E. Casari ( E d.)
Aspects of Mathematical Logic Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Varenna (Como), Italy, September 9-17, 1968
C.I.M.E. Foundation c/o Dipartimento di Matematica “U. Dini” Viale Morgagni n. 67/a 50134 Firenze Italy
[email protected] ISBN 978-3-642-11078-8 e-ISBN: 978-3-642-11080-1 DOI:10.1007/978-3-642-11080-1 Springer Heidelberg Dordrecht London New York
©Springer-Verlag Berlin Heidelberg 2010 Reprint of the 1st ed. C.I.M.E., Ed. Cremonese, Roma 1969 With kind permission of C.I.M.E.
Printed on acid-free paper
Springer.com
CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C. I. M. E. 3'
Ciclo - Varenna dal 9 a1 17 Settembre 1968
t
l
OF MATHEMATICAL ~ ~
LOGIC^'~
~
~
Coordinatore : Prof. E. C a s a r i
H. HERMES
:
Basic notions and applications
of the
theory of decidability. D. KUREPA
Pag. 1
:
On s e v e r a l continuum hypotheses.
'I
55
A. MOSTOWSKI
:
Models of set theory
"
65
A. ROBINSON
:
Problems and methods of model theory
A. SOCHOR - B. BALCAR :
181
The general theory of semisets. Syntaktic models of the s e t theory.
"
267
C E N T R O INTERNAZIONALE MATEMATICO ESTIVO (C. I. M. E . )
H. H E R M E S
BASIC NOTIONS AND A P P L I C A T I O N S O F T H E THEORY O F DECIDABILITY
Corso tenuto a
Varenna dal 19 a1 17 Settembre 1968
BASIC NOTIONS AND APPLICATIONS O F THE THEORY O F DECIDABILITY by
H. Hermes P r e l i m i n a r y Remarks.
( F r e i b u r g r Germany)
The f i r s t t h r e e l e c t u r e s contain an exposition of the
fundamental concepts of some main t h e o r e m s of the theory of r e c u r s i v e functions. One of the m o r e difficult theorems of the theory of r e c u r s i v e functions i s FI-iedberg-Muxniks theorem which a s s e r t s the existence of non-trivial enumerable d e g r e e s . In L e c t u r e s
4 and 5 we prove this theorem, following the t r e a t -
ment given by Sacks, but s t r e s s i n g somewhat m o r e the combinatorial p a r t of the proof ( L e c t u r e 4). Lecture 6
deals with problems in the theory
of
primitive r e c u r s i v e functions. As a typical example of the application of the theory of recursitivy we give in L e c t u r e 7
in detail a proof for the unsolva-
bility of the domino problem in the simplest c a s e of the origin-restricted problem and ehow the
in Lecture
8
how the domino problem i s connected with
c a s e of the Entscheidungsproblem. Lecture
has
6
been given before L e c t u r e s
4 and 5. The inter-
change is due to systematical reasons. The interdependece of the l e c t u r e s may be indicated a s follows: 1
2
3
4
5 , 2
6 , 2
7
8.
Contents L e c t u r e 1: Computability, Enumerability, Decidability L e c t u r e 2:
- Recursiveness, Turing Machines, Degrees of Unsolvability
Lecture 3: Kleenesl Normal F o r m Theorem; the Jump Operator Lecture 4: Theorem of ~ r i e d b e r ~ - M u g n i kP,a r t I
v .
Lecture 5: Theorem of Friedberg- Mucnlk, P a r t I1 Lecture 6: Primitive Recursive Functions Lecture 7: The Domino P r o b l e m Lecture 8: AvA- Case of the Decision Problem of Predicate Calculus. Bibliography
H; Hermes L e c t u r e 1: C o m p u t a b i l i t y
Enuii~erability,becidability
1. Algorithmic procedures and calculi always have been a n essential part of mathematics. In the l a s t thirty o r forty y e a r s a theory has been developed in o r d e r t o study the fundamental notions which a r e connected with this part of mathematics. Everybody knows algorithmic procedures for computing the sum of two decimals. The existence of such procedllres shows that the sum-function is a computable function.
