not put i n t o
r(g(z),r) I r(g(z),o)
f o r all
. Assume t h a t t h e r e a minimal s t a g e > . By t h e preceding no element is
Q
fro
such t h a t
I
r(g(z),co) < r(g(z),w)
w i l l be put i n t o r(g(s),Vo)
A
at some s t a g e
r(g(z),a)
4
where
t
w
5
y < r(g(z),.)
r c
i s d e f i n e d according t o c a s e 2 ) . Since
r(g(z),a) no element 5 wo :
y < r(g(z),uo)
w i l l be put i n t o
Otherwise assume t h a t
r(g(z),rO)
u1
A
> y
and
0' r(g(z),uo)
y
r
whereas
r ( g ( z ) , a o ) < cr at any s t a g e
i s t h e minimal such
i s defined according t o case 1 ) we have
= r(g(z),uo)
Q
i s t h e r e f o r e only p o s s i b l e i f
is defined according t o c a s e 1 ) of t h e d e f i n i t i o n o f
r
A
uo because o f c l a u s e b ) i n t h e c o n s t r u c t i o n .
It remains t o prove t h a t C '
255
c a n ' t be put i n t o
A
r(g
r
. Since
Q
l(z),o,)
at s t a g e
@ ,
as
it w a s shown i n t h e f i r s t p a r t of t h i s proof. Thus we have proved t h a t some X
i s an i n a c t i v e g ( z ) - f i x p o i n t at a l l s t a g e s i n
w0
[@,,OC) whereas t h e r e i s no i n a c t i v e g ( z ) - f i x p o i n t at s t a g e
Since we have
Txt ,C
definition of for all z LU Remark:
If
Q
stage
'L
au
r(g(z),r) a r ( g ( z ) , o )
u s a t i s f i e s t h e assumptions o f Lemma 4
x < sup f r ( g ( z ) , w )
ment
.
t h i s gives a contradiction t o the
and we have proved t h a t
.
Q
Iz
6 f
r\
dom g
. Therefore t h e s e s t a g e s
3
0
i s put i n t o
t h e n no e l e A
at any
play a r o l e i n t h i s proof
which i s similar t o t h e r o l e of " t r u e s t a g e s " ( s e e Soare1233) in t h e proof i n ORI. Lemma 5 :
a) b)
l c
S
~
=*
A
For every
e B a we have
and
.
Proof : For convenience we prove a ) and b) simultaneously by
256
MAASS
WOLFGANG
. Assume f o r t h e f o l l o w i n g t h a t
i n d u c t i o n on g"(e)
and t h a t a ) and b) a r e t r u e f o r a l l
el
g-'(e)
such t h a t
= z
g"(el)
.
< z
Observe t h a t t h i s a s s u m p t i o n d o e s i n g e n e r a l n o t imply t h a t
u
(A(e')
a3cfa
I g-'(e')
, which
z
5
u
=*
z1
S
. I n o r d e r t o show
f : rcf A +
). Lemma 5 b)
36 >
a.
.
=*
D r,A
7
.
.
1' 6 M
S
'trf it i s obvious t h a t A'
i f we s t a r t with some
r(g(a),u)
with negative neighborhood
can be expressed &-recursively i n
A'
.
259
HIGH a-RECURSIVELY ENUMERABLE DEGREES Case i i ) : a > w 2 c f a = The proof of Theorem 1
.
= w
U ~ D S
i s simpler i n t h i s c a s e s i n c e t h e
problem at l i m i t s of t h e p r i o r i t y l i s t d o e s n ' t occur ( s e e point 4 ) of t h e motivation). The c o n s t r u c t i o n i s c l o s e r t o t h e one i n ORT I 2 3 1 but we have t o be aware o f t h e o t h e r p o i n t s i n t h e motivation a n d t h e f a c t t h a t we c a n ' t use t h e r e g u l a r i t y of
as it i s done
A
i n ORT ( " t r u e s t a g e s " ) . According t o point 5 )
of t h e motivation we f i x a s t r i c t l y
increasing cofinal function recursive i n
B(O)
fw(X)J
3 %6 Q
If
:@
f : r c f D + a which i s weakly
with an index
. We d e f i n e t h e n
e
3 y 3 H (<X,y,H> E W
we go t o t h e l e a s t such
fQ(x) 1
e*c
r sw
minimal ( w i t h r e s p e c t t o a f i x e d c a n o n i c a l
4- o f Lot) say t h a t
such t h a t <x,;,fi>
$
f'(x)
and
E We,-
fi
A
A
H
5
a-
- A?)).
(01 I L,
and choose tx,$,$,
A 1 L,
well ordering
.
- A?)
c L,
We t h e n
i s t h e n e g a t i v e neighborhood of t h i s
computation. F u r t h e r we f i x a Z2 L, and
maps
g
approximation
3u
Vn
0
and
.
b) i C ~ g , ' ~A)
26 1
ENUMERABLE DEGREES
sup f r(g(m),w) I
which i s used f o r t h e proof of
Q
as usual.
implies t h a t
sup t l ( g ( n ) , u ) I u L Tn)
= rcf D
which i s absurd according t o t h e preceding.
The proof of Theorem 1 i s now f i n i s h e d . W e have proved Theorem 1
i n order t o get t h e following c o r o l l a r y :
Corollary : Assume t h a t incomplete high a-r.e.
u 2 c f a )v2poc
degrees.
. Then t h e r e e x i s t
262
WOLFGANG M A S S
Proof of the corollary: The case Shore C203
. For the other admissible
non-hyperregular a-r.e. sets D
OL
OL
= o2cfo~ is proved in
there exist incomplete
if o2cfa >, U 2 p a
according to
Shore [ I 9 3 (see also E l l ] for another proof of this fact). Apply Theorem 1
$2.
D and an u-r.e. set C
to this set
0'
.
0312
The depree For those
6
ci
where incomplete non-hyperregular a-r.e.
degrees exist there exists a distinguished a-degree between 0' and
for which we write
0"
0312
. We will show in the following
and in t 1 1 3 that there is a close connection between O 3 I 2
and the
jump of non-hyperregular a-r.e. degrees. Lemma 7 :
rx
Assume
is such that incomplete non-hyperregular 0312
a-r.e. degrees exist. Then there is an a-degree 03/2 cg
a)
01
b)
0312 is the greatest
0, L, C)
set and
D
such that
01'
A2
0312
L,
degree
(i.e. 0312 contains a
for every A ~ L , set D
0312 is the greatest tame- z, L, degree (i.e. 0312 contains
a set
S
such that { K c L-1 K
D B,03/2
Remark:
*
If a is
A, Lu degree and
for the
OL
0312
0"
1
is
z, L
o1
and we have
D with this property )
for every set
swa
d) u 2 ' a
S
6,
I,
for the set U 2 L a ~ O fand t any 8 , admissible then 0 '
is the greatest
is the greatest tamez, L
degree. Thus
of the Lemma they meet together in the middle, one
coming from below, the other coming from above.
H I GH
8-
P r o o f : $:= tL,,C) set
missib1e.A S
missible
r,
0
i s A2L,
xl$,).
and
r2 L,)
(tame-
i s ina d-
i f and o n l y i f
Friedman [ 3 ] obse rve d t h a t f o r inad-
A l Lp
a greatest
(3
between
w i t h C c 0' r e g u l a r and cl-r.e.
S B L,
A 1 '& (tame-
is
263
RECURSIVELY ENUMERABLE DEGREES
P-degree e x i s t s which l i e s s t r i c t l y
and which i s a n u p p e r bound f o r t h e tame-
0'
d e g r e e s . T h i s r e s u l t c a n ' t be g e n e r a l i z e d t o a l l ina d-
Lo
m i s s i b l e s t r u c t u r e s c L ,D) even i f D i s r e g u l a r o v e r L o : b and regular i s The s t r u c t u r e f+ = < L L,C > w i t h C I 0' a:-r.e. i n a d m i s s i b l e (we h av e
nu
A,$-
is the greatest
w = o l cf s x L u < w l p S
degree
. But
. A cco r d i n g t o Lemma
01
=
u2pw
but
)
0'
% = < Lp,B> where
w e h ave o 2 p a < at f o r t h o s e
1
where i n c o m p l e t e n o n - h y p er r eg u l ar
d we have u - l p at
= ?:t
F r i ed m a n' s argument works as
w e l l f o r those inadmissible s t r u c t u r e s
0 1 p ~ t l )< (3
:w
w-r.e.
degrees e x i s t . Since
5 =
f o r the considered s t r u c t u r e
t h e r e i s no problem w i t h t h e a d d i t i o n a l assum ption i n t h i s c a s e . Take a r
I
Al&
and d e f i n e
C v M
:=
set if
S
c L,
set
out of t h e g r e a t e s t
2x I x
E C
that
S
i s (weakly) $ - r e c u r s i v e
a-recursive i n
in
03'2
2 contains a
I n o r d e r t o p r o v e c ) i t remains t o show t h a t $ s e t . I n t h e case e 2 c f a z -2pw
Theorem 1
i n C91
.
If we have u 2 c f o t
s t r o n g l y i n a d m i s s i b l e and tamegree
0
zl &
4
may o r may n o t e x i s t f o r t h e s e
f i n e s t r u c t u r e of
t h i s f o l l o w s from
0 2 p ac
then
&
s i t u a t i o n where i n co mp l et e n o n - h y p er r eg u l a r
x
&
r2p
01
a
is
s e t s which a r e n o t of de-
, de pe nding
& as it i s shown i n 52 o f [ g ]
we h a v e an & - c a r d i n a l
i f and o n l y
by u s i n g t h e c o r r e s -
.
E
M
. T h e r e f o r e we c a n prove
C v M
ponding p r o p e r t i e s of t h e &-degree
El
set
1 u f 2 x + 1 I x c M 3 . Then we have f o r e v e r y
a ) and b ) f o r t h e s o d e f i n e d oc-degree
tame-
%-degree
Al$
t o be t h e &- d eg r ee of t h e A 2 , L
03/2
i s (weakly)
S
M c ct
s u ch t h a t
a-r.e.
on t h e
. However
i n our
degrees e x i s t
rr2cfLaw
= o2cf g
264
WOLFGANG MAASS
. Therefore we can apply the construction of and get tame- ,F,% set of degree ; . Lemma 5 in Property d) follows from Theorem 2 in C91 . according to Lemma 1 L9]
4
The greatest 4, L ,
Remark :
z2L,
and the greatest tame-
degree can be determined for the other admissible
well. The results might be useful for the study of
0~
r2 L,
as
de-
grees. For u with u2cfa < u 2 p a = oc we have that the greatest
d2 L,
degree is equal to
degree is equal to
0'
places compared with
and the greatest tame-
0"
(thus these two degrees have switched their
z2admissible at ).
For the other ac with the property that
0'
is the only
non-hyperregular x-r.e. degree we have that v2cf OL and in this case there is a greatest ween
0'
and
A,
L
4
u2p u s
Q
degree strictly bet-
whereas the greatest tame- Z, degree is either
0"
equal to the greatest A , degree
is equal to
r2 ,L
(if e2cfLa(e2por) = o2cfs )
or
(otherwise) as one can see by using Lemma 1
0'
and arguments of $ 2 in C93
.
For all a which are not 1, admissible we have that the greatest
r
I.,
A 2 L,
degree
for every
2 has the property that U2L"
Swag
+b
w-degree a _ .
The following rather technical Lemma will be the heart of the proof of Theorem 2
'. It generalizes an observation of Shore
(Lemma 3.3 inc181) which also has important applications in (3recursion theory (see Lemma 3
, $2
in L91
1.
HIGH a-RECURSIVELY ENUMERABLE DEGREES
Lemma 8 :
265
$= c L p , B ,
Consider a structure
and a limit 9i rL ordinal I -r (I such that ulcf (3 < g and vlcf a 9 1,P 1,P (see $0. for definitions).
*
J
If Dc- L A is regular over L A and [ K {K
L21K 5 LA
6
Proof:
C,
XI% then
D3 is
D3 is Zl& as well.
The same trick as in Shore C181 is used. F i x a
y
definition
-
L iK
E
*
of the set f K c LA[ K c D 3
zl3
a cofinal
zl%
function p : vlcf X + I and a cofinal Z,% function .L %% & q : ulcf p + (3 Define a set M S ulcf a r d c f (J by
.
bM
:t)
VX
nl$
c Lp(y)
(X 6
D + :=
2 is al-r.e., incomplete, regular and
non-hyperregular. Then we have
o(
olcf A&
3 i s Zl'&
n2 L,
.
)
YQ
is then
q
because according t o Shore L18] we have
cofinal i n
p,'L-9A'
i s obviously
A'
and r e g u l a r ,
all satisfied i n t h i s situation :
u l c f SGOL
i s A, L ,
A'
r2L,
is
A'
Zl < L q y A > map. We apply Lemma 8
:= tL,,C>
Lemma 8
i n t 2 0 1 ) . We g e t t h e n
by Lemma 7 b )
03/2
$a
.ik
We have
A
( i t i s t h i s fact
we show t h a t
A'
O3I2
it i s enough t o show t h a t
Z 2 L,
(g(f) = y
from Lemma 7 d ) .
A'
dq
Z
S CWa A '
which i s a c t u a l l y proved i n Theorem 2.3.
03/'
Then we have
Ix3 s A'
x tt
. T h i s implies
e
.
L,
VyPE
which i s defined by
S
n
and
glcf'La*A'a
1.
according t o
t l
26 7
HIGH a - R E C U R S I V E L Y ENUMERABLE DEGREES
$ 3 . Summarp Two factors determine the results about the jump of u-r.e. degrees : the relative size of u2cfcc
and W2poc
and
the existence of incomplete non-hyperregular u-r.e. degrees. Therefore we distinguish four different types of admissible ordinals oc :
( I I a2cfor
%
w2pw
a n d there exist
"0 incomplete non-hyper-
regular u-r.e. degrees which are
(these are exactly those
(2) o2cfa
3 v2pa
I, admissible)
and there exist incomplete non-hyperregular
a-r.e. degrees (these are exactly those
&
rr2cfoc < 02por
OL
which satisfy
o(
> a-2cfa 8 u 2 p u )
and there exist incomplete non-hyperregular
a-r.e. degrees
&
u2cfa
0 - 2 ~ and ~ there exist "0 incomplete non-hyper-
regular at-r.e. degrees , For the types (2) and ( 3 ) there exists the distinguished degree 03/2
between 0 '
been described in Lemma 7 For
that
a
@'
.
