AND
T H E F O U N D A T I O N S OF MATHEMATICS
Editors
A. HEYTING, Amsterdam A. MOSTOWSKI, Warszawa A. ROBINSON, New ...
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AND
T H E F O U N D A T I O N S OF MATHEMATICS
Editors
A. HEYTING, Amsterdam A. MOSTOWSKI, Warszawa A. ROBINSON, New Haven P. SUPPES, Stanford
Advisory Editorial Board
Y. BAR-HILLEL, Jerusalem K. L. D E B 0 U V 8 R E, Sanfa Clara H. HERMES, Freiburg i/Br. J. HINTIKKA, Helsinki J. C. SHEPHERDSON, Brisfol E. P. SPECKER, Zurich
N O R T H - H O L L A N D PUBLISHING C O M P A N Y AMSTERDAM
0
LONDON
MATHEMATICAL LOGIC AND FOUNDATIONS OF SET THEORY PROCEEDINGS OF A N INTERNATIONAL COLLOQUIUM HELD UNDER THE AUSPICES OF THE ISRAEL ACADEMY OF SCIENCES AND HUMANITIES JERUSALEM, 11-14 NOVEMBER 1968
Edited by
Y E 0S
UA B A R -
LLEL
Professor of Logic and Philosophy of Science The Hebrew University of Jerusalem, Israel
1970
NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM 0 LONDON
@ NORTH-HOLLAND PUBLISHING COMPANY - 1970 All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the Copyright owner.
Library of Congress Catalog Card Number 73-97195 I S B N 7204 2255 8
PUBLISHERS:
NORTH-HOLLAND PUBLISHING COMPANY -AMSTERDAM NORTH-HOLLAND PUBLISHING COMPANY, LTD -LONDON
PRINTED I N ISRAEL
PREFACE
on on
1968.
by
by
on
Y.
11,
on
on
no
on
no
WEAKLY DEFINABLE RELATIONS AND SPECIAL AUTOMATA* BY
0.
Nz= (Tyro,rl) $1).
L
a,p, y, A , B , C , ..., E P(T)” M2
...,
M2
.,
x, y , z, ..
definable F ( A , , ...,An)
F ( A , , ...,An)
weakly-definable by
[6]
on
Nz. special automaton on on H
H c P(T)” P(T)” -
c P(T)”
n.
by by P(T)” F ( A , , ...,
by 1,
G ( A , , ...,
MZ) Fz(A
..., F , ( A , , ...) A , )
,,..., 69 1
0192,
Office
MICHAEL 0.
2
[4]),
by [3] 24 1. Notations and basic standard definitions
0
= pl,
n n=
1 = {0} , 2 = (0,
...,n - l }
x: n +
by xi.
(xo, ...,
xy
l(xy) = l(x) xo
.
n-termed sequence on
x
x ( i ) , 0 5 i < n, l(x) = n
x
length
+ l(y).
.
x
y = (yo, ...,ym- 1) ...,x , - ~ ,yo, ...,y,- l).
x = (xo, ...,x,-
l).
(xo)
x = xoxl
... x,-
by A . i < o,
0
1.
projection pi
pi(x) = xi x = injinite binary tree xE immediate successors
i
by
< n. on (0,
=
xE
root of
A
x.
by A.
11.
10
.
3z[y =
x
subtree
xE
0
up
00
0
1.
1
on
5y
by x x # y roof x
6 y (x
initial x
9
u 2 ( 4 2 ( i ) ) ) E M(u(i), a>9
i < Ku) * up
'u
x
T.
O , l , ...,k - 1 ~ ( x )=
d,
M"((u,
(u, 4) E S", x
u(k) = so 4),a) = { ( d , d ) }
.
n
l(u) = k , u(O),u(l),. . . , u ( k - 1 ) . xE
= z,
23
23
d,
'u
v ( x ) = z,
'u d
so
x0
i, i < k , xl
'u
wl(i)
(w1(i),w2(i))~M(u(i),o). i l < i2 < ... < i,,-l wl(0), ..., wl(k- l ) , ij = wl(i) (u,,+~) 8 x0, I(ul) = p Z(ul) = p 1 ul(p) = ul(j)= wl(ij), < p . 4,:k+p ul(i) = wl(ij). ( U ~ , ~ ~ ) E xSl .
+
= wl(ij)).
41(i)=
WEAKLY DEFINABLE RELATIONS A N D SPECIAL AUTOMATA
9
23.
= {(A,@),(so,@)}
10. t = ( ~ , T ) V,, E rERn(23,t) r(A) E Sg r(x) # d , x E T. r(x) = (u , 4,), x E T . l(u,) = k , m . h : T + n + 1, h(A) = on (r,T)ER,nR, ncT by 11, n$ z) X E ~ m x , c ( { y [ x < y ~ n u, ' ( m ) ~F } ) < u. R =R,nRbnT(6). Proof.
R,
+
I
Proof of Theorem 9.
12, t = (0, T ) E D (r,T)ER.
WEAKLY DEFINABLE RELATIONS AND SPECIAL AUTOMATA -
Z = S” x Z.
(a, T )
A
B 3.
13
=
11
pofi E Rn(23,(plfi, T ) ) ,
I
T ) (pea T )E R } D = pl(A n B ) .
by
5. Automata on finite trees by EcT
%
a
P(S x S ) ,
=
so E S .
An r(Ft(E)) = {f}. [l, 5 , 71
A finite (u, E ) u : E - Ft(E) + X. on ( S , M.s,,f) M:(S- {f}) x Z + on % = ( S , M,so,f ) accepts e = (u, E ) r on e r ( A ) = so e by % by TI(%). on
14. Let % = ( S , M , s o , f ) be a Z-automaton. DeJne a set B c V, b y : (u, T )E B if and only if there exists a sequence (E,),,, of and ( ~ , G , ) ET I ( % ) , n < o , where frontiers of T such that En < G, = { x l x 5 y f o r some ~ E E , } .T h e set B is s.f.a. dzfinable.