If a
mathematical theory T is given based
on a finite number of axioms and on the r u l e s of first-order logic calculus, we may generate
one by one the theorems of T. Hence the s e t of theorems of
T is' a generable set. Using lexicographical principles it is possible co get nd theorems in a sequence, s o that we may speak of the (Ith, lst, 2 ,
...
theorem of T. In this way we get an (effective) enumeration of T, and we call T an enumerable set. The notions of generability and enumerability may be identified.
F o r any natural number it is decidable whether it is a prime o r
not. Hence the set of p r i m e s is called a decidable set. The concepts of computability, enumerability and decidability a r e narrowly related (cf.no.4).
In order to be able to develop a mathematical
theory concerned with these notions it is necessary to replace intuitive concepts by p r e c i s e mathematically defined concepts.
F o r each of these concepts
different definitions have been proposed and proved to be equivalent t o each other. Practically everybody is convinced that the precise notions correspond llexactlyllt o the intuitive concepts. This fact, the so-called
Church's Thesis
(1936), may be compared with the statement that there exists no perpetuum motiile. In the following (cf. no. 5 , 6 , 8 and Lecture 2 ) we give several p r e c i s e concepts which lead to definitions of enumerability and computability. Referring to such definitions we have notions like Turing-computability, recursir veness,
/u,-recursiveness
etc. But since these concepts can be proved to be
extensionally equivalent, we l a t e r on may interchange them arbitrarily.
H. Hermes 2. In o r d e r t o compute (calculate) it is necessary t o manipulate objects, i.e.
to t r e a t objects by manual means. Not every s e t
property that every element of ch1 s e t of r e a l numbers).
S can be used in this way
S has the
(e.g. the classi-
A s e t of objects which can be used for computa-
tion may be called a s e t of manipulable objects. Typical example f o r manipulable objects a r e the words composed of l e t t e r s from a given finite alphabet A. If A h a s only one element, these words may be identified with the natural numbers. An infinite s e t
S
of manipulable objects is denumerable.
If
S1 and S a r e two (infinite) s e t s of manipulable objects t h e r e exists 2 a 1 1 mapping f from S onto S which is effective in both ways, 1 2 i.e. : if any x S is given it is possible t o compute f(x), and if any 1 1 f (y) Such a mapping is often y S2 is given it is possible to compute
-
-
.
S coincides with the s e t of natural num2 (in this c a s e f(x) is called the Gtidel number of x). In principle it
called a ~'ddelization,especially if bers
is irrelevant on which
(infinite) s e t of manipulable objects the theory is
based. Very often (following Gb'del) we
choose for this purpose the
set
of natural numbers. But many applications may be much e a s i e r if other s e t s a r e chosen. fixed s e t
- We
speak of an enumerable o r of a decidable s e t S
St of manipulable objects is given and if
only if a
=St.
3. F o r most questions concerning computability if is irreleveant whether we consider
1-place o r n-place functions ( o r similarly 1-place o r n-
place predicates). L e t us consider e.g. n=2 t h e r e exist computable functions CS 2, (1.1)
d 2 ( 6)21(~)s %(x))
=
x
(1.2)
d2(x9~))
=
x
=
Y
(1.3)
GZ2(g 2 ( % Y))
. It
may be easily shown that
d21s &22,
S.t.
for each natural number x for each p a i r x, y of natural numbers
Using these functions we may associate with e v e r y 2-place functionj f a 1- place function g,
defined by
H. Hermes
(1.4)
Now we get (1.5)
f(x,y) As
place
g( 6'2(x, Y))
=
.
f a r a s questions of computability
a r e concerned we may
re-
f by g. 4. The following statements hold intuitively:
(1.6)
A s e t i s enumerable i f f it i s void o r the range of a computable function.