0''
at of type ( 4 ) we have
or-r.e. degree For
and
with the properties that have
p' = 0'
for every incomplete
& (Shore [20]).
a of type ( 3 ) we have for incomplete a-r.e. degrees k = 0'
if
is hyperregular respectively 2' = O 3 l 2
is non-hyperregular according to Theorem 2 For
.
if
a of type ( 1 ) and (2) there exist incomplete o(-r.e.
degrees & such that for type ( 1 )
1.
a'
= 0''
according to $1. (see Shore c2OJ
WOLFGANG MASS
268
In particular we have thus shown the following : Corollary:
Assume that
is admissible. Then there exist
~~
high incomplete a-r.e. degrees if and only if U2cfa
a2pa
.
We will continue the study of type ( 1 ) and ( 2 ) in c111. It turns out that ( 2 ) is the most interesting type as far as results about the jump of u-r.e. degrees are concerned.
FtEFERENCES : C1’J S.B.
Cooper, Minimal pairs and high recursively enumerable
degrees, J.Symb.Logic 39 ( 1 9 7 4 ) , 655-660 L23
K.J. Devlin, Aspects of constructibility, Springer Lecture
Note 354 (1973) [3]
S.D. Friedman,
[4]
A.H. Lachlan, A recursively enumerable degree which w i l l not
(3-Recursion Theory, to appear
split over all lesser ones, Ann.Math.Logic 9 (1975), 307-365
151 M. Lemnan, Maximal a-r.e. sets, Trans.Am.Math.Soc.
188
(19741, 341-386 r.63
M.Lerman and G.E. Sacks, Some minimal pairs of &-recursively
enumerable degrees, Ann.Math.Logic [7]
W. Maass,
4 (19721, 415-422
On minimal pairs and minimal degrees in higher
recursion theory, Arch.math.Logik 1 8 ( 1 9 7 7 ) , 169-186
[81 W. Maass, The uniform regular set theorem in at-recursion theory, to appear in J.Symb.Logic W. Maass, Inadmissibility, tame RE sets and the admissible
[9]
collapse, to appear in Ann.Math.Logic I101
W.
Maass, Fine structure theory of the constructible
universe in oc-
and p-recursion theory, to appear in the
269
HIGH a-RECURSIVELY ENUMERABLE DEGREES
Proceedings of "Definability in Set Theory" (Oberwolfach 19771, Springer Lecture Note [ll]
W. Maass, On
OL-
and
0-recursively enumerable degrees,
in preparation
[123
G.E. Sacks, Recursive enumerability and the jump operator,
Trans.Am.Math.Soc.
1133
G.E.
108 (1963), 223-239
Sacks, Post's problem, admissible ordinals and
regularity, Trans.Am.Math.Soc.
L143 Ann.
[15]
(1966), 1-23
G.E. Sacks and S.G. Simpson, The ct-finite injury method, Math.Logic 4 (1972), 323-367 J.R. Shoenfield, Degrees of Unsolvability, North Holland/
American Elsevier, Amsterdam/New York
116'1 R.A.
(1971)
Shore, Splitting an a-recursively enumerable set,
Trans.Am.Math.Soc. 204 (1975), 65-77
LIT] R.A. Shore,Minimal 4-degrees,Ann.Math.Logic ClSl
R.A.
4( 1972),393-414
Shore, The recursively enumerable u-degrees are dense,
Ann.Nath.Logic 9 (1976), 123-155
[191
R.A.
Shore, The irregular and non-hyperregular K-r.e.
degrees, Israel J.Math. 22, No.1,(1975),
28-41
f201 R.A. Shore, On the jump of an a-recursively enumerable set, Trans. Am.Math.Soc.
C211
217 (1976), 351-363
R.A. Shore, Some more minimal pairs of u-recursively
enumerable degrees, to appear
t223
S.G.
Simpson, Admissible ordinals and recursion theory,
Ph.d.dissertation, M.I.T.
[23]
(1971)
R.I. Soare, The infinite injury priority method, J.Symb.
Logic 41 (19761, 513-529
J.E. Fenstad, R.O. Gandy. G.E. Sacks (Eds.) GENEMLIZED RECURSION THEORY I 1 @North-Holland Publishing Conpany (1978)
E x t e n d a b i l i t y o f Z F Models i n t h e von Neuman H i e r a r c h y t o Models of KM t h e o r y o f classes W.Marek
0.
by (Warszawa) and A.M.Nyberg
(B0 i Telemark)
Introduction. The problem of e x t e n d a b i l i t y o f models o f ZF s e t t h e o r y t o mo-
d e l s of KM t h e o r y of c l a s s e s is as f o l l o w s : I f < M , E > is a model o f ZF, when c a n w e f i n d a f a m i l y r S p ( M ) s u c h t h a t <M u '3 ,M,E u € (Mxy)> is a model o f KM (Kelley-Morse t h e o r y o f c l a s s e s ) ? The problem is p a r t i c u l a r y i n t e r e s t i n g when M
r
is a t r a n s i t i v e set and E is t h e membership r e l a t i o n , €, o n t h i s set. As p o i n t e d o u t i n Marek-Mostowski [4] i n f a c t a t l e a s t t w o n o t i o n s of e x t e n d a b i l i t y a r e involved.
Namely, a p a r t from t h e above
mentioned n o t i o n t h e r e is a more r e s t r i c t i v e n o t i o n , f o r t r a n s i t i v e <MI€> o n l y , namely, one s a y s t h a t <MI€> is 6-extendable i f and s u c h t h a t < M u r ,MI€> is a 6-model of KM, i . e . one f o r which t h e n o t i o n of w e l l o r d e r i n g , W.O. ( - ) , is absolute. I n Marek-Mostowski [4] it is proved t h a t t h e n o t i o n s o f extenda b i l i t y and 5 - e x t e n d a b i l i t y d o n o t c o i n c i d e on t h e c l a s s o f denumer a b l e t r a n s i t i v e models o f s e t t h e o r y : o n l y if t h e r e is a n
0.1. Theorem. I f a. is t h e l e a s t o r d i n a l p s u c h t h a t L p is ext e n d a b l e and al is t h e l e a s t o r d i n a l v such t h a t L, i s 6-extendable, then
(1)
a0< a 1
(2)
a.
(3)
(4)
Both a. and
is denumerable i n
"Lao is e x t e n d a b l e "
al are denumerable.
On t h e o t h e r hand i t is e a s y t o prove:
0.2. Theorem.
I f M is a t r a n s i t i v e model o f ZF s e t t h e o r y such
t h a t c f (Onn M) >w
then
27 1
W. MAREK and A.M. NYBERG
2 72 <M,€>
is e x t e n d a b l e i m p l i e s t h a t
moreover, f o r a n y f a m i l y t u r e is a
B-model
such t h a t
<M,E>
is
pextendable,
<MAF,,M,€>bKM t h i s s t r u c -
( i n f a c t t h e n o t i o n o f w e l l o r d e r i n g is elemen-
t a r y over M ) . Thus e x t e n d a b i l i t y and
B-extendability c o i n c i d e on a class of
t r a n s i t i v e models w i t h c o f i n a l i t y of h e i g h t a t l e a s t wl.
This
l e a v e s open t h e case o f c o f i n a l i t y c h a r a c t e r w . I n t h i s paper w e a r e going t o t r e a t t h a t case and w e show t h a t t h e n o t i o n s of e x t e n d a b i l i t y and B - e x t e n d a b i l i t y d o n o t c o i n c i d e f o r t h e c l a s s of u n c o u n t a b l e models of ZF w i t h c o f i n a l i t y o f h e i g h t equal t o
To be more s p e c i f i c w e a r e going t o prove:
W.
I f a. is t h e l e a s t p such t h a t V p is e x t e n d a b l e and 0.3. Theorem. al t h e l e a s t p such t h a t V p is 6-extendable, t h e n
(1) a.
< al
> k"cfa
(2)
b
(M)
'I
(There i s a c h a r a c t e r i z a t i o n o f e x t e n d a b i l i t y f o r a r b i t r a r y models of ZFC ( n o t o n l y s t a n d a r d ) i n Marek-Srebrny [5] .) There i s a n a l t e r n a t i v e c h a r a c t e r i z a t i o n of t h e n o t i o n o f B - e x t e n d a b i l i t y i n terms of Ramified A n a l y s i s o v e r M ( f o r t h e d e f i n i t i o n see Marek-Mostowski
l e t R.A.M 0.6.
[4]
or Moschovakis [7] )
.
Namely
be t h e f a m i l y o f a l l r a m i f i e d a n a l y t i c a l s u b s e t s o f M.
Theorem.
6-extendable
Then M i s > s a t i s f i e s t h e second
L e t M be a t r a n s i t i v e model of ZFC.
i f and o n l y i f
Proof: By theorem 0.5 t h e r e is a t r a n s i t i v e model N of ZFC- such t h a t V,€
N and < N , E >b"Inacc(V,)". Thus € :V s i v e l y i n a c c e s s i b l e ) . W e r e a s o n now i n N:
N
( s i n c e N is r e c u r -
W e a r e g o i n g t o c o n s t r u c t ( w i t h i n N) s e q u e n c e s { S n l n e w ,
bl
s e t S(, = SonV,
and
QO
such t h t
so'€>- 4 < V iE, >, Sn+l of l e a s t p o s s i b l e r a n k and power, set Sn+l = and = t h e l e a s t p s u c h t h a t SA+lCVp. S i n c e V, is i n a c c e s s i b l e i n N we have t h a t nn+l 4 (V,, €>.
275
EXTENDABILITY OF ZF MODELS I N THE VON NEUMAN HIERARCHY
gWsn. clearly
Set s = S i n c e S;
<S,E> 4 have t h a t S n V, = n!wS,', = V 5 .
= S n n V,we
The set S is n o t t r a n s i t i v e b u t S n V ,
is.
The c o n t r a c t i o n f u n c t i o n IT f o r S is c o n s t a n t o n SnV,= V c and so a(V,) = V c . Thus < r * S , E , V5>: and so n(S) = V'. 5 nn and {q:}ncw is i n c r e a s i n g . Thus By o u r c o n s t r u c t i o n 5= cf(6) = w
which c o m p l e t e s t h e proof of t h e lemma.
W e r e c a l l t h e f o l l o w i n g w e l l known f a c t a b o u t models o f ZF i n . t h e von Neuman h i e r a r c y .
I f k ZF and c f ( a ) > w I t h e n t h e r e i s
1 . 2 . Theorem. such t h a t Proof:
Using a n y f i x e d enumeration
language of set t h eo r y we f i n d c r e a s i n g sequence
m formulae o f < V ~ , E >4 W w e a r e done. 0 6
The l e a s t a s u c h t h a t
k
ZF h a s c o f i n a l i t y
W.
W e g e n e r a l i z e t h e above theorem and c o r o l l a r y a s f o l l o w s : 1.3.
Theorem.
Yw
and V,
is
6-extendable
t h e n t h e r e is
such t h a t
i)
ii)
Vy
4
@-extendable
is
cf(y) =
iii)
Proof:
W
S i n c e V,
is t h e l e a s t cR.A.3,
n
TI,,€>
k @ ( x , T ) i f and o n l y i f x € 1 (T ~) A t t h i s p o i n t it i s i m p o r t a n t t o n o t i c e t h a t w e c a n make t h e c h o i c e o f 0 dependent of j u s t Y' and n o t t h e p a r t i c u l a r s t r u c t u r e m. e r more d e t a i l s t h e r e a d e r c a n c o n s u l t B a r w i s e [2]
.
EXTENDABILIPl OF ZF MODELS I N THE VON NEUMAN HIERARCHY
279
or Nyberg [ E l . i e t co%(T) be t h e formula l @ r x A i x ' ,TI is a c o d e and x is some s e n t e n c e i n . , L I f T is a f i r s t o r d e r d e f i n a b l e set o f s e n t e n c e s over w e t h e n have: To c o n c l u d e t h e proof
where
~ X ix1(EV,) A
By d e f i n i t i o n o f co%-.
IHYP
bcon~(T)
MYPm
k l@ ( rX A 1 tX'
3
S i n c e T w i l l be C1 ( i n f a c t A o ) o v e r pIYPm. By d e f i n i t i o n o f Y
ll
r x ~7 x1 q!
T$
T)
Iy(T)
!ATX
By t h e c o m p l e t e n e s s theorem 2 . 2 . T has an m-model.
But t h i s is e x a c t l y t h e c l a i m o f theorem 2.1.
111. The R e s u l t .
W e w i l l now prove t h e theorem which is o u r purpose f o r t h i s paper
namely:
L e t a. be t h e l e a s t p such t h a t bZFCand V p Theorem. is e x t e n d a b l e and l e t al be t h e least p such t h a t < V p r € >kZFC and V p is 6-extendable, t h e n : 3.1.
(a)
ao t h u s of . Hence Vs is ext e n d a b l e and so
ao)= z
iff
can be defined from
rn-’
.
(a,o,z) E
Xu. p(n-l,u)
is
0-
in the
same way as
rn
n
This shift operator can easily be constructed from
to
n-I.
is defined from
283
So it is needed to pass from s
JOHN MOLDESTAD (1.4.2
i n 111).
What h a p p e n s i f w e l e a v e o u t
s
" r e c u r s i o n ' t h e o r y " where
Below w e g i v e a n example o f a
s ?
i s n o t r e c u r s i v e ( q u o t a t i o n marks because
one c a n a r g u e w h e t h e r s u c h a c l a s s of f u n c t i o n s s h o u l d be c a l l e d a
It h a s o t h e r p r o p e r t i e s t h a n r e c u r s i o n t h e o r i e s
recursion theory). with
s .
One o f t h e main r e s u l t s i n P l a t e k ' s t h e s i s [ 5 1 ( F i r s t Recur
s i o n Theorem 5 . 3 . 2 )
f a i l s i n t h e example.
s
is
I n [21 I gave a proof o f P l a t e k ' s theorem
assumed t o be r e c u r s i u e . Omitting
I n P l a t e k ' s proof
There i s a mistake i n t h a t
t h i s assumption (theorem 29).
p r o o f , and by t h e example below t h e "theorem" i s f a l s e .