F)
fl = S x S ; by ((sl,si),(s2,s;))E (s1,s2)€ M ( s , a ) , (s;,s;)E M(s’,o) s‘ # f , s’ = f. T(23) = B . t = ( u , T )E T(23) i :T + S (E,),,, T i(E,) G F ,E, < En+ , n <w . G, = { x I x S y f o r some y E E n } , n < o. r’: G,+ + S r’(x) = po(i(x)) x E G,, r’(x) = pl(?(x)) x E G,+l- G,. r‘ on (u,G,+J IT,,+^) = { f } . (u, G,+ 1) E TI(%), n < o. (u, T )E B . (u,T)EB. (E,),,, (u,G,),,, r,: G, + S , n < o , on (u, G,) r,,(E,) = {f } . T, = { x Z(x) = n}. (n(i))i. S, S = (0, m < o. r = (B,, ..., B,)EP(T)" r:T-+S r(x) = ( x ~ , ( x ...,zBm( ), x)), X E T . 1-1 P(T)" ST. r(x) = s , ES ,x T, r(x)EF G S , r ~ R n ( ' % , z ( A , ..., , A,)) A r(A) = so , r = (Bly..., B,,), N
WEAKLY DEFINABLE RELATIONS AND SPECIAL AUTOMATA
VTc[In(r
1
17
$1
by frontier of T, and V y [ y E a + r ( y ) E F ] ] G(B,, ..., B,)
S2S z ( A , , ...,A , ) € T ( % ) , S2S by
3a. (Al7...,A,,)~R, 3B1 ...3 B , [ H ( A l , ...,A , , B , , ...,B,)
A G(BI, . . . , B , ) ] .
7. Applications to decidability and definability [ 6 ] by
on [6], [6].
23. There exists a n efectiue procedure of deciding f o r a special &automaton % = ( S , M , s , , F ) whether T(%) # 0. If c(S) = n , then this procedure requires n4 computational steps.
M'(s,cr)
Proof. T ( ( S , M ' , so, F ) ) # 0 .
E c T. R(H) r:E
=
UmELM(s,o)),
{a}. T(%) # M'(s,a) = M ( s ) .
%
H ES, E # {A}
SES
-,S
0
r(A) = s R(H)
by
r(Ft(E)) E H . H , = 0,
i <w,
Hi+1 i <w, Hi= H i + 1 Hi= H i + k = R ( H ) , k <w. Hi E S, H , = H,+ = R ( H ) . Hi, Hi+1 n2 c(Hi) 6 n . H, = R(H), R(H) n3 F, = F , i w , Fi+,= Fi n R(Fi). F i + lE F i i w . Also, F i = Fi+, Fi = Fi+k k < w . F, = F , , , ; F, = G . G c(F) 6 n R(H), n4 T(%) # 0 so E R ( G ) . T ET(%) so E R ( G ) by 27 H,
E
,
-=
-=
MICHAEL 0.RABIN
18
s,ER(G) r: T +S, T ET(%). to = ( r o , E o ) ro E E , ) , ro(A) = so, ro(Ft(Eo)) sEG t , = (r3,Es) r , e Rn(%,E,), r,(A) = s, r,(Ft(E,)) G. to T A; ( r o ,T ) . x EFt(Eo), (ro, T ) x; (rl,T). El c T rl r,(Ft(E,)) E G. xeFt(EI), tr,(*) ( r l ,T ) x; T). (r,, T ) , i < w Ei = D(ri), i <w. E, # {A} sE G, Ft(Ei) < F Z ( E ~ + ~i ))* :
by
f :X + 9
p
g : R x X + 'I)
= X
g(P,x).
a
X
o
:X + 'I) f:X
p.
type(X) = 0
X
X
+
9
continuous
o ,type(X) = 1 type(X) = type('I)),
R . One n:X+'I)
on X x X
X
type('I)) = 0 , then a partial function f : X+ 9'J is continuous if and only if Domain(f) is open and f is continuous on its domain. If type@) = 1 , then a partial f:X + 'I) is continuous if and only if Domain(f) is a Gd set (a countable intersection of open sets) and f is continuous on its domain.
on w
w
(the analog of enumeration). For each X, 9 there i s a recursive partial function @: R x X + 9 such that each continuous partial f : X - 9 i s given b y
27
DETERMINACY AND PREWELLORDERINGS OF THE CONTINUUM
f(x)
=
W , X )
To
f o r some
@(E,X)
=
{&}(X)
N
{&}xqX),
ITERATIONTHEOREM. For each X there is a recursive function S x : such that f o r all 9,3 {El
(x, Y )
=
{E}XX993(X,y)
{SX(E,
=
+
(Y) 9
{SX(E, x)}9*3(y).
RECURSIONTHEOREM. For each X there is a recursive function F X : such that if E* = FX(&),then f o r every 3 { E } (E*,
x)
2
A,
(XI.
X,
A
A (1-1)
[X]A = { A c X: f o r some B E A and some recursive
f:X [X]A “ A A”
“ A EA”
A =f-’[B]}.
+
by A
E
X
X
X
=
A
on
&A = { A n B : A , B E A } (conjunction, &)
VA
=
i A
=
{ A U B : A , B E A } (disjunction, V) 1
- A : A E A } (negation
dual, 1)
XOA = { A :f o r some B E A , B s o x a E A o (3n)[(n, a ) B ] } (existential number quantification, 3 m )
-
X’A = { A : f o r some B E A , B 5 x a EA (3p) [(B, a ) E B ] } (existential real quantification, 3a) IIOA =
II’A
=
1 x 0i
A (universal number quantification, V m )
TA (universal real quanrwcation, V a ) .
28
YIANNIS N. MOSCHOVAKIS
EA = { A c R : A is open}
=
Xi+'
=
A;
=
172; ,
c'n: , f: R
A A =
--f
o
light-face (0) by recursive
R +w,
R : f o r some recursive partial
Zk = ( A
A e f ( u ) zO]
C' , 1, n
72;
f:R
B k > 0,
no.
+R
&, v ,
af-'[B]
C'
A second order number theory =,
analysis.
', +, * .