(1.7)
A
1-place function is computable, iff
the
2-place relation
R is
enumerable, where R holds for y and x iff y=f(x). (1.8)
A set
(1.9)
A set
S
is decidable iff S and i t s complement a r e enumerable.
S is decidable iff i t s characteristic function f i s computable.
f (x) has the value
0 or
1 according a s
x ES
or
x 4 S.
5. Here and in no.
6 we give two definitions of the notion of enu-
merability. Here we a r e concerned with s e t s whose elements a r e words over a finite
alphabet. Let be given four mutually disjoint alphabets A, B, C, D. The ele-
ments of
A a r e called constants, the elements of
,*.
B variables, the elements
of C predicates. With each predicate i s associated a natural number a s i t s place number. the words over
AvB
a r e t e r m s , then
Ptl;,
D =
);
The words over
t e r m s . If P is an n-place predicate and t
. .;t n
by substituting a proper word for a variable, system Rule relation
and a formula p+F
in a formul:
system
@
Rule 2
p 1' 2'"" etc. a r e called
F to a formula G
the transition from an
t o the Formula
(Smulyan) is given by a finite s e t
derivable
p14p2w3
Rule 1 p e r m i t s the transition from a formula
atomic formula p
.,tn
is called an atomic formula. If p
a r e atomic formulas, then the words pl, pl+p 2, formulae.
nl 1 A called proper words,
F. A formal
4 of formulae. A formula is
, if it can be obtained by applications of
1 and/ o r Rule 2, starting with the elements of
R between words over a finite
@.
An n-place
alphabet A. is enumerabld (in the sen-
H. Hermes
s e of Smullyan) iff there i s a formal system
A, B, C, D, where A n-tuple w ble in
4
0
C
@,
belonging to the alphabets
A, and a n n-place predicate P, s. t. for each A the formula Pw o w n
w of words over 1'"" n iff R holds for w1,
.;w
.
...,
is deriva-
n
6. Another way to define enumerable relations is given by F i t c h f s
minimal logic. We s t a r t with
4 ( , ),*) . A
the 3-letter alphabet
word over
this alphabet is called a n expression if it coincides with i , o r if it may obtained, starting with words a and b
;W.
, by the rule which permits to go over from
to the word
sion. Take the s e t of all
(a, b).
(r (*+))
is an example for an expres-
expressions a s the underlying s e t of manipulable
objects. We choose certain expressions and call them = 12, 13,.
.. .
,/\,V,V,
11,
With these expressions a r e connected certain rules. We confi-
ne ourselves h e r e to indicate the r u l e s connected with (1.10)
,4
v,
=,
V, and
11:
F o r each expression a we may write down the expression = a a ( t h i s is an abbreviation for ((=a)a) (parentheses to the left, also in the
following). (1.11) F o r a l l expressions
a, b
(1.12) F o r all expressions
a, b
(1.13) F o r all expressions a b
we may go over from a t o we may go over from
sult of substituting c llabc
for
Va. the r e -
a in b, we may go over from d to
.
expression is called (lerivable (w
to
a is variable, and d
(These rules a r e s i m i l a r to r u l e s of logic, hence derivation
.
b to v a b .
we may go over from ab
(1.14) F o r a l l expressions a, b, c, d, where
Vab
if it can be obtained by the rules. E.g. the
(d), (Vjt),V\/) dhows, that
T
between expressions
r
s. t.
f o r each n-tuple
r a l . . . a n i s derivable iff
is
"minimal logicm.) An
(vV)
is derivable
.A
relation
(Fitch-) enumerable iff there i s an expression al,
R
...,a n holds
of expressions, for
a
.. , a n .
the expression
H. Hermes
7.