Let
u
d e n o t e t h e h e r e d i t a r i l y c o n s i s t e n t o b j e c t s of t y p e
HC(u)
o v e r t h e n a t u r a l numbers
HC'
= U{HC(u): u
i s a t y p e symbol of l e v e l 5 ' 1 .
f : N + N.
p a r t i a l functions
B 5 X,
such t h a t
FP('+u)+u
If
X
f o r a l l types
u
if
f,g E X
f E Rw(B).
that
and t h e t y p e s o f
W e let
k.
RQ(B)
X
Let
a b o v e i s of t y p e
Rw(B)'
are
f,g
is i n
X.
u +
Rw(B) b e t h e l e a s t s e t
X
R w ( B ) flHC'.
=
f(v)
T,
respectively,then
u
i s r e c u r s i v e from
f
b e d e f i n e d as
0+0+1+1+0+0
DC(x,y,f,g,v)
let
i s closed under composition, i.e.
i s assumed t o b e i n
FP('+u)+u
a t most
T )
HC
0 , some c o m b i n a t o r i a l f u n c t i o n s , and
$
( d e f i n i t i o n by c a s e s ) , a n d
(which i s o f t y p e
c_
contains t h e fixed point operators
DC
f(g)
B
S e e [ 5 1 or 812 i n HC(1) = t h e s e t o f
HC(0) = N .
[ 2 ] f o r n o t a t i o n s and d e f i n i t i o n s .
X c_ H C
HC = U { H C ( u ) : u i s a type symbol}
Let
N.
B
if
with t h e exception
Ru(B)
o n l y when t h e l e v e l o f u i s The f u n c t i o n
DC
and it i s d e f i n e d by: if
x = y
if
x
+
y
.
mentioned
285
THE SUCCESSOR FUNCTION I N RECURSION THEORY
Platek's First Recursion Theorem (5.3.2 in [ S ] ) : Suppose
B 5 HCLt2, and Then
cessor function s . (In [51
B
B
contains the number
Rw(B)a+3
0
and the suc-
Rat1(B)"'.
is supposed to contain pairing and depairing func-
tions for HC(0).
When
N
HC(0)
functions which are recursive in
there are pairing and depairing 0,s
,
hence this assumption can be
dropped. 1 Let
=
0.
By Platek's theorem:
If
RU(Bl3 = R1(B)3.
In the example below
B 5 HC2
R2(B)*
*
.t
R,(BI2.
that particular
0,s
E
B y B 5 HC s $
2
, then
B y and
So the conclusion in Platek's theorem fails for B.
The example: Let
fi,gi
(TEN), S,T
(of types
1,1, 1 + I , 1 +(l+l)
respec-
tively) be defined by: fi(X)
{x0
gi(x)
{x
S(f)
0
=
T(f,g) =
where
+
lfi 4
I:"
...,2i
x = OYZy4,
if
otherwise x = 1y3y5y...y2i+1
if
otherwise if
f = fi+,
otherwise
if fo
gitl if f = fit, and otherwise
denotes the partial function of type
undefined.
g
1
I
which is totally
Let
B = {0,152,---3fo3f~y f2y...yg0~g~~g2,...~SyT). Then
B 5 HC2.
gi
286
JOHAN MOLDESTAD
Theorem: Proof:
R1(BI2
+
R,(B)'.
U E HC(l+I)
Let
be defined by: f = fi
if
otherwise U = FP('+u)+u(AUufl
Then u
is
2, hence
to prove that if
Proposition:
gi's If
H
Hfi = H'fi
such that
because
H'
E
Ro(B).
R1(B) nHC(1+1),
E
H' E Ro(B) nHC(1+1)
{H'fi: iEN})
(=
such that
u = l+l.
The levelof
U $ R,(B)'
it suffices
then there is an
H E Rl(B) nHC(1+1)
{Hfi: T E N }
finitely many
where
To prove that
U E R,(BI2.
H' E R,(B) nHC(1+1) Then the set
.T(f,U(Sf))),
for
i = 0,1,2,.
.. .
will contain at most Hence
U
H.
$
then there is an
Hfi = H'fi
for a l l
i E N
Before going into the details of a complete proof we present a characteristic example of how to get HZ = Ax. FP[Xp uv. t1](4,x) tl(u,v)
,
Let
where
DC(u,v,hw. 0 , h w . t2,0),
~,(u,v) = DC(fl(u),f3(U),Xw'. t3(v)
HI, given H.
P(f2(V),f4(V)),XWr.
t3"I),
Xw". 0, Xw". l , O ) ,
= DC(p(q(l,l,O,v),O),O,
q = FP[Xq abcd. DC(a,O,Z,Xw'". t4,d)1, t,(a,b,c,d)
= DC(p(l,E),O,hr.
S ( X s . q(b,c,a,s))(d),Xr.
~,~,~,~,~',w",w"',a,b,c,d,r are variables of type 0
variables of type 1
,
O+l,
O + (0+(0+1))
,
0,o)
Z,p,q
are
respectively.
Let
Z = fQ
for some number
Q.
see what
p(u,v)
and
are,regard the following diagrams:
q(a,b,c,d)
Let
p = FP[Xp uv. t,].
To
287
THE SUCCESSOR FUNCTION I N RECURSION THEORY
0
u=v ?
q(a,b,c,d):
The v a l u e s o f
p(u,v)
and
q(a,b,c,d)
depend on t h e a n s w e r s
t o q u e s t i o n s , and c a n be f o u n d t o t h e r i g h t of t h e a r r o w s when no more q u e s t i o n s a r e a s k e d .
i s t o t a l l y undefined.
po for
p
(with 2
t,).
For e a c h
i s t o t a l l y undefined,
qa'O
qa
in
i s i n d u c t i v e l y d e f i n e d as f o l l o w s :
p pa''
= huv. t l
pa
q
pa
substituted
is i n d u c t i v e l y d e f i n e d a s follows:
qa7"'
substituted for
p a , q""
(with
Xabcd. D C ( a , O , f p,q
9
respectively in
,hw"'.
t,+,d)
t4). Let
qa,",
By t h e d i a g r a m : patl(u,v)
=
o
if
u = v
= pa(f2(v),f,+(v)) if = o
if
u
= I
if
u
+ +
v, fl(u)
v, fl(u)
* *
u
*
v
E
fl(u)
f,(u)
f , ( u ) , p a ( q a ( l , ~ , ~ , v ) ,=~ o) f,(u),
pa(qa(l,l,O,v),~)
+ o
288
JOHAN MOLDESTAD qa’Btl(a,b,c,d) = f (d)
a = 0
if
Q
= S ( A s . qa”(b,c,a,s))(d)
= 0
if
+
a
if
+
0, pa(1,8)
a
$
0, pa(1,8) = 0
0
By some calculations one can prove: = f (d)
qO”(a,b,c,d) q1 = qo
p’(1 , 8 )
because
2
p (u,v) = 0
p3(u,v) =
o
+
u
if
p2(1 , 8 )
because
u
if
= fo(u)
Q = 0
q3’2(a,b,c,d) = f
If
Q = 2’
+
0, b
or
fl(u)
9
Q-1
if
(d)
then
3
a
if
+
0, b = 0.
a
$
f3(u).
q
q > 4.
q5 = q4
when
Define HI
If
Q
4
(
= q1
Q > 4 Q.
2
then
q3” ‘ 9 ,
If
Q
>
1 then
q3y3(a,b,~,d)=fQ-,(d) 0 or 1
Q
p4 = p3.
then
o Q
v,
= 0
p4(u,v)
p5(u,v) when
+
u
if
then
>
for all p4
4,
if U I V , u,v
if
when
Substitute by the term
q 3 , where
is the symbol for undefined), qn
Three iterations are sufficient because
Then replace
2
u 9 v, fl(u) 9 f3(u),
if
qntl = Xabcd. DC(a,0,Z,~w~’~.t,,d) with t,.
q
< 2.
as follows:
4
Q = 1
q3 = q3”.
b = 0.
pm = p5
Q.
FP[Xq abcd. DC(a,O,Z,Xw“‘. t,,d)] qo = Xabcd.
if u = v,
f2(v) = f3(v).
f,(v),
$
p4(u,v) = 0
then
= q 3 for any
for any
2 5 Q 5 4, = p 3
0,
p4(u,v) = 0
f3(u), f2(v)’= f3(v).
fl(u)
If
q 3 = q 3 y 3 . If
If Q = 4 4
fZ(v)
q 3 = q2.
then
0, c = 0.
fl(v) = f3(v).
o
=
is undefined.
v, fl(u) = f,(u),
$
q3”(a,b,c,d)
$
q o y l . p’(u,v)
is undefined.
If
a
0
v, fl(u) = f3(u), f2(v) = f4(v).
p3(l,8) = 0.
if
a = 0, q
if
Q
FP[Xpuv. tl]
by
substituted for qa = q a y 3
p 5 , where the term
p,
q
in
for any a
.
is defined
similarly. Each fQ-l
’
fQ-2
pa, q a y B is defined by cases involving and the numbers
0,l.
fl,f2,f3,f,, fQ,
Only finitely many cases can be
defined by these functions, as compositions of them can be reduced,
T H E S U C C E S S O R FUNCTION I N RECURSION THEORY i.e.,
= fk(x), where
fm(fn(x))
k
max(m,n).
289
This is why the in-
ductions stop after finitely many iterations, and the number of iter-
Q
ations is independent of
+
s(n).
X.
If a skift operator fornumbers
ntl, things would have been different, as
for instance s(n) s(s(n))
and
In that case there might be no uniform upper bound
to the number of iterations.
Proof of the proposition:
Suppose H E R l ( B ) f l H C ( 1 + 1 ) .
H
can be
expressed as a composition of combinators, D C , fixed point operators FP(a+a)+"
where the level of
is
a
B.
1, and objects from
replace the combinators by the corresponding A-terms, i.e. by
Xu'.
u
,
Ka+T+a
by
XU'
vT. u, and
(the superscripts denote the types). reductions in the
X-term for
(Xu'.
s
s)ta, where
tained from
s
and
t
H
by
S
are
Ia+'
Xuvw. u(w)(v(w))
Then we perform the following
thus obtained:
by substituting t
We
Replace a term
A-terms, by the
X-term
s'
in all free occurrences of
obu in
Only finitely many reductions can be done because the terms are typed.
s.
Let
E
be,the expression thus obtained.
z,?,g
Let
be the objects from
B
So
H
= XZx.E.
(except
S
and
T) which
E.
occur in
Suppose t is a subterm of E of type 0 , with free variables + among x,u. Then t is one of the following expressions: I
U
I1
x
I11
n
U E b
nE.d
JOHN MOLDESTAD
290 VIU IX
T(Xv.sl,Xv.s2)(s 3 DC(sl ,s2,Xv.s3,Xv s , , 7 s 5 )
X
FP(,Xp?.s)(sl..
...'k) s,s,, ..., sk
(the level of p is 1 )
.sk
p(s,
XI where
are subterms of
Y
We will associate a tree we assign a term
E
Assign t
t
to this node. one node for
t
VII
is of type sp.
and one for
s1
t
is assigned to a node.
there are no nodes below.
there is one node just below.
If
To each node in Y
E.
to the term
Suppose
I , I1 or III
IV, V or VI
type
of type 0 .
as follows:
to the top node.
is of type
t
three nodes just below.
Assign
t
is of type
s 2 , s3
t
Assign
sl,
s 2 , s 3 , s,, s 5 respectivelytothese nodes.
there are
these nodes respectively. p(s,,
...,sk),
then
p
Y
Assign
is of type s
nodes just below
to these nodes respectively. In general
t
is a subterm of k t l
this case there are
If
VIU
thereare
there are five nodes just below.
nodes just below.
ktl
s
respectively to these
If
X
IX
If is of
Assign the term
nodes.
type
is of type
t
there are two nodes just below,
If
sl,
If
If
in some t.
is of
...k'3
s,sl,
XI, i.e.
t
t
to
is
FP(1pG.s).
Assign
s,sl,
In
... "k
This completes the description of
Y.
will not be wellfounded, as there may be infinitely
many iterations of
p
when
t
is of type
X.
We will assign finite lists of equations and inequations to some of the nodes in
Y, and for each list
L we also assign a term r.
Each list will consist of equations and inequations between terms of number variables, constants which occur in
the following type : 0,
fi(v),
gi(v),f.g.(v), 1 1
where
will also be of this type. called the value of
t
v
is a number variable.
The term r
Each list is called a condition.
under the condition
L
E,
r
is
( t is the term as-
29 1
THE SUCCESSOR FUNCTION I N RECURSION THEORY
signed to this node of Y).
r
and the value
L
dition
The number variables in a condition
will be among those which are free in
t.
L
A con-
is consistent if there exist numbers such that each equa-
tion and inequation in
L
is satisfied when these numbers are sub-
stituted for the variables,
Each condition will be consistent, and
the conditions assigned to a node will be disjoint, i.e. no list of numbers will satisfy two of them. node of m
r
is a constant
c, m
gi(v), figj(v)
and
t
L, r
t, and suppose that the list of numbers
be the value of
of
Suppose
are assigned tothe -f
n
satisfy m = c
under this substitution, i.e. n,fi(n), gi(n),
f.g.(n) 1 1 is substituted for v.
n
if
is
r
Then
m
L. Let
r
of
v, fi(v),
is the value
under this substitution.
The conditions and the values will be assigned in steps. taneously we will define a binary relation is a triple terms in
X'
(s,L,r),
Y, L and
L'
is a triple
r, r'
respectively.
X
low
Y, and the condition
L
in
s'
.
If
(s',L',r'),
X
X'
0,
when
M
v.
q'
is
is satisfied. 0
if
q
q'
is
is one ofthe
29 3
'THE SUCCESSOR FUNCTION I N RECURSION THEORY
constants q'
is
0~2~4,...~2(j-I)~ q'
f,(v)
if
fj-lgi(v) if where
is
is
q
(s,,L,q)
< (s,,L,q).
If
(t,LUM,q'),
and
Case V I I I
fi(v), where q'
gi(v),
is
f9.gk(v)
X
(t,LUM,q'); and
..,2i again (we let
downwards in
fied for
is obtained as
fi+l(v) = 0, because then we can let
0,2,4,..
chain
fi
fils
gifs can be defined by constants, or be obtained as
T(fitl,gi)
( =
gitl), in which case
fi+l must be obtained first.