0
a,P, ... ,
n, m , ...,
a ( n ) , p(a(n)) 3a,Va.
on w
by on R no bound
by
k
=- 0 ,
A
R
3alVaZ
0(P,a,al,az,a3)
Ei 3a30(p0,
by
is a,
no
aZ, a 3 )
Po go
third order number theory F, G,H ,
...
by
Rw,
29
F(a), P(G(a)) 3 F , VF A A
.
Z:
R
by
a, F )
3F O(B, a, F ) do
ll:
As
=
Po
no bound
Po, Z:,A:
by
(k > 1)
72;
3 F VG O(Qo, a, F, 92.
X:.) Et,
ll:.
= Z:n
i
Structure properties. R
. X. A, R
A (1-1).
R xX
2.1. Universal sets. A
=
G,
[X]r
G
{G,: ~ E R ) ,
G,
a-section
G, [X]r GI.
(R-)unioersal
= (x:(~,x)EG}.
G
(R-)parametrized
c R x R.
by
X,
h(x) = A t ( Z ( t ) ) ;
h:X
--f
R +X E X,
R
h-' on =
A
E
~
=
=
f), H = ((a,x ) : (a, hx) E G }
(2-1)
[X]r.
ir,
nor, Z,,!, ll:
MOSCHOVAKIS
YIANNIS
30
CA
by
G = {(&,a): { & } ( a ) =
by
A
6ir. EL- II; #
k,
hierarchy theorem,
2.2 Separation and reduction. separation property A,BEr AnB= C E r n i r A E C, B nC = reduction property A, BE E A,B, B,A,nB, Reduction(T) 3 Separation (ir)A Reduction(T) 3 not Separation i
CA, II: on C
Cl II
k
>= 3 1959b
k 2 3.
Reduction (Ci)
A ER x R
2.3 Determinacy. I1 a(O),
...
p(O), a ( l ) ,
(a, p)
(a, p) $ A , I1
A,
A
on 11
strategy
I
p.
a * [,4] = a where a(n) = 11 [a]
A 11
r~
*T
winning
a , (a, [ a ]
=
p where p(n) = r@(n +
* z) $ A .
p, A
(a*
p) E A ; z
determined
I 1953,
I1
DETERMINACY AND PREWELLORDERINGS OF THE CONTINUUM
31
by 1964
R x R is determined.
Axiom of Determinacy, A D . Every subset AD
Countable axiom
choice f o r
sets of reals, Vn3a(n,cr) E A + 3aVn(n, (a),)
( A E o x R)
EA
AD R
by A D AD
for R
AD-
by L [ R ] = the class of sets constructible f r o m R AD if there exist measurable cardinals, then every II: set is determined. 2.4
A;
1968 dejinable f r o m a real is determined.
every subset of R x R ordinal no
by A Determinacy (A): every A E A is determined, A E A for A E R x R of Determinacy(A)
A E [R x RIA. A
YIANNIS N. MOSCHOVAKTS
32
2.4 Prewellorderings. A prewellordering of a set A (with field A )
x
4
4
5Y *
5 4(Y),
5
on
4 :A + 5
5,
5 on A x, y A ,
on
5
5
5
length
5).
canonical surjection
5
A
-
5, lack of infinite descending chains, Vn[xn+, 5 xnl
=>
3n[xn 6 xn+1] s
{x~}.”=~
5
A.
dependent choices -
X,
A EX
A ER x R,
Dependent Choices, DC. V U ~ P P) ( ~ ,A
=>
3aVn((U)n, (a), + 1) E A
AD,
AD. DC L[R])
DC,
R
.
AD
DC. by G .
Prewellordering - and 5 in and 5
There is a prewellordering 5 on G and relations
i r respectively, such that W E G => V z { z 5 w z 2 w e [ z ~ G & z5 w ] } .
(2-2) G’
yo. (El,
PI> 5 ‘ (UZY P 2 ) * 5 ’, 3 ’ .
5’ on
on
(UlY P I ) )
5 ( Y o J (U2Y P 2 ) )
Prewellordering
Y
DETERMINACY A N D PREWELLORDERLNGS OF THE CONTINUUM
33
&,
Prewellordering
e- Reduction
Prewellordering
, Prewellordering (ZJ),Prewellordering (II:) Prewellordering (ZA) by
in 1964 Prewellordering (Xi)
II:
Z;.
[o]Ai,
by Suzuki 1967
if i s parametrized, closed under recursive substitutions, &, v, Vcc and Prewellordering then Prewellordering (C'T)
.
1967
of
1969.
II:
-
do
if i s parametrized, closed under recursive substitution, &, v, 3m,Vm and Vcc and some relation in wellorders R with order type K,, then Prewellordering (C'T).
k 2 2 , Prewellordering (Xi). 1968
If
1968.
i s parametrized, closed under continuous substitution, &,
V,3m, Vrn and 3a, if
Prewellordering then Prewellordering DC Prewellordering (Xi)
(II;) X
odd k .
II
k
D C a n d Determinacy (rni r ) ,
k
Prewelfordering on
34
(II:))
(IV), on
X: (111)
L[R],
Prewellordering ( X I ) , do Prewellordering ( X i )
53. The basic lemmas. A &
R,
A
.
[
i
S
realized
R
[.
A
A [ R x RIA)
A
5 is realized in A & < [ is realized in A a [ [>0
&c
*5
+ 1 is
is realized in A ,
realized in A ,
is realized in A 5 5 is the length of some prewellordering R in A .
X.
on
0th) = supremum {[: 5 is realized
o(A),
A}.
A is
=
in this theory.
All results stated thus f a r in this paper a r e provable AD DC, o(A)
= supremum =
{[: there exists a surjection ~ : R - B [ )
supremum {[: there exists a n injection
by
+ ‘2);
35
DETERMINACY AND PREWELLORDERINGS OF THE CONTINUUM
X;
A (3-1)
... 3a,8,
3a,3a2
8
no
Et
X , by X. Ct
o.
on R
j(
by
m” ,
N
= m”
(3-1) subformulas R
occurrences
0 no the f o r m
.
only positive
&(x)
m. by X : ( x )
N
x N i
N
m
m ) o 3n[n # m &x(S)
N
n].
formula 0 there i s a Z: formula Y(a,P) 1. For each (with no occurrences ofx) such that
0 * 3a3P{Vn[x(a),
(3-2)
i.e. the universal closure
=
P(n)l 8z ‘y(a,P)}
9
(3-2) i s true in the standard interpretation.