The l a s t example shows that the enumerable s e t s (of expressions)
a r e manipulable themselves, because they may be given by expressions, and each expression determines such an enumerable set. Unfortunately we do not have this pleasant fact for the computable functions. In o r d e r to show this l e t us assume that we have an enumerable set s . t . (a) each element of ctinn
and that
the elemets of
determines effectively a unary
of words computable fun-
(b) each such function may be given in this way (think of S
get a contradiction n
S
S
being descriptions of the computing processes). Then we a s follows : We get in a effective way for each
a prescription how t o compute a certain unary function
.
We introfn duce a new function f by postulating that f(n) = f (n)+l. According to n our assumptions t h e r e is a n m s.t. f=f This leads to a contradiction m' for the argument m (A diagonal argument of this kind is often used in
.
the theory of recursive functions). It is possible t o remedy this defect by enlarging the s e t of functions hitherto
considered. Until now we only have admitted total functions.
The domain of an
n-ary total function consists of all
n-typles of objects
in question. We now consider partial functions. The domain of an
n-ary
total function consists of an n-ary partial Tunction does not necessarily have all
n-tuples a s elements, it may even be void. Intuitively a partial
function is called computable, if there is a procedure which terminates for a given argument iff the function has a value for this argument which determines in that c a s e that value. With partial function we do not get the contradiction of no. is only possible to conclude that
f is not defined f o r the argument
If we admit a l s o partial functions, the statement
7 . It h.
(1.7) remains
true. (1.6) may be simplified : (1.15)
A s e t is enumerable iff it is the range (or the domain) of a computable partial. function.
H. Hermes
8. There a r e different important p r e c i s e
ty
definitions for computabili-
for partial functions. F o r Turing-computability and
Lecture 2.
/1L. -recursivity
cf.
Here we mention only the concept of Markovfs algorithm.
Let be given a finite alphabet A A Markovfs algorithm
and words
A 1., B.1 ( i = l ,
...,p) over A.
i s given by sequence
where "(. )I1 indicates that t h e r e niay be a dot behind the arrow o r not. (1.16) determines a unary partial function. f
. The domain
and range of f
tained in the s e t of all words over A. F o r any word mine
... of
uniquely a sequence
w=w(O), ~ ( l ) w , (~),
iff the sequence terminates, and in that case
W
over
words.
a r e con-
A we deter-
f is defined
f(W) is the l a s t element of
the sequence.. If w ( ~ + ' ) is defined we will have a uniquely determined number ( l < ~ , + ~ < p )which , describes in the sequence (1.16) the rule which responsible for the transition from We call a word
K
a
w ( ~ to )
w ( ~ + ' ).
is
part of L iff t h e r e a r e words K1, K2 S. t.
L=K KK Given K, there may be different decompositions of L of this 1 2' kind. If K has minimal length, the decomposition of L is uniquely deter1 mined and called the normal decomposition.We now procede to define w ( ~ + ' ) and p
-
n+l ' w(n+l) and pn+l a r e only defined if there i s
an i s. t.
A . is a part 1
.
n=O o r (n>O and the pth t e r m of (1.16) has no dot ) In n this case let be p the smallest i, 5.t. A. i s a part of w ( ~ ) Let be n+ 1 w ( ~ ) = KA K the normal decomposition of W(') relative to Ai. Now W(n+ll l i 2 = K1 BiKZ. of w ( ~ )and if
.
A unary partial function &whose domain and range i s contained in
the s e t of dl words over a finite slphabet) is called computable by a Markovls algorithm over an alphabet A, iff
A CA and if for each word 0
W
- 10 H. Hermes over A (a) if f (the function determined by this algorithm) is defined f o r W 0
if
f(W) is a word over A g is defined for
(b) if
0'
W
then g is is defined for W and g(W)=f(W), and then also f is defined f o r W and again f(W)=
=g(W). R E F E R E N C E S: ------------
Davis. [l], of Fitch). Kleene lyan
L e ct u r e
-------we--
2:
111 ,
117
[I,
Hermes [23
.