This proves that there is an upper bound, independent of the number of
fils
and
gi's
Q, x, to
which can occur in a condition or a
value.This also completes the proof of the proposition.
THE SUCCESSOR FUNCTION I N RECURSION THEORY
30 1
References
111
J.E. Fenstad, Recursion theory: Springer Verlag (forthcoming).
121
J. Moldestad, Computations in higher types, Lecture notes in mathematics no. 574, Springer Verlag 1977.
[31
Y.N. Moschovakis, Abstract first order computability I, Trans. Amer. Math. SOC. 138 (1969) 427-464, and 11, 138,465-504.
[41
[51
An axiomatic approach,
Y.N. Moschovakis, Axioms for computation theories - first draft, in: R.O. Gandy and C.E.M. Yates (eds.), Logic Colloquium ‘69 (North-Holland, Amsterdam 1971 199-255. R.A. Platek, Foundations of recursion theory, Ph.D. thesis, Stanford University 1966, not published.
J.E. Fenstad, R.O. Gandy. G.E. Sacks (Eds.) GENERALIZED RECURSIffl THEORY I 1 0 North-Holland Publishing Company (1978)
Set Recursion Dag Normann 1. Introduction.
In this paper we will define a computation-theory called E-recursion, which will be a theory of partial set-recursive functions, defined on sets and with sets as values.
We will use natural numbers
as indices. The original purpose was to develop a theory on the companion of a normal functional of type k+2 in F
such that semirecursion over type k
and the theory are the same.
The motivation for this was that
this set-recursion theory might accept priority-arguments, arguments giving results about degrees of functionals.
Some results from that
program are given in Normann [I21 and [13]. The recursion theory we developed for that purpose, happened to be of a more general nature, and not quite unnatural even if one does not have the applications on degrees of functionals in mind. Moschovakis [ l o ] has constructed essentially the same theory, using inductive schemes and fixpoint operators. A computation theory on a structure must satisfy certain funda-
mental properties, composition of recursive functions gives a recursive function, you may diagonalize or compute on indices ([e)(eq,x>r (e,,](x>>.
There will also be some trivial manipulations
which are so deeply connected with the structure that they obviously must be computable.
In addition there may be some finiteness-proper-
ties, search-operators, stage-comparison etc. giving the theory its particular flavour.
These properties may either reflect the purpose
the theory-maker had with his theory, or they may reflect what the theory-maker thought was a natural notion of computation for the par303
304
DAG NORMA"
t i c u l a r structure. E-recursion w i l l be l i k e
a - r e c u r s i o n i n t h e sense t h a t we make
a c l e a r d i s t i n c t i o n between r e c u r s i o n i n an o b j e c t and r e c u r s i o n r e l a t i v e t o a relation.
This i s i n c o n t r a s t w i t h Kleene t h e o r y f o r r e -
cursion i n h i g h e r types.
Scheme 8 , which i s t h e only p l a c e where t h e
n a t u r e of h i g h e r t y p e s i s u t i l i z e d , i s a scheme of r e l a t i v i z a t i o n . It
i s j u s t n a t u r a l t h a t a c l e a r d i s t i n c t i o n h e r e should g i v e a s t r o n g e r theory. On t h e o t h e r hand, E-recursion w i l l be l i k e Kleene-recursion i n t h e sense t h a t t h e computations w i l l be a b s o l u t e , and not dependent on t h e domain.
Ir,
a , while i n
a - r e c u r s i o n we may s e a r c h through
E-recursion we may e s s e n t i a l l y j u s t s e a r c h through w i l l t h u s be weaker t h a n
The domain f o r
E-recursion
w.
a-recursion.
E-recursion w i l l be t h e u n i v e r s e of s e t s .
When
we use a s e t as an o b j e c t we w i l l t r e a t t h e s e t as a f i n i t e e n t i t y , we may use a l l information about t h e s e t o r information uniformly deI f we accept a
r i v e d from t h e elements o f t h e s e t a t t h e same time. set
a s a r e l a t i o n , we may j u s t ask i f a given s e t
R
o r n o t , and expect an answer from an o r a c l e f o r
x
is in
R
R.
Our scheme &may be d e b a t a b l e , b u t t h a t i s t h e one t h a t r e f l e c t s o u r d i s t i n c t i o n between o b j e c t s and r e l a t i o n s .
E-recursion i s nothing more than t h e schemes f o r t h e rudimentary f u n c t i o n s augmented w i t h a d i a g o n a l i z a t i o n scheme.
We hereby g i v e
the definition: D e f i n i t i o n of Let
E-recursion.
R 5V
sive relative t o i -
ii -
f(x
be a r e l a t i o n . R
,,,...,xn )
f ( xl , . . . , x
n
w i t h index = xi
) = x.\x
1
We d e f i n e t h e p a r t i a l f u n c t i o n r e c u r e
by t h e f o l l o w i n g schemes. e
3
=
(l,n,i)
e = (2,n,i,j)
3 5
SET RECURSION iii iv -
-V
f(Xl
,...,xn 1 =
f(XI,. f(XI,.
vi -
f(xl,
vii -
f(el
. ,xn> 2 ..,x,>
=
EXi,X.1 J U
- - ,Xn>
h(~,x2,
Y Ex1 h($1(Xq,
*
e = (3,n,i,j) e = (4,n,e') where e' is an index for h
- - - ,xn) - - ,gm(xq - - - ,xn>> 9
...,xn) x i n R ..,xn,yl,...,sm>=. [el 3R (XI,E
3
..,
e = (5,n,m,e',el, em) where e' is an index for h and e,,,...,em are indices for @ , ,gm ;Iresp.
...
e = (6,n,i)
,XI,.
e = (7,n,m) In scheme & it is understood that the computation terminates only if h(y,x2,
...,xn)
terminates for all y E xl,.
The partial functions defined by these schemes are called Erecursive relative to R
and they are denoted
{elR.
2. Some properties of E-recursion.
All functions that are rudimentary in R will be E-recursive relative to
R
stant function n sive.
Since for each n E
(E(R)-recursive).
UJ
the con-
is rudimentary, these functions will be E-recur-
Combining schemes
and
we may commute the arguments in the
functions. The schematic definition gives us canonical concepts of
-i ii iii
length of a computation
11 [
subcomputation computation tree
By standard proofs we obtain the recursion theorems and the S z - theorem.
DAG NORMANN
306
The following lemma will prove that arguments really are 'finite' in the usual sense of generalized recursion theory; and justify the term 'E-recursion'. Lemma 1. In E-recursion there is an index
e
such that for arbitrary
-I
R,x,el,x :
o
if ~
1
if V y E x
y
3y E x where
&
Proof.
and
~[ellR(y,Z> x z
o
(elIR(y,G)&
and
(e,I,R(y,;)
+o
means 'has a value'. There is a rudimentary function cp
(@I
y(x) = 1 =
takes values 3
and
for all x # 0. 1
y(0) = 0 = @
So we may assume that
{el)
only.
-
Let
such that
u
{el R (X,el,x) =
cellR(y,f>
YEX A s a corollary we will have stage comparison.
u
does not terminate, we write
u f and
llU[I
If a computation
= a.
Lemma 2. There is an E-recursive function p and only if ul&
or
Indication of proof.
u2J
such that p(u1,u2)
4
if
and then
We define p
by the recursion-theorem. Essen-
tially there will be 64 cases, one for each pair of schemes used in o1
and
02.
Similar results are well known in the theory of normal
functionals, and we regard the methods involved well-known. Moldestad
[9] gives an argument similar to our case (iv,v).
307
SET RECURSION
As a consequence of stage comparison (Lemma 2) we obtain:
-.
Lemma 3 . In E-recursion there is an index 3nE w
(e)R(el,;)J (el lR( EeIR(e,
e
such that for any R,el,x:
{elIR(n,;)&
, and
then
,GI , f >&
This is proved in Grilliot [ 2 ] , see also Moldestad [91. This kind of selection operator was first investigated by Gandy [I] and we call it Gandy selection. Definition. Let
R 5V,
We say that index e
E Vm.
Let
rp
be a partial map from Vn to V.
&
J
relative to R
is recursive
cp
if there is an
in E-recursion such that
vf
E
vn (rpG) zz
R-{el (x,Y)>
We then obtain natural definitions of sets recursive and semirecursive relative to R.
in
Definition. Let
A
5V, R 5V.
({elR(G> ; e E w If A
,n E w )
,f E
is E(R)-recursively (MB(R))BEfA
If A
C
fA
xEC is A;
M~(R)
.
closed we may split up A
closed, B
is s(R)-definable
parameters from MB(R)
cA
=
closure of A
be
as follows:
where fA is the set of finite subsets of A.
is E(R)-recursively
say that
C
Let the E(R)-recursive
if both
,
a finite subset of A , we if for some Ao-formula cp with
3yEMB U 1x3 rp(x,y) C and A\C
are $-definable.
3oa
DAG NORMA"
Lemma 4. Let A
be
(MB(R))BEfA Let cp Let
u E
E(R)-recursively
closed and transitive.
Then
satisfies C*-collection, i.e. be a
%.
A,-formula
with parameters from MB(R).
Assume V x E u 3yEMB U [X)(~)~(~,Y,~) *
Then 3vE%
Proof. Let
B'
V x E u 3yEv v(x,y,R)
.
be a listing of B.
By assumption R Vx E u 3e E w q(x, [el (B,x),R) By Gandy selection we choose one
e
.
to each x
and use the union
scheme to find v. Lemma
5.
Well-foundedness is C*-definable. Proof.
(i.e.
C;-definable)
By the recursion-theorem we find an index e
&
is a well-founded relation on x
,
the rank-function of y.
y is a well-founded
relation on x
h
3fEM
So,
then
[e](y,x)
such that if y
and
[e](y,x)
is
( f is a rank function for y).
b,Y)
Theorem 1. Let A
CZ A
be
E(R)-recursively
is E(R)-semirecursive
closed, B I
in B
Assume that
index such that
C is E(R)-semirecursive
xEC
ceg
(elR(x,:)&
A.
if and only if C is %(R)-
definable. Proof.
a finite subset of
.
By the recursion theorem we may prove that if
-
in B.
Let
{elR(x,g)J
e be an
, then
the
309
SET RECURSION computation tree will be in MBu (x~(R). xEC
Y
So
I T E MB U Ex3(R) (T is well-founded and T is a computation-tree for
By lemma
(el (x,;))
5 , this is a CI;(R)-definition of C.
On the other hand, assume C
is
CI;(R)-definable.
A s in the
proof of lemma 4 , we use Gandy selection to find a function that terminates exactly on C
.
Definition. R-admissible if each % is rudimentary closed in R , for each B , C E fA , MB 5 MC cs= B 5 MC,and We call a family (%),€fA
the family satisfies C*(R)-collection. Lemma 6. Let
(MB)BEfA
be R-admissible.
Then each
'%
is closed
under E(R)-recursion. Proof.
By induction on,the height of a well-founded relation we
prove by
C*-collection that if y
then the rank function is in M definable over (elR($)
, then
(MB)B EfA.
is a well-founded relation on x ,
-
So well-foundedness is C*b,YJ By the same method we prove that if
the computation tree is in
%.
The value of a com-
putation is rudimentary in the computation-tree, and lemma 6 follows. -l
This also shows that the relation (elR(B)= able over
(MB)BEfA.
By lemmas 4 and 6 we see that if A then (MB(R))B
x is C*(R)-defin-
is E(R)-recursively
closed,
fA is the finest splitting of A into an R-admissible
family.
3. E-recursion and Kleene-recursion. We are going to prove that E-recursion in a sense generalizes recursion in normal functionals. We will restrict ourselves to a set
DAG NORMA"
310
I with a canonical pairing operator.
I
=
Typical examples will be
the total functionals of type k ( = tp(k)).
I is a transitive set rudimentary closed in R. We may then identify finite subsets of I with elements of When we are investigating the part of E(R)-recursion by
I.
generated
I , it is natural to seek the least R-admissible family contain-
ing I U (I].
It is, however, an advantage to restrict the set of
indices to a smaller set.
This is covered by the following defini-
tion: Let
I be as above, R a relation.
By Ma(R;I)
we mean M(a,Ij(R)
over I we mean
By the spectrum of R
I
Spec(R;I) = (Ma(R;I))aE
,
U Ma(R;I) aEI
will be rudimentary closed relative to R.
are called R-admissible over I. theory that Spec(R;I) over I.
M(R;I) =
.
I will satisfy Z*(R)-collection over I , and each
(Ma(R;I)), Ma(R;I)
*
Such families
It will follow from our general
is the minimal family that is R-admissible
A key to this observation is the following definition:
Definition.
a
Let A 5 I x I ( = I)
Let
a
2
b
be a transitive, reflexive relation.
if A(a,b)
code for a set x
and A(b,a)
if A/=
.
We say that A
is isomorphic to
(TG(
{XI),€
is a )
(TG = transitive closure)
-b
Let
(Ma)aEI
be a family over I.
locally of type I if for any set x x E Ma
V
x
and
has a code in Ma
Lemma 7. Spec(R;I)
We say that
is locally of type I.
a E I,
(Ma)aEI
is
SET RECURSION
Proof.
By the recursion theorem we define an index el such that if
is a code for x , then
A
e,,
theorem one may use e E w, then
311
...,An
codes A1, (e,) R (e,A,,,
...,An)
{e,](A,I)
=
x.
Again by the recursion
to define an index e2
...,yn,
for
yl,
if
is a code for x.
such that for any R (el (y, yn) = x ,
...,
The definition is by
cases according to the schemes, and involves trivial but tedious constructions of codes. We are now ready to prove Theorem 2. is the minimal family R-admissible over I.
Spec(R;I) Proof.
We already remarked that
To prove that Spec(R;I)
Spec(R;I)
is R-admissible over I.
is included in any family (Ma)aEI
R-ad-
missible over I , we prove by induction on the length of the comput-
...,xn,
ation that for any x,,, R
(el (xl,.
..,xn) &
...,xn
if xl,
have codes in Ma
and
then both computation-tree and value will be in Ma.
Where we in lemma 6 would use here use a code for x and
C*-collection over a set x , we will
Z*-collection over I.
' Define the functional E
by
IE(f> f : I -IN
where
BY lemma Let
I
=
I,
IE is E-recursive in I.
tp(k)
,
E'
=
k+2E.