0
Proof.
by up
x
no
&, v , 3 m , V m ,
m by
N
no
3 a 3 P { V n [ ~ ( ( a ) , )N P(n)] 8c
m
0 o Vt@*(t) ,
by by
= 6 8c P(0) = m }
by
@*(t>
* 3 w q V n [ x ( ( a ) , ) = P(n)l 85 Y * ( %P, t>>
Vt@*(t)
*
* ~ r ~ ~ { v n [ x ( ( r=) ,6(n)l > V t 3 ~ ~ P { V n C ( a= ) n (Y)I &k W P ( n ) = s ( ( t , n > > l
Y * ( %P,
o>> Y
.
YIANNIS N. MOSCHOVAKIS
36
by (up LEMMA2. Let G c R4 be universal f o r [R3]3c:, the ternary relations in x;,f o r each partial x: R + o put (4@ G(X)
(3-3)
* 3a3P{Vn[x((a),) 21 P(n>I & ( E , 8, a, P ) E G ) ;
then G ( x ) is universal f o r
X
S n: R 1:R
type 3E
=
1,
+X
5
-, w
S, R
S ~ ( L x&) n~ = 01
~(aN ) n e-
(3-4)
v [n(a)o S n(.>o
S
&
&
n(co1
S 71(a>,l
1 >.o(.[
& n = 11.
W S )
(3.5)
=
x;w,
x,
5
4
5 4(x)
x~Field(S)
on
5
4: F i e l d ( S ) -+ 5 5 3, =q
E,W,
A&,W = G g k w ) = { ( x ,Y ) : (g(E, w), x, Y ) E
by
4 z)
i(w
x
13
,
o
E E R ,W E X ,
T.
GI * by
G Zi(x)
&, v,
~CX,
on
w EX,
E*
{&*I (w) _N
on R x 3 ,
[g(E*, w ) ] * T
21
ME*, 4 1*7
5
g
as =
SUBLEMMA. For each w E F i e l d ( I ) there i s some choice subfunction g,(,) of frcw,such that
5
= I;(w) and some
DETERMINACY A N D PREWELLORDERINGS OF THE CONTINUUM
39
and by Field ( S )
Proof
on 4 ( w ) .
4(z) < 4 ( w ) ,
E
A&*,,
u
=
{Cod(g,(,);3 :4 b ) < 4(w)l
C o d k , , ; 51,
=
q = supremum { [ ( z ) : 4(z)
[
< q, g,K)
u {&(,)(o:
4(z> < 4 < w > > .
=
f,,;
g,, q
4(z)
< +(w)}
-= 4 ( w )
Y
2 4(w), 5(z) > 4 ( z ) *
q =2, q q 2 $(w) [g(e*,w)] * z = { E * } ( w ) by
fr.
gc E*,
=
u
WE
Field(
S)&(w,(O
9
fA
gA
( x , y ) ~ C o d ( g L5; ) *
6 w &({E*}(w),x,Y)E
~ W [ W
Zt(5)
Cod(gA;5 )
5
;i
5
5’’ -,‘4’2
4 : F i e l d ( 5 ) + 5.
on
5
‘1).
Cod(f;5 ) =
{(XI, * * xn,
Y):~
1*,
xn E
P on Cod(P; 5 ) = {(x,,
.. .,x,):
Cod(A; 5 )
=
xl,
Field( 5 ) & Y ~ f ( 4 ( ~ 1 * * ) ,4 ( x 3 ) } *
5
A
5,
.. ., x, E Field( 5 ) & P ( 4 ( x l ) ,...,$(x,))},
{ x : x E F i e l d ( g )& ~ ( x ) E A } .
YIANNIS N. MOSCHOVAKIS
40
Iff:
5" -92,
g : 5'' -+ s2
choice subfunction
ql,
g,,
n x '1). g* Cod(g*; 5) E i ( 5 ) .
G
f* (xl,.
=
qn)
*
n
< 5,
. . , x n , x, y )
* 3a[(x, 4
3,
Cod(g*; S ) 8~(a, X I , . . .,x,,,Y>E
Ei(S)
A
A
=
C o d ( g ; S)
g
5 . Assume A D . Let S be a prewellordering with jield a subset of some X and length 5 , let P be an n-ary relation on 5 , A a subset of 5 . Then Cod(P; S ) , Cod(A; 6 ) are 4
Proof.
P on f(ql,..*,gn>
a,,,al is f (xl,
so by
5 = {ao)
f'(ql,**.,vn),
= ( ~ 1 )
lP(g1,***,qn),
4 Cod(f;S )
...,x , ) ~ C o d ( P ; s )0 ( x l ,...,x , , a , ) ~ C o d ( f ; 6 ) . 3
41
6 . Assume A D . Let 5 be a prewellordering withjield a subset of some X and length let x be the partial function associated with - via (3-4). Let be a class of sets containing all singletons, parametrized I and closed under continuous substitution (preimages), &, v, 3m, tlm and 3a and containing ( ( 6 , m ) : ~ ( 6 E ) m } . Suppose A = where each A,, is a
Proof. ‘2).
E
subset of some ‘2). Then A E ~ . x ‘2)
f(V)
by Cod(g;S )
Uq
by E
Pr otherwise. Assume A D . I f 5 < 0 ( ~ 2 )then
COROLLARY 1.1.
Proof.
5
+ 5' A , if A
x(A) = order type
E
5
5' < 0 ( ~ 2 ) .
t+h : R -+ct;xe)2.
5 w,
5 5
F
5
. 9 1 + A, S q, (6zm-l) S q .c S:,+l 6; = 6: = K1. AD DC 6: = K: for
tAS
1
6,,
=
.
1
=
K:,+,
tA, q
0
1.9. If a a n y nonzero degree then there are 2" degrees b such that the g. 1. b. o f a and b exists and is 0 . by
$51 1.7 1.10. There are 2" degrees b such that b is not the join of two incomparable degrees. 1.7,
d< b
b
d
3.2 bound s ’ b .
2 D.
0 < a < a , < ... . a , < u2 < ...
no
do by
b O(l).
3.1 [4].
by
EDU
A? 0) nD(< a ) t;
D ( > 0) C D ( < a ) “ ( V x ) ( 3 y ) ( y < x)”
i (EDL
t-
(s)
++
i ‘3); i (s
EDU t ( 3 x ) @ ’ ( x ) ,
by
W(x) @’
x.
i (s
3.1
EDL
(s,
EDU. EDL
[2]
EDU
EDU
IV
D ( 5 0‘”). $2
D.
odd
$4.
Initial segments of D ( 5 O(’))
As
5 O(’);
no
D;
D
on 6.1. I f on
" K , 2 D".
of
no
D
K,
(P, 5 ) (P, 5 ) (P, 5 ) (P, 6 )
_-+
by
D
.
D.
K,
81
N,
(P, 2 ) (P, 6 ) S D . ( P , S) (P, 5 )
K,
D.
If ( P , S )
(P,
fs,
s)
D. U.3
U.l
do. K,
by
U.l
> 57.
by
a.
Some general problems of isomorphism and indiscernibility
on
4.2.
( 2a)
D ( 2 b)
( 2a )
D ( 2 b)
a
b?
conjecture)
5)
5) ( I , S)
5 ’) 5’) D
f
D . Unfortunately,
f’ ( I r ,5 ’) this conjeclure false. (I’, S ’ ) is O(’)
f‘ a
M. YATES
C.
82
(I, S )
D(
I’
a
’)
s 0“’)
> O(l).
a
1.u.b.
a
(I’,
D ( s 0“)) by
a,
not
a2
2 O(l)
$51). by f ’
D.
no
4.2. Q.l
by do
M = {X
c D : (3a)(Vb)(b2 a
-P
bEX)] X
X
D -X
M
M.
7.1 T h e Axiom of Projective Determinateness implies that there is a degree a such that D ( 2 a ) and D( 2 b) are indiscernible all b 2 a . T
Proof. by
@E
T
-
5?< (3a)(Vb)(b2 u @
+
$3)
D ( L b) k @).
5?< -
{ b : D ( z b) k @}
Addison-Moschovakis [l] actually formulates a more general axiom but it is the the axiom above which plays the crucial role.
83
{ b : D ( L b) k
A4
i (D}
M
{b:(V(D)((DeT * D ( 2 b ) I= a)} a
M.
9, -
D ( 2 b)
b 2a ,
for
w
T.
of
Q.2 by is
of
of and Y. Some consequences of the axiom of definable determinateness, 59 Undecidability of some topological theories, 38 Initial segments of Turing degrees, of unsolvability,
fur
19
Distributive initial segments of the degrees und 14
Some non-distributive lattices as initial segments of the degrees of unsolvability, 34 D. Category, measure and the degrees of unsolvability D. The axiom of determinateness and reduction principles in the 74 analytical hierarchy, Theory of recursive functions and eflective computability G. 416419.
minimal degree less than O"), Degrees of unsolvability,
67
no. 55
of
Measure-theoretic uniformity in recursion theory and set theory, Foundations of Mathematics On the degrees of unsolvability, of 69 theorem on minimal degrees, 31
On recursive well-orderings, On degrees of recursive unsolvability,
20
of
theorem on initial segments of degrees, Untersurhungeniiber die Struktur des Kleene-PostschenHalbverbandes der Grade der rekursiven Udvsbarkeit, zu 1962. Density and incomparability in the degrees less than O"), 31 Recursively enumerable degrees and the dcgrees less than O"), Set, Models and Recursion Theory Initial segments of the degrees of unsolvability, Minimal degrees,
SOME APPLICATIONS OF ALMOST DISJOINT SETS R. B. JENSEN
R. M. SOLOVAY*
AND
Seminar fur Logik, Bonn
University of California, Berkeley.
Z$
[7],
1. Introduction. (I).
If of
[9]).
At
A:.
O‘,
lTi
A:.
a,
A:
[3]
lli uses
of o $2.
93,
+ V = L.
M M. M[x] p [1 x J
* (1)
6 x,
=
6.
by sets, see (21.
(2)
84
85
SOME APPLICATIONS OF ALMOST DISJOINT SETS
6
M MCxI$4,
$5,
A:
by
$4.
2. Almost disjoint sets
2.1. 01
:
Let M be a countable transitive model of ZFC. We suppose that K f = K, , in M . Let A E K, , A E M ( 3 ) . Then there is a Cohen extension of M , M [ x ] , obtained b y adjoining the sei of integers x such that A is constructible from x and M and M [ x ] have the same cardinals.
[4]. x {Q,( ), u < K,}
2.2.
is
A.
x
do R(x, y ) : x n y
As y w , R( , y )
Ec, R( ,y )
2.3. R( ,z ) A
y
z
R( , y ) y
y
z
z
almost disjoint.
R( , z )
y nz
us.
There is a pairwise almost disjoint family, 9, of subsets of w of power (3) We frequently use set-theoretical concepts relative to a model without explicit mention. Here for example, A is, in M , a set of countable ordinals.
R. B. JENSEN AND R. M. SOLOVAY
86
{si}
Proof.
w
. j
S(f)
w,
{j:sj
S(j’) w
f , g S(f ) n S ( g ) , s j S(g)
f ( n ) # g(n).
w.
5 n.
if S(f )
S(,f) n S ( g )
% = { S ( j ) : f.w“)
w
2.4. 2.1.