We assume that the reader is acquainted
with the basic facts about Kleene-recursion. Theorem 3 . Let F be a functional of type k+2, C 5 tp(k+l). following statements are equivalent
-i
C is Kleene-semirecursive in k+2E,F
Then the
312
DAG N O R M A "
ii
iii C
_.
Proof. -i
is C;(F)-definable.
We already proved that
a
j
in I
is E(F)-semirecursive
C
and
iii are
equivalent.
By the recursion theorem for E-recursion we find an index e
such that F {el (el,?>
=
K1eene (,;
[ell
F,k+2E
The definition is by cases according to the Kleene-index. For scheme 8, we use schemes & and v i , the other cases are rudimentary.
ii
h
-D
Since C
f E C
W
tion
(e)F(f,I)
is E(F)-recursive,
{eIF(f,I)J.
there is an index
e
such that
The method of proof is to copy the computa-
as a k+2E,F-computation on codes.
In doing this we
need: In Kleene-recursion there is an index el such that if f and are characteristic functions for codes for x and y respectively, then
el is found by using the recursion theorem and induction on min (rank(x), rank(y1). We then use
e,, and the recursion theorem to find an index e2
...,fk
such that if fl,
x1 ,...,xk, and
(elF(x1
are characteristic functions o f codes for
,...,xk) = y ,
XaEI {e21 (e,fl
then
,...,fk,F,k+2E,a)
is the characteristic function of a code f o r
y.
The construction
is by induction on the length o f the E(F)-computation. This theorem shows that Kleene-recursion in normal functionals is a special case of E-recursion in relations.
We will later see
that if we restrict ourselves to regard semi-recursion over I , then
SET RECURSION
we may reduce E(R)-recursion thus by theorem 3
to
313
for some F
E(F)-recursion
,
and
to Kleene-recursion in k+2E,F.
We will need recursive approximations of the spectrum: Definition. be an ordinal, A
Let a
; B E fA
( [e)R(i)
,e E w
a set.
By %(R)
we mean
and the length of the computation is shorter than a )
(%(R))B
We obtain definitions of
From now on, assume that
fA
,
.
I etc.
(e(R,I)>a
I is a set with a canonical pairing
and that N 5 I. Definition. Let
(Ma)aEI
be a family admissible over I , C
C is weakly C'-definable
We say that
in
5M
=
U
a €1
Ma.
(w-Ci) if f o r some
a
with parameters from M a ,
Ao-formula cp
Vb(xEM(a,b)
xE C
*
(xlY))
3Y M(a,b)
The concept is relativized to an arbitrary relation R. C
is w - A:
if both
C and the complemel.5 of C are W-C;.
Lemma 8. Let R
If C 5 M Proof.
I
be a relation, (Ma)a is
Ci(R),
-
then
Assume xE C
Let x E Malb.
xEC
h
C
=
Spec(R)
.
is w-Zi(R).
~ Y E M cp~(x,y) , ~
.
Then 3a EMa,b(3yE%,b)(cp(X,Y))
It is sufficient to show that the relation z = that a E Malb , x E Ma,b 3 A E M a ( R ) ,
s p e c t r a a r e l o c a l l y of t y p e I ,
and t h u s , s i n c e b o t h
3 16
DAG NORMA”
Spec(F;I) 5 Spec(R;I). From claim 3 it also follows that if some aEMa(R) Fa
,
Fa(f)
F(f) = 0
Ma(F)
,
then Tor
in a, I and by lemma 8
.
w - A*
we then have c5
3a E Ma (F,(f)
V
Va E Ma (Fa(f) is defined
Thus F is w-A*(R) Remark.
=
is defined.
is E(R)-recursive
For fEMa(R)
fEMa(R)
,
=0)
.
+. F (f) =0)
and theorem 4 is proved.
If well-foundedness is E-recursive in I
(e.g. I=tp(k)
for some k > O ) the proof of theorem 4 is much simpler.
Then just
define F by 0 if 1
f
is a code for a set x in R
otherwise
4. A hierarchy for the w-X*-relations in Spec(R1.
We will now restrict ourselves to recursion over I where S
satisfies pairing and contains N.
=
s u“ S ,
Moldestad [9] devel-
opes a notion of recursion in a normal functional over I , and by o u r results we may as well do E(R)-recursion
Let R be any relation. We write
%
Spec(R;I)
=
(Ma)a
I ,M
=
for %(R;I).
If C is a mations
Let
over I for some relation R.
X*(R)-subset
of
M , we obtain recursive approxi-
Ca by restricting the definition to xECa
x€fl
A
U Ma. aEI
E
L(a)-recursive r e l a t i o n
Y=(I,(Y)*)
5 ( a ) = { B ( x ) ;x ~
L(a)-recursive r e l a t i o n ,
Thus f o r example
L1(M,a).
F i n a l l y we u s e t h e n o t a t i o n t h a t i f "I,, t h e n
L(a)-recur-
p
such t h a t
C1-definitions
M U (L(M,@);B ' a l .
for
x
324
OAG NORMANN and VIGGO STOLTENBERGHANSEN
Let
ii
u
of
L* L(a3
,...,xn)
ky(xl
Furthermore
a(L(M,B))
definable s e t
iii
-
be an automorphism on ‘h.
A
5 L(a)m t h e r e a r e p l , ...,pk
Let
u
-
be an isomorphism from ?n onto o f * L*
Y
L(a) b ~ ( x ~ , . . . , x ~ )
m
Furthermore Y
-
and
there are
For A,B 5 L(a),%
,
on
we l e t
A
where “ r e c u r s i v e i n “ i s t h e
such t h a t
p l , ...,pk.
.
%7!
Then f o r
-
u
p l , ...,pk
u
E M
= L L u ( M , @ ) a r e dependent on
I,(L(M,B))
mined by t h e v a l u e s of
s o f o r each
I L ( a ) t y “(I,(X~>,
b u t n o t on t h e p a r t i c u l a r
5 L(a)m
A
on
0
Y
(xn>>.
@ < a
L(M,@) f o r each
=
i s determined by t h e v a l u e s of
each formula
B
L(a>, t=Y(a(xl), ...,a
5“A
“n’
Then f o r each formula
,
s o f o r each d e f i n a b l e s e t
E M
i s deter-
I:A
such t h a t
p l , ...,pk.
5
B
mean t h a t
L(a)w
i s recursive i n
A
analogue of “ a - r e c u r s i v e i n “ .
Theorem 1 Assume A
5
B
in
Proof:
L(a)
i f and o n l y i f
We f i r s t show t h a t i f
i n fact by a
‘ht i s imbeddable i n t o
-
Y
5
W
A,B
L(a)%
is
5 L(a)
L ( a b b Y(X,Y~,...,Y~)
C1
If
in
B
5 L(a)
u s i n g parameters
But t h e l a s t r e l a t i o n i s
is
W
So suppose
L(a)-r.e.
L,-formula
XEW
A
L(a).
5 L(a)
then
.
L(a)m-
then W is
r.e.
i s d e f i n a b l e over
y l , ...,yk L(a)
E L(a)h
.
L(a)m Then
I= YI(I(X),I(S~),...,I(Yk)).
over
L(a)
since
- 0 )
where
sp
Y
is
C1 and I m a )
L(a)-recursive. Let
tion.
Pu(a) = [ x E a ; s p ( x )
Suppose
A
i s t h e support func-
5 B i n L(a) v i a a r e d u c t i o n procedure
W,
i.e.
325
A NON-ADEQUATE ADMISSIBLE SET WITH A GOOD DEGREE-STRUCTURE
f o r each
al,a2 E L(a)
,
& a2nA =
0
C A
al
bl S B & b 2 n B = W
where
is
procedure
V1
Then
is
5
in
B
L(a)m-r.e.
Clearly
A
5
L(a),,,,,-degrees
&
5
in
B
L(a)m
v i a t h e reduction
B
and a subset of
in
given by
W3
.
L(a),,,,, v i a t h e reduction procedure V .
L(a)
via
There i s an imbedding
Corollary.
A
A
=bl
E(Pu(al>,Pu(a2),Pu(bl),Pu(b2));Pu(bl)
V1
r.e.
Then
= ( ( a l ,a2,bl ,b2) ; (al ,Pu(a2),bq ,b2)
W1
=
E W
0)
L(a)-r.e.
Now suppose A Let
3bl,b2((al,a2,bl,b2)
L(a)
,
(9,a2,bq,b2)EV1.
so
i s L(a)-
V1
Vl.
of t h e
i
L(a)-degrees i n t o t h e
i(L(a)-deg(A)) = L(a&,,,-deg(A)
where
5 L(a). Using t h e r e g u l a r s e t theorem for
f o r a regular
L(a)%-r.e.
L(a)
and t h e deficiency s e t
s e t we have
Theorem 2 The imbedding only i f every
i s an isomorphism on t h e r.e.
i
L(a)*-r.e.
degree contains a r e g u l a r
I n p a r t i c u l a r we have t h a t grees i f
L(a)%
i
degrees i f and L(a)m-r.e.
set.
i s an isomorphism on t h e r.e.
de-
i s adequate.
Theorem 3 Assume t h a t
?n'
i s a n a t u r a l r e p r e s e n t a t i v e of %ILet ..
be a f i x e d isomorphism of /hz onto ding of If
and
L(a)h A
I"A
Corollaq.
into
L(a)%
,
I = I,
L(a).
0
t h e derived imbed-
5 L(a)rhz i s 1st order definable over L(a)m,
have t h e same Let
.
then
A
L(a)%-degree.
a , 9 7 ~ and
an isomorphism between t h e of
'bt'
u0
'h.' be a s i n theorem 3. Zn-degrees of
L(a)
Then t h e r e i s
and t h e
Cn-degrees
326
and VIGGO STOLTENBERGHANSEN
DAG NORMA"
Proof of corollary: It follows from theorem 3 that the imbedding of degrees described in theorem I will be onto the definable
L(a),,,,-
degrees. Proof of theorem 3 : p1
Let A
,...,p~,L~M,a1),...,L(M,an)
be defined by a formula using parameters only.
Let
,
o
for any isomorphism u : 3n- m' I"A
=
if
=
T
1 [p,,
,...,pkI .
u o r (PI
...,pk] =
T
,
Then
then
I "A.
The following relation will be recursive: R(x,y)
There is a
T'
of parameters from M
such that
T
isomorphism u : 'h-
is defined on the set
T'
used in one of the definitions in
is as in lemma 1 (i)),
(where p on
1
w'
and
T I
p(x)
may be extended to an ( Io(x)
Iu(x) = y.
depends only
).
T'
Let R1(x,z)
h
3y(R(x,y)
& z = {w;wEy)).
Then x
sA
x
n
~z(Rl(x,z) & z 5 1"A)
A
=
0
This reduces A
v ~z(R,,(x,~) & z
to
1"A.
I"A
=
0).
To obtain the other reduction, define
u = (~;3yEvR(x,y)]
R2(u,v)
n
and
.
R2 will be recursive. Moreover, given v , the set of x for some y E v,R(x,y) in L(a)h R2(u,v)
.
holds, will be of bounded constructible rank
Thus there will be a set u
in L(a)m
will hold.
But then v 5 I"A
v
n
I"A
V
= 0
such that
3u(R2(u,v)
&
u 5 A)
v ~U(R~(~,V) & unA =
This ends the proof of theorem 3.
0 ) .
such that
327
A NON-ADEQUATE ADMISSIBLE S E T WITH A GOOD DEGREE-STRUCTURE
Theorem 4.
Let
=
field Q.
?(Q)
be a countably-dimensional vectorspace over the
Then L ( U ~ ) ~will be admissible, resolvable but not
adequate. Proof: a
.
0 ) ( 3 C ) [ 0 < C < A, B
v A, B
Thus no p a i r o f e l e m e n t s j o i n i n g t o
< C < 11. 6'
c a n b e a minimal p a i r .
If
one i s t o c a r r y on t h e d e c i s i o n p r o c e d u r e one must a l s o answer t h e q u e s t i o n l e f t open i n [6] of whether such a p a i r can have any (The r e l e v a n t s e n t e n c e i s
infimum a t a l l . [D(A,B
+
(VA,B,D)( 3 ~ )
Now i n ORT a l l t h a t [ 6 ] t e l l s
D < C(A,BVA,B(C(l].)
us (by r e l a t i v i z a t i o n ) i s t h a t t h i s i s t r u e i f (i.e.,
A'
such a s
=
or
Ni
incomplete
B'
For many
@I).
with
a
A
or
u 2 c f ( a ) < u2p(a)
t h i s i s r e a l l y a l l t h a t we need, f o r by d e g r e e i s low.
NL-r.e.
Now Lerman
t h e non-diamond theorem h o l d s f o r e v e r y t h a t t h e r e l a t i v i z a t i o n t o low r . e . r e s u l t p r e s e n t s no s e r i o u s problems. simplify the
v]
a
i s low
B
[34]
every
[16] h a s shown t h a t
and we a r e c o n f i d e n t
d e g r e e s needed for t h e g e n e r a l
This should t h e n g r e a t l y
t h e o r i e s of t h e a - r . e .
d e g r e e s for many
a
b r i n g u s t h a t much c l o s e r t o a d e c i s i o n p r o c e d u r e for them.
and
Indeed
we s u s p e c t (or b e t t e r hope) t h a t t h e problem of embedding t h e nond i s t r i b u t i v e l a t t i c e s i n t h e a-r.e.
degrees w i l l t h e n be t h e only
s e r i o u s one l e f t .
I n a s l i g h t l y d i f f e r e n t d i r e c t i o n w e have been a b l e t o e x p l o i t t h i s same p r o p e r t y of
i s c r e a t e d or
ac,i
OF a-RECURSION THEORY
ON THE &SENTENCES
Proof:
349
< €,i>.Let
We proceed by i n d u c t i o n on
u E be t h e
bound given by i n d u c t i o n f o r
< €,l>.
We wish t o c o n s i d e r
We may of course assume t h a t
f'hu(c)
= fh(c)
A-correct p r e s e r v a t i o n of index < E , O , 6,y> stage
>
T
u e only i f we enumerate a
z
for
>
a
< ctl,O>.
can be cancelled a t some in
a t stage
B
i s i n some p r e s e r v a t i o n o f index < c f , l , 6 ' , y ' >
with
i s a bound in
< c , O , 6,y>
W = {6< fh(c)
Next consider t h e s e t inactive a t stage
3.
a-finite. become
T>
<e,O>
T
is
W
As
witness
ue
and
i s regular there
B
no A-correct p r e s e r v a t i o n of index
T~
i s ever cancelled.