Let be a countable transitive model of Let Y E be a subsei of P ( o ) ( ~ ) .Then there is a subset, x , of o such t h a t : 1 . M [ x ] is a model of with the same cardinals a s M . 2. I f y E Y , y n x is finite. 3. Suppose y E P ( ~ )and ~ ,y nx isJinite. Then y E y , U ... U y , u F where y i E Y f o r i = 1,.. ., n , and F is finite. 3 2.)
by 2.5. on
y
M , y nx
2.4 [6]. :
1.
on P
Y
p
>= p’
p
2.
p’.
on P
G [6].
P
Y. 1)
s
P,
(s,t), t.
p
t
w,
s
2 (s‘,i‘) snA =s’nA,
(s,t)
cs’; 2) t s t ’ ; 3) on 1) s E x
P,
p
An
P,, t)
x. 2 ) x nA
G
x
A E ~ .
=
M c
s nA , on P,.
=
{n13(s,f)EG
p A
=
nEs]
EM[G].
M[x] [6] (4)
10.2
P ( w ) is the power set of w , and o is the set
non-negative integers.
87
1. P , satisfies the XI-chain condition. p1 = ( s , t , )
p , = (s,?,) p, p, ( p , t , u t,).
Proof.
X
P,
M
P,,
X X
2.
Let A E Y . Then x n A is jinite. (s,t)~G
Proof. x n A =snA.
AE?.
3. Let A E M . I f x n A is jinite, then lhere is a jinile set s and a finile subset 1 of Y with A E s u
(Ut).
A nx
Proof.
“A n x
t)E
=
=
F”.
A n E A - (F u s u u t ) . t) “ n E X nA
(s, t ) 5 (s’, t ) , so .) 1
s‘ = { n } U s . x n A E F”,
2.4.
3
2.6.
2.1.
{f,,a < XI} x
w,
0)
y , = S( f,) w by 2.4. 2.4 and the fact that {y,,a < K,} A = {a y , nx 2.1
Y
=
{y,: a E A ) . 3)
1
3. Cardinal collapsing with reals 3.1.
4
6,
f:a+ 6,
a < 6,
regular
p < 6,
y+
5
(f) b . y.
weakly inaccessible.
is strongly inaccessible, A < 6, 6.
y
y+
6,
R. B. JENSEN AND R. M. SOLOVAY
88
6.A {f ( 4 ) :
S,
S,(a, x, i ) )
+
A;.
S,
4.6.
{(ai,i
< a)}.
by i
< o.
Qi
Qi
T,(ai).
ai
t)
o
s
t
a < X,}.
on Qi by
$2.2. P
Qi’s.
f
o i.
. f ( i ) Qi P by
i,
f(i)
=
P ($,$)
S
i,
S f2(i). LEMMA1. P satisjies the K,-chain condition. EP
Proof.
,
w by
n,(f)
n,(f)(n)
P
f(n). n,( f)
f) = 1
g
$2.2.)
G , on P . a i = (n
w
I
=
( s , t ) A n s)> . $2.
LEMMA2. Let N ,
=
M [ ( a i , i E w ) ] . Then f o r all i , N , k 7;:(ai). T,.
LEMMA3. Let N ’ be a model ZFC with M c N ’ c N , . Let x N ’ . Then N , k iff T,(x). (This result is a special case a general absoluteness result f o r statements due to Shoenjield [7].)
95
4.7.
$81).
T
P,,P, P , x P , by
( p , , p,)
5 (pi,pi)
5 pi
p,
p2
on
5 p;.
x P,
on PI
x
on on n , on P , x ... x P , . M [ G , , . .. ,
... x
x
i , Gi
on P i . x
P
on P”.
{ 1, ...,
7c
P”,
... x .
II
Gnfl, x ... x G,(,, = ,
9.11)
on P”.
7c
M [ G , ,..., G j - , , G j + , , ...,
Gj
by
4.8.
N,. N’ = M [ ( a f , i < w ) ] .
#j .
a; = a,
a; =
aj
4.
<w)
(a;,
a:.)
Rj
P
=
P
=
Qj x R j
Gj on R j .
so Gj
=
Proof. X E N’ 1 p
Nj, x
c o.
Njk
N’ 1 q(x).
$4.6,
z z
G
by M[Gj].
i Tj(x).
.
x.
p
E
Ti(z).
Gj
. M’
1. M’ i s Proof. f:w + 5 (6)
nR j . N’ Rj
N,
THEOREM.
(4,4).
L,,
p, z
(6).
M’ .
5 w work
5
L,, ,
5
5.
5 is
R. B. JENSEN AND R. M. SOLOVAY
96
L,,
f
9
4
=
M‘.
M’ .
{f ( n ) : n w }
w G M‘.
M’ M‘,
I/ = L
%,,
M. Rj
“n
“nE z”
T”
F,, %,,
%,,EL,,. 8,E M ’ .
$4.6, F,, is p z,
1
L. L,,
R j be generic over M . Then G‘ np,, is non-
2. empty.
Proof. X = { q E R j : q’ 5 q
q’
4.
M. G’nX # ~’EG‘, M’.
q 2 4’.
4
x
F
p’
~ E G ’ ,q ’ E P , , Tj(x)
N’,
n S(fc,j) = F
Gj
z n S(f{,’)= F .
fA,k
p‘
P’ 2 P . p’(n) = ( s , t )
S( fL,J
t
.
f,,’
n , s, t , n=k. P p‘, p‘ ER j .
3. Let g l , . . . , g m be the set of functions which appear in p ’ but do not lie in M . Then fC,’ is generic over M ’ [ g l ,. . . , g , ] (with respect to the set of conditions, P , of 94.5).
{ h l , . ..,h,,,+l}
Proof.
{ g l , ..,,g,, fC,’}
$4.5. M ’ [ h l ,..., h j - J
94.5
1
j
$4.7. N’
M ’ [ g l ,. . . , g , ] .
Y
=
{nEw: i p ‘
ngz}. z
Y 4.
YEN‘.
p’.
m + 1.
hi
SOME APPLICATIONS OF ALMOST DISJOINT SETS
97
2 Y
=
{ n : (3q' €9,) (q'
q' Itn € 7 ) .
p'
{ F n~,E
M'.
W }
M'. p'
L,, N'
N'.
p' q'
5. Y n S(f
K,} .
K, x
K,
x Kl
M, ,
.
fu,a
pa.
M.) Kk --+ Kk.
g , , g, g,
.
K, < O R M ‘ < K,
T
1) M , 2) (a,p) Kt K; (Of 5.6.
g,
alnzost disjoint
I
{a gl(a)
= g,(a)}
< Nf .
h : Kf; + Kf; S ( h ) : Nf -+ N; c(
f Ia
< K, .
c(
I
.
f a
L,,
W>(.) = 5. Kf;
g
S(f)
Kf.
S(g)
T,.
5.6.
f
$5.2,
K: f: Nk --+ Kk. (Va) (f
So
i
S(fU,J
/? = i).
Pi,
M,
positive
f
7i(f)”.
A
Qi
qo
(qo,ql).
Kt. (fu,i;y)
jinite
qo
q,
a
*
Then g : Kf;-+ Kf;.Moreover
5.7.
q g 1
m).
f~ Qo
f
Qo
f: K O
E K f. KO, f
Qo by g = vG,
Kf. on
,
Kt.
KO
g
Qi’s.
GI, Gi
on go = U G , . 5.6.
on Q i .
i > 0,
g i : K’;+ Kf
N , = M[(gi, i e w ) ] . 4.8.
4.8.)
LEMMA.For each. > 0 , we have N , C T,(gi). Let now j > 0 . Put gi = g i , f o r i # j . Let g i = 4 . Let N ’ = M [ ( g i , i < w ) ] . Then, N’ C (Vg) i q(g). 5.8.
M [ g o ] , Kf;
ai
2)
-
i
a, E w
ai~M[go,gi]:
go(r) < g,(s). gi(go(r)) = g d s ) . by M[ai] = M[g0,gi]. N,, a $4.9. N = M[a]. a,~M[a]. 5.6,
5‘7~ai
n€ai
= { j : g j E N } = {j:(3x)q(x)
=
01.
{ai>,
R. B. JENSEN AND R.
102
SOLOVAY
a
4.9.2, There is predicate S ( i , x ) , such that for i > 0 . 1) N k If N k S ( i , x ) , and i > O , then i E a . 2)
5.8.
as two place predicate,
5.7.
1) i 2) 3)
R = { ( r , s): 2'3' g : w s' h
=
S(i,x)
(o,R) a} .
{ ( g j r ) , g ( s ) ) : 5'7s T(a). S,
h
Nf. N;.
o
R
a}.
5.7
1)
Il:.
R
R.
R.
[l].)
L,
c1
1S Nf.
R. = c1
Nf,
A',
R' LA,,
= @(g(n,
n =0 X
7
by
.
7
@i(X)
g ( x , X , , ..., X , )
2k -1.
n :
q + ' ( x >=
@ ( S ( ( @ , o ( X ) , . . . , @ ~ ( X 7> >@i,"(X>>, )
(@:(X), ..., @ : ( X ) ) (i =
0,..., n ) n
mg(X)
Qg(X) ( i S n). {(n,i,j): i < n&jEXi}.)
(XO,...,Xn-,)
:
DEFINITION. Q Q ( x , , ..., x,, X , , ..., X , Y,, ...,
x-j)
x-j :
Y,, ...,
Q c wk x (2")', local in ( X l , ...,X j ) R i , Ric w x m i x = x l , . ..,xk X , , ..., X j ,
~ ( xX ,, Y ) 0 32, m [ z = 2 4
4
-
-
-
4
i Q(x,X,Y
a
)
+
-
+ + +
+
4
3z,m [z = X
- 4
-t
G & ~ , ( x z, , Y ) ] , +
r m&
+ + +
R 2 ( x , z ,Y ) ] .
HAIM GAIFMAN
116 +
+
‘3, ‘X’
‘Y’
‘x,,
+
-t
‘Y,, ..., q - j ’ ; ‘z’ ‘m’ ‘mi, ...,mi’, ‘ z = X I‘ rn’ (i = 1, . . . , j ) . + Q(G,X1,..., X j , Y ) Q*(G,X, ?) X, Q* Xl,...,Xj X. 4
-
..., xk’
‘ X , , ...,Xi’
‘ z , ,. . . , z j ’
+
‘zi = X i p m,’
( X , , ...,X i ) Q
U,,..., U j . @(xl,...,x k , X 1 ,. . . , X j , Y,, ..., q - j ) is in -+++ + + + l z x X Y [ z ~ @ ( x , XY,) ] X , , ...,X , .
( X I , ..., X i )
1, 2, 1* QD,(X,Y )
@,(X,Y )
3 (X,Y )
@;
@Z @;“(Xy
r>
@Z+
Y ) = @,(@XX,
V
Y
= @l(@XXY Y ) Y
n, X
@:(X, Y ) ( X , Y ).
Y
@W, r>>
Y
@W, r>)
Y
Y 3
by
92. Construction of minimal types.
$1 X,
f: w + w . X,
X’,
X’ X‘,
X,
X‘ x’s
X
X’ = {z:
.
w,
f(x)=w y x X , x Iy # f ( z ) . zo zo > y f(x) # f(zo) xE X E X y < z => # f(z)}, X’
P,
X
z z
x
< zo .
X’
P
ON LOCAL ARITHMETICAL FUNCTIONS
117
f:w x
w +w,
y,
...
w,
Lxf(0, x) w,
& f ( O , x ) = w,};
= {x: x
Axf(0,x)
,
,
+
f(n
x
= w,,,,
w,+,
w
if
w
Ixf(n F(n,
=
no
w
+ .
F
P f(x,y),
z(u,u),
~ X Uu,, U),
,
F(y,
P. ( ),:
( ),: x
x + (x),
w (x), = y
+ (x),
,
w
x
(x), = w (x),
>0
&f((x),,
o zE
@(x,
=
(x), - 1.
y< =n
+1
= @(x, w
+
(x), = w 1, = w} (x), = 0 ,
&f(n
+
(x),
x
.