A-correct)
By
c'< e.
on t h e s t a g e s a t which such elements a r e enumerated
T~
Thus a f t e r
B.
u ~ . As
which
T
assumption a l l such p r e s e r v a t i o n s a r e c r e a t e d b e f o r e s t a g e t h e y only c o n t a i n elements less t h a n
Now any
ae.
A
T >
T
and a subset of
fh(E)
i s h y p e r r e g u l a r t h e map t a k i n g
-inactive.
created a f t e r stage
)(
~
is
6
< €,O>-
a n d t h e a s s o c i a t e d p r e s e r v a t i o n requirement i s
C1(A)
i s bounded by
T~
(3
After
'rl.
no
Thus it i s
~
Wt o i t s
E
6< f h ( e )
S o any p r e s e r v a t i o n of index
(or existing a t
T~
T~
< p.
T~
with
6
can
< € , O , 6,y>
-active)
must have an a s s o c i a t e d computation g i v i n g t h e c o r r e c t value of
If t h e r e were such p r e s e r v a t i o n s f o r every
B(y).
B
compute
a-recursively i n
a-recursive i n
A)
A
y
t h e n we could
( a s checking f o r A-correctness i s
f o r a contradiction.
Let
x
be t h e l e a s t
number f o r which t h e r e i s no such p r e s e r v a t i o n requirement and l e t T~
2
T~
for a l l
be a s t a g e by which we have such p r e s e r v a t i o n requirements y
T ~ .
Thus
i.e.
U>_T> T ~ .
has s t a b i l i z e d , i . e . k
k(T,C,O,x) =k(E,O,x)
t h e sentence coded by
for
h ( T , €,0)=&(T2,
has a l s o s t a b i l i z e d say a t
VT> T2.
k(E,O,x)
By our choice of
i s f a l s e and s o
By our E,O)
k(E,O,x), x
we see t h a t
g ( k ( c , O , x ) ) = 0.
350.
RICHARD A. SHORE
We can t h e n choose a g(O,k(c,O,x)) = 0
>_
T~
so that
T,
f o r every
>
U,T
g(o,k(T,e,O,x))
=
.
T3
F i n a l l y we claim t h a t no p r e s e r v a t i o n requirement of index <E,O,
6,y>
any
y
i s c r e a t e d a t any s t a g e
T
only i f i t happens f i r s t f o r
x.
. This
~
can happen f o r
Suppose we a r e c o n s i d e r i n g
0 ( x ) = BT(x) a t s t a g e
we can never f i n d a
T~
T
A T V C'
some computation choice o f
>
o>
T
with
>
T
T ~ .
By our
g(a,k(T,c,O,x)) =l.
Thus we must f i n d t h a t t h i s computation i s A-incorrect when we do t h e search a t stage
Of course we t h e n d o not c r e a t e any p r e -
T.
s e r v a t i o n and go on t o t h e next s t e p . This concludes t h e argument f o r required
CI~+~,
o.
but we begin a t ordinal
h
I+
there i s a
a,
bound on t h e range of t h i s map a s required.
We can now conclude t h e proof of theorem 4 . 1 by e s t a b l i s h i n g t h e following: LEMMA
Proof:
4.4.
B$
A V C ~for
i=o,1.
I f not we can compute
r e d u c t i o n procedure
b
uc,i
A
and
then t h e associated Ci.
p r e s e r v a t i o n would be c a n c e l l e d at some s t a g e a f t e r tradicting
,the d e f i n i t i o n of
.)
( I ~ , ~ ) . Now i f (Otherwise t h e Uc,i
con-
We would t h e n have a
c o r r e c t computation from t i o n about
< ~ , i > - i n a c t i v ea t a s t a g e
g i v i n g i n c o r r e c t informa-
>
bo,i
T
>
u ~ , ~ A.s
{a},
such t h a t
ATV
6.
Thus
i s never
6
C are regular A VCi i ( x ) = [6] ( x ) = B(x) =
A
cT
and
and t h e l e a s t such computation i s i n f a c t t h e c o r r e c t one
B'(x) via
T
6
c o n t r a r y t o our choice of
B
there i s a
via
AVCi
35 1
OF a-RECURSION THEORY
ON THE @-SENTENCES
We would t h e n t r y t o c r e a t e a p r e s e r v a t i o n requirement of
6.
.
index < c , i , 6,x>
A s t h e computation i s A-correct
and
t h e f i r s t p o s s i b i l i t y i n our s e a r c h can never happen.
T
>
uc,i
We must then
c r e a t e such a requirement a g a i n c o n t r a d i c t i n g t h e choice of
o ~ , ~ .
References [l] C.T.
Chong, Generic s e t s and minimal a-degrees, t o appear.
r 21
and M. Lerman, Hypersimple a - r . e .
s e t s , Ann. Math.
Logic 9 (1975), 1-1r8. [3]
S. Friedman, A counterexample t o P o s t ' s Problem, c i r c u l a t e d
manuscript. [4]
S.C. Kleene and E.L.
P o s t , The upper s e m i - l a t t i c e of degrees of
r e c u r s i v e u n s o l v a b i l i t y , Ann. Math. Ser. 2, 5 g ( i g 5 4 ) 397-407. [5]
A.H.
Lachlan, The elementary t h e o r y of r e c u r s i v e l y enumerable
s e t s , Duke Math. J . 35(1968) 123-146.
,
[61
Lower bounds f o r p a i r s of r e c u r s i v e l y enumerable
degrees, Proc. Lon. Math. [7 1
,
SOC.
(3) 16(1966) 537-569.
A r e c u r s i v e l y enumerable degree which w i l l not
s p l i t over a l l l e s s e r ones, Ann. Math. Logic g(1975) 307-365. [8]
A.
Leggett, Maximal a - r . e .
s e t s and t h e i r complements, Ann.
Math. Logic 6 (1974) 293-3557. [91
[lo1
,
a-Degrees of maximal a - r . e .
and R.A.
s e t s , t o appear.
Shore, Types of simple a - r e c u r s i v e l y
enumerable s e t s , J . Symb. Logic 41(1976) 681-694.
RICHARD A. SHORE
352 [ l l ] M.
Leman, On t h e elementary t h e o r i e s of some l a t t i c e s of a-
r e c u r s i v e l y enumerable s e t s , t o appear.
,
[ 121
I d e a l s of g e n e r a l i z e d f i n i t e s e t s i n l a t t i c e s of a-
r e c u r s i v e l y enumerable s e t s , Z . f . math. Logik und Gmnd. d. Math. 22(1976) 347-352.
, Maximal
[131
a-r.e.
s e t s , Trans. Ann. Math. SOC. 188
(1974) 341-386.
~ 4 1
, On
suborderings of t h e a - r e c u r s i v e l y enumerable a-
degrees, Ann. Math. Logic b(1972) 369-392.
,
1151
I n i t i a l segments of t h e degrees of u n s o l v a b i l i t y ,
Ann. Math. 93(1971) 365-389.
,
[161
Least upper bounds f o r minimal p a i r s of a - r . e .
degrees, J . Symb. Logic 39(1974) 49-56. [171
and G.E. Sacks, Some minimal p a i r s of a - r e c u r s i v e l y enumerable degrees, Ann. Math. Logic 4(1972) 415-442.
[18] W. Maass, On minimal p a i r s and minimal degrees i n h i g h e r r e c u r s i o n t h e o r y , t o appear.
,
1193
I n a d m i s s i b i l i t y , tame r . e . s e t s and t h e a d m i s s i b l e
c o l l a p s e , t o appear. [20] J . MacIntyre, Minimal a - r e c u r s i o n t h e o r e t i c degrees, J. Symb. Logic 38(1973) 18-28. [21] R.W.
Robinson, I n t e r p o l a t i o n and embedding i n t h e r e c u r s i v e l y
enumerable degrees, Ann. Math. 93 (1971) 285-314. [22] G.E. Sacks, Degrees of u n s o l v a b i l i t y , Annals of Mathematical S t u d i e s n0.55,
Princeton U n i v e r s i t y P r e s s , P r i n c e t o n , New
J e r s e y , 1963. [231
, The
r e c u r s i v e l y enumerable degrees a r e dense, Ann.
Math. 80(1964) 300-312. r241
,
Post ' s problem, a d m i s s i b l e o r d i n a l s and r e g u l a r i t y ,
Trans. Am. Math. SOC. 124(1966) 1-23.
ON THE B S E N T E N C E S OF a-RECURSION THEORY [251
and S . G .
353
Simpson, The a - f i n i t e i n j u r y method, Ann.
Math. Logic 4(1972) 323-367.
[ 261 R. A. Shore, a-Recursion t h e o r y , i n Handbook of Mathematical Logic, J. Barwise ed., North-Holland,
Amsterdam, 1977.
, Minimal a-degrees, Ann. Math. Logic 4(1972) 393-414. , S p l i t t i n g an a - r e c u r s i v e l y enumerable s e t , Trans.
[271
re81
Am. Math. SOC. 204(1975) 65-77.
, The
[291
r e c u r s i v e l y enumerable a-degrees a r e dense, Ann.
Math. Logic g(1976) 123-155.
,
[301
degrees, Z .
,
[313
Some more minimal p a i r s of a - r e c u r s i v e l y enumerable f . Math. Logik und Grund. d e r Math., t o appear.
On t h e jump of an a - r e c u r s i v e l y enumerable s e t ,
Trans. Am. Math. Soc. 217(1976)
,
[321
351-363.
On t h e elementary t h e o r y of t h e r e c u r s i v e l y enumer-
a b l e degrees, i n p r e p a r a t i o n . [33] S.G. Simpson, P o s t ' s problem f o r admissible s e t s , i n Generalized Recursion Theory, Proceedings of t h e 1972 Oslo Symposium, J.E. Hinman e d s . , North Holland, Amsterdam 1974,
Fenstad a n d P . G .
437 -441.
,
C341 [35] R . I .
Degree t h e o r y on admissible o r d i n a l s , i b i d . ,
165-194.
Soare, The i n f i n i t e i n j u r y p r i o r i t y method, J. Symb. Logic,
41 ( 1976) 513-53 0
-
[36] S.K. Thomason, S u b l a t t i c e s of t h e r e c u r s i v e l y enumerable degrees,
Z.
f . Math. Logik und Grund. d. Math.
[37] C.E.M.
Yates,
17(1971) 273-280.
A minimal p a i r of r . e . degrees, J. Symb. Logic
31(1966) 159-168.
J.E. Fenstad, R.O. Gandy. G.E. Sacks (Eds.) GENERALIZED RECURSION THEORY I1 ONorth-Holland Publishing Conpany (1978)
1 SHORT COURSE ON ADMISSIBLE RECURSION THEORY
Stephen G. Simpson Department of Mathematics The Pennsylvania State University University Park, Pennsylvania 16802
CONTENTS
§ 5.
........................................... ...................... The In selection theorem.............................................. Admissible ordinals: some examples...................................... Degree theory on admissible ordinals........ ............................
I
Reflecting o r d i n a l s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
§ 1. § 2.
§ 3. § 4.
6.
§ 7. § 8.
s
9.
$10.
Introduction.................
355
Primitive recursively closed ordinals.............
356
...................................................... Oracles, fans and theories..... ......................................... The basis problem: applications to logic................................ Conclusion...... ........................................................ ..................... References......................................... The Sn hierarchy.
361 367 371 374 377 380 383 387 387
$1. INTRODUCTION.
This paper consists of slightly expanded notes from my course of three lectures at the Second Symposium on Generalized Recursion Theory, held at the University of Oslo, June 13-17, 1977.
I would like to thank the organizers and sponsors
of the Symposium for making it possible for me to give these lectures. Although the lectures dealt only with admissible ordinals, the discussion was far from complete. My aim was to give, not a definitive treatment or even an overview, but rather a quick introduction to the subject as a whole and to some topics of current interest.
I am preparing a book-length treatment which is to be pub-
lished by Springer-Verlag in their series, Perspectives in Mathematical Logic. 'Preparation of this paper was partially supported by NSF Grant MF'S 76-05993.
STEPHEN 6. SIWSON
356 $2.
PRIMITIVE RECURSIVELY CLOSED ORDINALS. I n this section we define- the notion of a primitive recursively (p-r.) closed
ordinal. We then show how to generalize some theorems from ordinary recursion theory (on the natural numbers) to an arbitrary p.r. closed ordinal. Let OR be the class of all ordinals.
An n-ary function F : ORn
-+
OR i s
said to be primitive recursive if it i s generable from the initial functions F(x l.....x F(x)
=
=
xi,
15i4n;
x+l;
F(x) = 0;
F(x,y.u,v)
=
c
x
if
y
otherwise;
u < v,
by applications of the schemata of substitution
and primitive recursion + + F(y,Z) = Gbup F(z,x),y,x). Z2
and
=
There exists
p.
Theorem 5.13 (Shore [43] generalizing the Splitting Theorem of Sacks 1371).
B
If 8 is a nonzero a-RE degree, then there exist incomparable a-RE degrees and % such that
R U
6 = a.
Theorem 5.14 (Shore I441 generalizing the Density Theorem of Sacks [38]).
a
and
R
such that
are a-RE degrees with
a is an end extension of L Let zo be the first failure of stability for p within <M,Ez.
R (p ,z ) 0 0 0
holds.
B+'
Then Ro
Hence zo
admissible. Hence zo
and po
are witnesses for zo
is not less than a.
is not less than p.'
But
Hence p
p+ 5
is
so
in particular
a since a is
p+ stable. Hence by
3 77
SHORT COURSE ON ADMISSIBLE RECURSION THEORY 6.3
fi
is rI:
reflecting.
This completes the proof. The next theorem was first stated in some unpublished notes of P.
21.
The
proof can be found in Abramson-Sacks [ Z ] . Theorem 6.6 (H. Friedman and R. Gostanian [9])
Let a be countable, ad-
missible, and locally countable (i.e. every ordinal less than a is countable in
1;
the sense of L ) . Then a is
reflecting if and only if it is not Gandy.
(An admissible ordinal a is said to be Gandy if each ordinal less than a+ is the order type of an a-recursive well-ordering.) We end this section by presenting a new theorem which gives a sufficient (but not necessary) condition for a p.r. closed ordinal to be
reflecting.
In $7 we shall see how this theorem can be used to construct counterexamples in a-recursion theory. Theorem 6.7 (Simpson [50])
Let a be a p.r. closed ordinal which is
greater than w and equal to its
An
projectum (Definition 3 . 8 ) .
We use a Skolem hull argument similar to the one in the
Proof (sketch). proof of Lemma 3.5.
k ~[p].
Let cP(x)
Using the
7>n
be a
In+z
formula and p a parameter such that
selection theorem we can construct a certain Skolem
hull X 5 a such that X is the range of a p < 6 c a. LB
k ~[p].
Theorem 3.9
Let fi = type(X).
fi
o which
a be the first admissible ordinal such that La
projectum is
0.
is a model of the
l2 comprehension axiom. (The is equal to its own l2 projectum.) It is not hard
axiom of infinity, the power set axiom, and the last condition means that a
sentence Jr
to show that these conditions are expressible as a La
Jr.
Thus a is not
than its A3
projectum.
show that the A3
l5
reflecting. Hence, by Theorem 6.7,
such that a
is greater
(By going back to the proof of 6.7, we can actually
projectum of a is w.)
It remains to show that a is its own Sn projectum, for all n.
To see
this, note first that
a = sup N
keo
La k '
i.e.
a is the supremum of the first o infinite ordinals which are cardinals in L the sense of La. It is not hard to see that the function k + Nka is 12(La).
Thus the
,&
cofinality of a is o.
Now suppose that B 5 p c a is Sn+l.
XB(x) = lim 1' =
lim kl
where g is a-recursive.
We have
... lim g(x,yl, ...,y ,) n'
... lim g(x8 La,...,N La1 kn
kl
Clearly the predicate
kn
380
STEPHEN G..SIMPSON
L La g(x,U a,...,Uk ) = I kl n is A*.
Hence B is a-finite since
=
a.
Thus a is its own Sn+l
pro-
jectum, Q.E.D. Remark 7.6.
For any admissible ordinal a, the following inequalities
-:G
q3 C
S3 projecturn 5 p
2'
By Example 7.5 the first inequality can be strict. the second inequality can be strict while the S3
A similar example shows that projectum is
W.
Thus Lerman's
criterion for the existence of maximal a-RE sets could not have been stated conveniently in terms of the Jensen projecta qa and
p:.
In other words, Lerman's
Theorem 7.4 is a genuine application of the Sn hierarchy. $8.
ORACLES, FANS AND THEORIES. We are going to discuss a certain recursion-theoretic topic which has appli-
cations to logic on a countable admissible ordinal. was pioneered by Barwise in the 1960's.
The study of these logics
It is reasonable to look for applica-
tions of a-recursion theory to a-logic since, after all, ordinary recursion theory originated in the study of logical systems (adel IS]). The basic recursion theoretic notion which we shall require is the notion of a "fan".
Roughly, an a-fan is a nonempty subset of the powerset of a whose
complement is a-recursively enumerable.
In order to make sense out of this, we
must first answer the question: what do we mean by an a-recursively enumerable collection of subsets of a? Actually, this question is rather subtle and has at least three answers corresponding to three different intuitively natural notions of a-recursive oracle computation.
Each of the three is useful in certain contexts. We shall simply
list the definitions and refer to them uncritically by number (1,2,3). FROM NOW ON, a IS A FIXED BUT ARBITRARY COUNTABLE ADMISSIBLE ORDINAL.
Note
381
SHORT COURSE ON ADMISSIBLE RECURSION THEORY that the hypothesis of countability is satisfied in m s t of the examples of 54. We identify a subset A of
a with its characteristic function XA so that Za,
{O,l},
the set of all functions from a into Definition 8.1 an a-RE set W 5 a
A set
5 2a
a.
is said to be a-RE in sense 1 if there exists
such that
xcs
+-+
3u,v[~(u,v)
Definition 8.2
S
is just the power set of
cw
A set S
&
5 2a
a primitive recursive relation R
D~
5x
D,,
&
nx
=
d ~ .
is said to be a-RE in sense 2 if there exist
5 2a
x
a
x
a and a parameter p such that
Here we are using the notion of a primitive recursive functional F : amx (2a)n+a which is defined in the obvious way using the initial functional
F(x,A)
=
XA(x).
An equivalent formulation is that there exist a meter
p
ll
formula ~ ( x ) and a
para-
such that
6,
L [XI = fodo( call an acceptable prewellordering.
which admit what we
As in the case of
B-recursion
theory such structures fall into three disjoint classes determined by how "infinite" the domain is with respect to the recursive power of the structure: admissible, weakly inadmissible and strongly inadmissible. Not every
5-r.e.
f o r inadmissible
rate the
!-finite
! it
set is tame when
is inadmissible, i.e.
is not always possible to effectively enume-
subsets of an
!-r.e.
set.
Since the notion of
"finite" is basic in our generalization we are primarily interested in the tame(1y) r.e. (t.r.e.) adequate structures F l ,
sets.
Our main result is that for
the structure of the regular t.r.e.
39 1
2-
39 2
VIGGO STOLTENBERG-HANSEN
degrees is non-trivial (and rich) if and only if
M
or weakly inadmissible ($.e. weakly admissible).
Furthermolt a r i i g u -
i s admisr,iblt
lar set theorem for t.r.e. sets holds for adequate weakly admisLLhle
M.
N
The arguments used for weakly admissible structures are recursion theoretic in nature.
Thus, although we here restrict ourselves
to nice set theoretic structures, we believe that these arguments can be formalized in the axiomdtic framework of Moschovakis [ 9 1 and Fenstad [ 3 1 by weakenini their notion of "finite" to cover the inadnissible case.
This should be contrasted with the positive and nega-
tive results of Friedman ( [ 5 1 , [ 7 1 ) where
0
for strongly inadmissible
SB
and Fodor's theorem, or effectivized versions thereof, con-
stitute important ingredients.
1.
Preliminaries We will restrict ourselves to transitive rudimentarily closed
structures
E =
cM,E,R>.
Clearly the inclusion of urelements, im-
portant in applications, would not invalidate any of the results below. For a treatment of the rudimentary functions we refer to Devlin [ 2 1 where a proof of the following lemma can be found.
= Ici,xl,...,xm>
i:th tn formula 4 M is m-dry and E I= ~(xl,...,xm)l is uniformly 1 - for transitive n rudimentarily closed E . Lemma 1.1.
I=>
:
The
N
An immediate consequence is the existence of a universal t l ( M ) formula:
$(e,x) v c(e)o,x,(e)l>E
El
k M
.
N
The recursion theoretic notions are defined as follows:
For
393
WEAKLY INADMISSIBLE RECURSION THEORY
A 5 M
we say
: by
over
a
is
.XI
formula with parameters from M .
A
M-recursive if
N
M-A
Finally we index the
N
where
and
there i s an
!-r.e.
! admits
!-recursive
E-r.e.
A
is definable
A
If
is
A E
M
then
A
if its graph is
sets by putting W e = k:@(e,x)}
an acceptable prewellordering if
prewellordering
EM: y < x }
(i)
L~ =
(ii)
121
= ordinal of
L6n
181 =
ELY)$(e,x,z)l
is
q(e) = v o g y [ r ( e ) = y
and
be a u n i v e r s a l
r ( e ) = least
Let
0
q ( e ) + d We+
I i s z1 (_M).
There i s an
(Selection operator).
Let
N
sible
f o r m u l a where
El(!)
v = w n tx:(3z E w)$(e,x,z)l
&
such t h a t
q
Proof:
is
be a u n i v e r s a l
Then
,-
function
STOLTENBE RG-HPNSEN
Thus
g(x)
.
For
xE F
i s defined.
f(f3,;)
.
let
( D e f i n i t i o n by r e c u r s i o n ) . Then
I
,
k"x
i s bounded by
0
Suppose
= g(b,;,{:y
g:K
51)
x
Mn"
is
+M
9 5
WEAKLY INADMISSIBLE RECURSION THEORY
The proof is the usual one (see e.g. Barwise Ill),the point being that we have
Il-replacement below
K
.
The reducibility notions we consider are the wellknown ones from the admissible case.
B,
A
sM
Thus for sets
B, if there is an
A and B, A is
M-r.e. set
W
M-recursive in
such that
N
0
a c A & bnA =
gc,d(EW
w
&
0).
c c_ B & d n B =
A partial function f is weakly M-recursive in B, f sw; B , if -+ f(z) = y w 3c,d(<x,y,c,d>EW & c 5 B & d n B = 0) for some :-r.e. set
W , and
B
A
if
cA
swM
B
where
cA
is the characteristic
N
function of
A.
!-degrees
are defined in the usual way using
'M. N
As in the admissible case, we need consider two notions of projecta.
141*
Let
= least
M-r.e. set
verified that
151'
A c_ L6
151*.
s
acceptable prewellordering If
M = Sg
then
2
= 151
.
! is
such that
We call
4
2
A
M
6!
.
lzl*
=
:-recur-
for which
6
It is easily
adequate if
for which
: admits Ln
151' = limit ordinal.
is adequate.
It follows from lemma 1.5 K
151' = least
> M , and let
sive function q : L~@ there is an
for which there is a partial
6
that
fl
is admissible if and only if
said to be weakly inadmissible if
and strongly inadmissible if
K
e a ) ]
&
a 5 B
B,
Thus
a n B = @-3u,bEN[r(a)
f o r some
403
a E M .
F i n a l l y suppose such t h a t
K
(ni'a
x
h-'[n;'aIn
.
B,
Then
U { < T , h ( X ) > : XE B:ll,
u
h"B;)
fl
Using t h e r e g u l a r i t y of
1 n B:
h-'[T;'a
1,
T : x n A * @ l U { < l , x > : x t l (M-A)
set
B,
5
K
such t h a t
clearly t.r.e. via
W,
.
We show
Viewing
a 5 A o
p
e
A,
p
-1
o
-1
[Al
I =or B,
A *M B 4. .
(x)
.
as i n
p - l ( < i ,a>)
$01
.
Let Choose r e g u l a r a - r . e .
B = h"Bl
and l e t
First
suppose
.
p-'[Al
B
I
zocB1
t h e p r e v i o u s c a s e w e have
n p " [ ~1 =~ 0
0
3b E N(*p-'( ) , b > E W,
&
b f l B,
0
3 b E N ( < p - l ( < l ,a>) , b > E W,
&
h"b
nB
=
0) = 0) .
is
404
VIGGO STOLTENBERG-HANSEN
The analogous reduction holds for negative neighbourhood conditions of
B,
E
W,
&
o 3b E N[i<w t o sequences <xi>i<w
-
where x i = h ( s i )
.
"he aim i s t o b u i l d up a "good" r e c u r s i o n
t h e o r e t i c l i m i t s p a c e r e l a t i v e t o t h i s map, wherein t h e "index" o f
e a c h r e a l g encodes a c a n o n i c a l ( p r i m i t i v e ) - r e c u r s i v e - i n - h sequence determining g .
Our e a r l i e r c o n s i d e r a t i o n s , and t h o s e i n
[4] ,
s u g g e s t t h a t i f g = l i m gi i s i n t h e s p a c e , t h e n so should be t h e s e t D g = [il 2j> i ( h ( g j ) f h ( g i ) ) {
.
Thus L ( h ) w i l l be g e n e r a t e d from
h by c l o s i n g o f f under r e l a t i v e r e c u r s i o n and a p p l i c a t i o n s o f t h i s scheme. (Note however t h a t t h e a r b i t r a r i n e s s o f h means t h a t t h e need n o t c o n v e r g e , s o t h e s e t s D may someg There a r e now two o p t i o n s open t o us i n d e f i n i n g L ( h ) :
sequences i<w times = N ) .
e i t h e r we w r i t e down Kleene- t y p e schemes f o r p a r t i a l f'unctions and t h e n e x t r a c t t h e t o t a l o b j e c t s , o r we g e n e r a t e d i r e c t l y a s e t o f c o d e s f o r t h e t o t a l f u n c t i o n s we a r e i n t e r e s t e d i n .
We w i l l t a k e
t h e s e c o n d , m o r e h i e r a r c h i c a l , approach a s i t seems r a t h e r more c o n v e n i e n t for our p u r p o s e s .
In o r d e r t o g e n e r a t e t h e r e q u i r e d cod-
e s w e f i r s t need soine t e c h n i c a l machinery for c o n s t r u c t i n g canonic a l sequences. Definition.
Let [xIh, x<w
, be
a s t a n d a r d enumeration o f a l l f u n c t -
t i o n s p r i m i t i v e r e c u r s i v e i n h . Then a l i m i t i n d e x f o r g i s a numbh e r e such t h a t f o r each 1 , [ e l ( I ) i s a f i n i t e sequence g i , a n d g = l i m gi
.
Lemma i (Canonical S e q u e n c e s ) .
( I ) There i s a p r i m i t i v e r e c u r s i v e
4 10
STANLEY WAINER
d such t h a t i f z i s a l i m i t i n d e x f o r f and i e l f i s t o t a l t h e n
d ( e , z ) i s a l i m i t i n d e x for- ( t h e c h a r a c t e r i s t i c m c t i o n o f ) t h e s e t f f D = In1 3 m > n ( h ( [ e l m m ) f h ( l e l n n ) ) l
.
(11) There i s a p r i m i t i v e r e c u r s i v e f u n c t i o n 1 such t h a t i f z i s a f
l i m i t i n d e x f o r f and f o r each x , e x = [ e l ( x ) i s a l i m i t i n d e x f o r
( t h e c h a r a c t e r i s t i c f'unction o f ) a s e t E x , t h e n l ( e , z ) i s a l i m i t index f o r t h e s e t E = ) < n , e
Proof. -
I
n E Ex]
.
( I ) L e t si b e t h e f i n i t e sequence s u c h t h a t f o r n < 1, f f s,(n) = 1 i f 3m(n< m < i'&h ( ) e k m ) # h ( ) e I n n ) ) and 0 o t h e r w i s e . S i n c e h h f m = [ z ] (m) f o r each m , Xi.Si = [ d ] for some d p r i m i t i v e r e c u r s i v e l y
computable from e and z .
C l e a r l y D = l i m si
.