@(x, x
g z
(x), = n &f(n, =
(x)~ (x), = w
no
y
y
+ 1, y
w (x), = 0.
g.
x,
= =
3,
F(n,X) = 3(I) I
118
HAIM GAIFMAN
F 0 P. f(n,X ) f ( n , y X ) =f ( n , F(n, X )
P
3(II)
f*(n, X ) xo xo = x F(n,X)
F(n, X ) =
x X
F(n,X)
x
X F(n,X), f*(n,f(n, X ) = x
x
=f ( n , X ) ,
-
f
F ,f
F(n,X),
f*. P.
f*
X X , F(n,X) F(O,X),
3
F(n
..., F ( n , X ) , ...
+ 1,X) by on.
F(1,X),
D,(X),
:
P
D,(X). D,(X), n , D,(X)
D,(X)
X
on by
f
n , F(n, X )
P F(n, X ) 3 F(n 1, X ) . f”(n,X) x E D,(X) x F(n,X ) . :
x
+
3
X
n = f ( n ,X )
f
F(n, X ) . P,
X
x>z,
D,(X)
(11)
.
F ( z , X ) & y < x}
F ( z , X ) & { y: y
x E D,(X) o
. x, f*(n,f(n,
A(u, U ) P:
x >f”(n,X), X)
=
x.
z(u, u) f ( u , U ) , z*(u, u, U ) , z #(u, U ) , D,(X)] f,f * f .
”
119
LOCAL ARITHMETICAL FUNCTIONS
[ V U ~> U uU(u)]
-+
[ V U ~> U uAr(u, U ) ] ,
Ar(u, U ) A u > z # ( u , U ) -+ [z(u,u) = ?(u, U ) v T * ( u , T ( u , u ) , U ) = U ] .
(11*)
2, P , 23,
I ~ ( a , ..., , a,)
a,,
ui
ai
A, z(uI, ..., u,)
B. =
..., a, E B , 1, ..., n ) .
b EB
... ,
a,,
1 ~ ( a , ..., , a,, b ) 1% ,
A u{b}
n
T
a,,
= 0,1,
23,
T
by
..., a, E A .
..., a, z
A
by
A
by b .
u+(u),
?+(u), ?(u, U ) , z*(u, u, U ) ,
b
Ar,+(u)
EA,
-+
~ ( ab,) = ?+(a) V T ; ( U , ~ ( U ,
1% E A ; 1%
A, tz(a) b' = z(a, b ) ?+(a) A . A b',
I
Vu30 > uAr,+(u).
23 k A r , + ( b ) ,
b > zz(a)
1
+(u)
23 C
(11*)
I Ib
T ( U , o)
Ar(u, U ) by
Ar,+(u) U(u), u T$(u,u) z$(u) T#(u, U ) . b
(12)
+(u)
a A,
b)) = b .
I
b' = ?+(a) A, by
b' b. by A u { b ' } .
1%
23 1 Ar,+(b),
u)
x> Ar,+(u) Vu3u > uA,,+(u)
'u.
23,
T:
23 CAr,+(b)
2l.
'u
1%.
by A U { b }
,
23,
I
b > t$(a)
I z$(a, b') I%=b. b
T(U,
+(u)
23, 113:
Vu3u > u+ ( u ) d(u)
z(u,u),
Ar,+(u) T(u,u), To, TI, ...)T", ...
.
HAIM GAIFMAN
120
40(u)
P !- Vu3u > U
~ ~ ( U ) .
n , P t Vu3u > uA,(u) P t Vu[A,+,(u) {A,,(u):n = 0, 1, ... } U { u > 6:(r P.
+ A,,(u)].
P P'
P
no
0
P' {A,(u): n = 0, 1, ...} . P. 'u 93 P' , by A
1,
P, P
'u ,
U{b}
$ 1 , so
P' , P. by
$1, {t'}'
E {Ai(u): n = 0, 1, ... ,[ E E} A$(u), ...,Af(u), ... Ao,A,, ..., A,, ... [ # t,~
z(u)
n
A: 3,
,
P:
by
9
1,
121
[l] R. Proc. Symp. on Foundations of Mathematics 1961 pp. 257-263.
Infinitistic Methods,
1959), Sets,
Models and Recussion rneory,
1965),
by J.
1967, pp. 122-155. up
[3] H. 1967,
on
connecLos
Set
Soc.,pp. IV.
1-IV. R 16.
DEFINABLE SETS OF MINIMAL DEGREE RONALD JENSEN Bonn
ZF
$1.
ZF,
by
ZF
4 L[x] ,
x E L[a]
x
by on A:.
A:
ZF
ZF
ZF
+ I/ II:
by
A:)”.
A: and on
L,
V
L.* w1
L; :
*
Cf. Souslin’s Hypothesis is incompatible with V = L, by
inJ. S.L.
122
Jensen; to appear
123
If o1 > m i , then there is a sel a c o of minimal degree of constructibility such that a is the unique solution of a II: predicate in L [ a ] . 52.
(I 1 I.
P= P ,5 P ‘P 5 a P P
I
I I
) Q’
on ZF A
forcing conditions
5
Q’ . P , Q P compatible A A c P dense P A. X P pre-dense P P P X . X c M-dejhable X P P-generic ‘P
P
, over P 2 P’ A G n A # Pr
4
PEG A.
n X # Pr
X .
on M.
ZF
X i( i < o)
P.
A
V i X ic A . M
ZF
X P,
M
+ CH ,
P
CCC
w, > m y ,
5 o1
.
ZF M[G,].
A nX
P
A X ,
cX
is P,,
product lemma M
G , x G2
M[G,] nM[G,]
=
P, x
..., P,
RONALD JENSEN
124
by by by
w,
#
I
u, U ' I u
u u' on,
'5'
E
on
w
a a
u, u'
uE
!ij
T
I
w a
{a
SE
=
P
T"
--+
P
M
.
E
G = Ga = M[G] = M[a]. P; G, M. X l = l Gai P a (T, s s c s r -, 2
1
a co
M, a
{ T EP I u
. a cw (ul, M M,
ZF a
n tl
r