( i i ) L e t si b e t h e f i n i t e s e m e n c e such t h a t f o r < n , x > < i , si((n,x>) i s computed as f o l l o w s : F i r s t f i n d e x , i = i e ] I i ( x ) .
Then f i n d t h e
g r e a t e s t j < i ( i f t h e r e i s one) such t h a t [ e
I h ( j ) can be computed x, i h 1 ( j ) i s a sequence, w i t h i n i s t e p s . If t h e r e i s such a j and [ e x, i g i v e o u t i t s n-th component a s t h e v a l u e of s i ( < n , o ) , Otherwise h p u t s i ( < n , g ) = 0 . C l e a r l y h i . s i = [I] f o r some 1 p r i m i t i v e rec u r s i v e l y computable from e and z , and i t ' s e a s i l y checked t h a t E = l i m s1 '
We w i l l show in Lemma 2 t h a t t h e s e q u e n c e s b u i l t up b y means of Lemma 1 have t h e c r u c i a l p r o p e r t y t h a t i f < s i > i s any such sequence and
El-)
g t h e n a "modulus" for <si>
c a n b e computed from g .
But
f i r s t , the d e f i n i t i o n of L(h) :
D e finition.
The system o f codes Ch and s e t s D:
f o r a € Ch, a r e defin-
ed i n d u c t i v e l y a c c o r d i n g t o t h e f o l l o w i n g t h r c e c l a u s e s (we hence-
f o r t h omit t h e s u p e r s c r i p t h )
.
For each code a , ( a ) , w i l l b e a
l i m i t i n d e x f o r D a , and we s h a l l d e n o t e t h e correponding sequence h < [ ( a ) , 1 ( i ) > i < w of a p p r o x i m a t i o n s t o Da by i<w. ( 1 ) s C and D D
and
9,=i n ( 3 m > n ( h ( [ e ] m a 9 m ) f h ( [ e ] n a y n ) ).t
( 3 ) If a s C and @ = [ e l D a i s a t o t a l f u n c t i o n such t h a t @ ( O ) = aand $ ( x ) E C for every x , then ~ = < 2 , l ( e ~ , ( a ) e~, a) >, E C , where e l i s D such t h a t [ e l l Da ( x ) = ( ! e l a ( x ) ) l , and D c = [ < n , D l n € D $(XI] * F i n a l l y we bet L ( h ) = [ f l f < Da f o r some a € Cl
.
"he following i s an e x t e n s i o n o f S h o e n f i e l d ' s Modulus Lemma t o L ( h ) Lemma -2.
.
( i ) There i s a r e c u r s i v e f u n c t i o n M such t h a t i f a € C then f o r every haa x and every i > lM(a)] ( x ) , D,,,(X)=Da(X). Da ( i i ) There i s a r e c u r s i v e f u n c t i o n N such t h a t if a € C and i e ] h,Da i s t o t a l then [ N ( e , a ) ] i s t o t a l and for each n and every h,Da D D ( n ) , [ e l i a Y i ( x ) = i e ] a ( x ) f o r a l l x i n , D [ e ] i a 9 i ( x ) = [ e l a ( x ) for each x < n . This i s done a s f o l l o w s : f i r s t (iib
f i n d the l e a s t yn such t h a t w x < n l y < y , T ( % ( y ) , e , x ) . "hen compute h,Da zn= max ihl(a)] ( y ) s o t h a t f o r every 13 z n , D,,,(y)=Da(y) f o r Y< Yn I t then s u f f i c e s t o p u t i n = max(yn,Zn). C l e a r l y in i s each y < yn
.
r e c u r s i v e i n h,Da w i t h index N(e,a) p r i m i t i v e r e c u r s i v e l y computable from e and M(a). Although the spaces L ( h ) may be of some independent i n t e r e s t we a r e henceforth concerned only with t h e case where h = h F f o r some non-normal
type- 2 F .
(Thus each
9, d e f i n e d
by c l a u s e ( 2 ) w i l l be
finite). The C h a r a c t e r i s a t i o n o f Non-Normal I - s e c t i o n s . Theorem 1. For any non-normal type
-2
object F
,
I-SC(F) = L ( h F ) . This i s proved by means of Lemmas 3 and
need a convenient d e f i n i t i o n of I - s c ( F ) .
4 below,
b u t f i r s t we
We w i l l assume from now
C l e a r l y no
on t h a t F i s such t h a t , f o r e v e r y r e a l g , g ( 0 ) = ( F ( g ) ) o .
g e n e r a l i t y i s l o s t by t h i s , s i n c e we can always deform F t o F' = h g . and choose q ( a ) t o be an index of the zero f u n c t i o n . b If a = 2 E @ then by the i n d u c t i o n h y p o t h e s i s , f b i s r e c u r s i v e in D P(;)
w i t h index q ( b ) and s o for each x , f a ( x ) = F ( [ x I f b ) ::
F ( [ e x ] p ( b ) ) where e = t r ( x , q ( b ) ) f o r some p r i m i t i v e r e c u r s i v e funcX
tion tr.
Let $ ( O ) = < O , k > a n d $ ( x + l ) = < I , d ( e ~ p ( b ) ) , )e, x , p ( b ) > .
Then q5 = [ z l h for some z p r i m i t i v e r e c u r s i v e l y computable from p ( b ) and q ( b ) , and every $ ( x ) i s a c o d e c C . Now s e t p ( a ) = < 2 , 1 ( z l , k ) , z, .
.
It remains t o show t h a t fa i s
r e c u r s i v e i n Dp ( a ) = i < n , x > l n ~D$(x)] But for each x , f ( x ) = D D l i m $ ( [ e x ] i P ( b ) 9=i h ) F ( l e x ] m P ( b ) * m )where m i s the l e a s t element of
D$ ( x + l ) = [ n l Ql,x+lMD i<w
p(a)
]
.
Therefore, s i n c e the sequence
i s primitive recursive i n
$
and hF = D$ ( o ) ' fa
w i t h index q ( a ) p r i m i t i v e r e c u r s i v e l y computi s recursive i n D p(a) a b l e from p ( b ) and q ( b ) , Note t h a t $ i s r e c u r s i v e and each D 4 ( x ) is (uniformly) r . e . i n h F , and s o D i s r . e . i n hF. P(a) f F i n a l l y , i f a = 3 C 5 e ~ @then f a ( x ) = fq(. (x, ) where @ = [ e and by the induction h y p o t h e s i s , each f with index q ( * ( y ) ) .
y,
0
*(Y)
i s recursive i n D P(*(Y)
Thus from c , e , and an Sndex o f
D ,
1
we can f i r s t
p r i m i t i v e r e c u r s i v e l y compute p ( a ) s o t h a t Dp ( a ) = i ( d e f i n i n g
$1
and c = < 2 , 1 ( e l , ( a ) l ) , e , a >
(effective joins) are straightforward. Suppose c = < 1 , d ( e , ( a ) , )
, e,a>E
C.
I n d u c t i v e l y we can assume
D a = i r ( a ) j F and s o the f u n c t i o n g = ) e j D a i s r e c u r s i v e i n F with i n -
.
For each i dex p r i m i t i v e r e c u r s i v e l y computable from e and r ( a ) D s o Xi.gi i s p r i m i t i v e r e l e t g be the f i n i t e sequence l e ] i i 'Be must show t h a t the s e t Dc = c u r s i v e i n hF and gi-t g
.
[ n 1 3 m > n ( h F ( g , ) f h F ( g n ) ) l i s r e c u r s i v e i n F , uniformly i n c (Dc i s For each n compute t h e f u n c t i o n g:
o f course f i n i t e ) .
from h F , Da
a s follows : given x , f i r s t compute i x =i N ( e , a ) l h 9 D a ( x )by Lemma&i). Then see i f t h e r e i s an m such t h a t n < m < ix and hF(gm) # hF(gn)
.
If t h e r e i s one l e t mo be the l e a s t such and p u t g*(x) = g ( x ) , If n m0 Notice t h a t if x i s t h e r e i s no such m by s t a g e x s e t l c c O& x d D c ] ~ ~ < C , X , I > I C E C ~ : X E D ~ ~ . Corollary.
By Sacks' Density Theorem ( s u i t a b l y r e l a t i v i z e d ) every
t o p l e s s I - s e c t i o n o f a type-2 o b j e c t c o n t a i n s dense c h a i n s of
4 16
STANLEY WAINER
degrees.
Thus no t o p l e s s well-founded
i n i t i a l segment o f degrees
can form a I - s e c t i o n . Remark.
I n [ 6 ] Normann c o n s t r u c t e d a continuous G with a non-
c o l l a p s i n g h i e r a r c h y , by i t e r a t i n g B e r g s t r a ' s [ 1 ] "small jump" b along a r e c u r s i v e ordering of r . e . o p e r a t o r Fa s e t s whose maximal 1 I Theorem 3 s u g g e s t s well-founded i n i t i a l segment i s 111 b u t n o t A l
.
t h a t for each non-normal F i t might be p o s s i b l e t o c o n s t r u c t a continuous G F , along s i m i l a r l i n e s t o Normann's example, such t h a t I-sc(F) = 1-sc(GF). Weak A s s o c i a t e s
.
J u s t a s an a s s o c i a t e aF e x t e n s i o n a l l y encodes the behaviour o f a continuous f u n c t i o n a l F on a l l r e a l s
,so
a "weak a s s o c i a t e " aF
w i l l i n t e n s i o n a l l y d e s c r i b e the continuous behaviour of a non-normal f u n c t i o n a l F on the r e a l s { e l F , b u t only with r e s p e c t t o c e r t a i n F canonical sequences approximating { e l
.
CE
Given [ e l F E I - s c ( F ) we c a n , b y Lemma 3 , compute from e a code F DC and an index e l such t h a t { e l = { e l ] .From e l and c we can
ChF
then compute a l i m i t index j ( e ) such t h a t f o r each i , [ j ( e ) ]hF( i ) = D hF F { e l l i C p i and hence l i m [ j ( e ) l ( i ) = {el The f u n c t i o n j i s primhF the canonical 8equi t i v e recursivc. and we c a l l < [ j ( e ) ] ( i ) > i < w
.
ence f o r -
[ e l F . A "modulus" f o r the sequence < $ ( [ j ( e ) l hF ( i ) ) > i < u
F approximiting F o e ! )
(1
,d(el (c), ),el
,C>.
then any number n i n F But Dc, = ) r ( c ' ) ] by Lemma i s
q , where 4,and
c' =
s o we have:
Lemma 2. There is a f u n c t i o n mF p a r t i a l r e c u r s i v e i n F such t h a t whenever { e l F i s t o t a l then Xn.mF(e,n) i s t h e c h a r a c t e r i s t i c f u n c t i o n of ! n [ V i b n ( % ( [ j ( e ) ] hF ( i ) ) = P ( { e l F ) ) j . Definition.
A
weak a s s o c i a t e f o r a non-normal type-2 f u n c t i o n a l F
i s a p a r t i a l f u n c t i o n a such t h a t whenever [ e l
F
i s t o t a l then
( i ) Xn.a(e,n) i s t o t a l and I n ( a ( e , n ) > 0 ) , and hF F ( i i ) if a ( e , n ) > 0 t h e n \ d i > n ( $ ( [ j ( e ) l (i)) = F ( [ e ] ) = a ( e , n ) - I ) ,
THE 1-SECTION OF A NON-NORMAL TYPE
The f i n a l r e s u l t r e - c a s t s
- 2 OBJECT
417
Theorem 1 i n terms o f weak a s s o c i a -
t e s , and s u g g e s t s t h a t from a p u r e l y r e c u r s i o n - t h e o r e t i c p o i n t o f v i e w , t h e i n t e n s i o n a l n o t i o n o f " a s s o c i a t e " i s p e r h a p s t h e more a a p r o p r i a t e one. Theorem
4.
There i s a p a r t i a l r e c u r s i v e f u n c t i o n a l @ such t h a t f o r
e a c h non-normal
type-2 o b j e c t F , a F = X e , n . @ ( F , e , n ) i s a weak a s s o c -
i a t e f o r F and l - s c ( F ) = l - s c ( a F ) .
Proof.
Cur work throughout h a s been c o m p l e t e l y uniform i n F .
In
p a r t i c u l a r t h e r e i s a f i x e d i n d e x u such t h a t f o r e v e r y non-normal F , I u l F i s t h e mF o f Lemma 5 . Thus we need o n l y d e f i n e @ so t h a t @ ( F , e , n ) = 1 + hF([ j ( e ) ] hF ( n ) ) i f mF(e.n) C I , O i f mF(e.n) ~ 0 .Then e v e r y t o t a l f u n c t i o n r e c u r s i v e i n aF w i l l be r e c u r s i v e i n F and F F c o n v e r s e l y , s i n c e F ( i e ] ) = a F ( e , p n ( a F ( e , n ) > 0 ) ) -1 f o r t o t a l lei
,
a n a p p l i c a t i o n of t h e R e c u r s i o n Theorem w i l l y i e l d for each e a n a F F such t h a t if [ e l i s t o t a l t h e n [ e l = [ e ' ] F *
e'
References. J.A.
B e r g s t r a 1976, C o m p u t a b i l i t y and c o n t i n u i t y i n f i n i t e
types, Dissertation, Utrecht R.O.
.
Gandy and J.M.E. Hyland 1977, Computable and r e c u r s i v e l y
c o u n t a b l e f u n c t i o n s o f h i g h e r t y p e , in: Logic Colloquium 76, North-Holland,
.4msterdam, p p . 407-1138.
T. J. G r i l l i o t 1971, On e f f e c t i v e l y d i s c o n t i n o u s type-2 o b j e c t s J.S.L.
36, 245-248.
J.M.E.
Hyland 1977, F i l t e r s p a c e s and c o n t i n u o u s f b n c t i o n a l s ,
t o appear. S.C. Kleene 1 9 5 9 , Countable f u n c t i o n a l s , i n C o n s t r u c t i v i t y i n Mathematics, North-Holland,
Amsterdam, 81-100.
D. Normann 1 9 7 6 , A c o n t i n o u s type-2 LYmctional w i t h a nonc o l l a p s i n g h i e r a r c h y , J.S.L. S.S.
t o appear.
Wainer 1 9 7 4 , A h i e r a r c h y f o r t h e I - s e c t i o n of any t y p e
two o b j e c t , J. S.L.
3 9 , 88-94.