.
(also f o r the minimal logic
Markov [I],
+Recursiveness, Enumerability, ......................
Rogers
[I) , Smul-
Decidability.
1.
In no. 1 we use natural numbers a s manipulable objects. Let be 1 the 0-place function with value 0, S the 1-place successor-function and n U. the n-ary function whose value coincide with the i-th argument 1 O l n (i = 1, ,n). The functions Co, S , U. a r e called initial functions. The
...
initial functions a r e computable total functions. The process of substitution leads from function a function f = g(h (2.1)
1'""
f(xl,
h ),
r
....x,)
g, hl,
..., hr t o
where = g(hl(xl,
...,xr ), ..., hr(xl. ...,xn)) .
Substitution p r e s e r v e s totality and computability. The process of primitive recursion leads from functions g, h t o a function f, where (2.2)
f(xl..
..,xn ' 0) = g(xl.. . ., xn),
H. Hermes
Primitive recursion preserves totality and computability.
r-
The process of application of the tion
g
to
a function f,
( p y = the l e a s t s.t.
for all
z
y. p y
k o s.t.
o r not, e t c .
But if t h e r e a y
"checkstt whether Ckl
be
s. t.
r ( x ) = G(Nl(x))
-
M(x) - cx, 1
.
Now
If t h e r e e x i s t s no y s. t. Rxy, M(x) will halt.
M(x) , s t a r t i n g with
(3.12) Now l e t
1
< g,
0
.
RxO
o r not. If
M ttcheckstf
R x y , M(x) d o e s not
Hence we have
c ~ ( ,~ h a) l t s iff t
Vy Rxy.
r i s a computable total function. F r o m (3.4)
we infer: (3.13)
~ ( x ) s, t a r t i n g with c ~ ' ~ ) h, a l t s iff t
VY T
1
G(M(x))) t y
.
H. Hermes Comparing (3.12) and (3.13) we get (introducing r(x)) (3. 14)
1 V y T r(x)ty , g
V y 'Rxy ; iff
which gives (3. 11) for
.
t = r(x)
6. We now apply Kleenels Theorem in o r d e r t o prove (3.6) and (3.8). Proof of (3.6):
We introduce R by postulating RxY
iff
. Hence
It is obvious that
R< g
ble total function
r
(3.15)
vy R x y
according to Kleene we have a computa-
s. t.
The left side i s equivalent t o obtain
g ( G z l ( x ) ) = Gz2(x) A Y = Y.
g < g l ( r ) , and trivially
V ~ T ' r(x)
iff
g
g(
r(x) y
cg 2 1(x)) = 6/ 22 (x) .
gl(r)
which shows that
7.
f1 = g f ( r )
O we d i s t i n g u i s h C a s e 1 w h e r e we h a v e t h e defi-
(5. l o ) , (5.11) a n d C a s e 2 w h e r e we h a v e t h e definition
FS, H',
and g(s)
. In
both c a s e s
T
. 0
F
(5.6)
= H
0
s is g i v e n by ( 5 . 1 3 ) a n d
= T o = 0 (void s e t ) ,
g(0) = 0.
s- 1 In o r d e r t o define FS , H' and g ( s ) f o r s > O we s u p p o s e t h a t F , 1 A : a n d A' a r e g i v e n by a n d g(s-1) a r e defined. H e n c e a l s o 1 (5.2). Let b e f , f 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 s of A,: A:. L e t b e
-
~ ( s )= 0
(5.7)
if
s is e v e n ,
e ( s ) = 1 if s i s odd,
e ( s ) = t h e n u m b e r of p r i m e f a c t o r s
(5.8)
3 occuring in the pri-
m e n u m b e r r e p r e s e n t a t i o n of s. Consider
t h e following condition ( w h e r e p is t h e
kth p r i m e n u m b e r ) :
We h a v e
C a s e 1 if (*)
C a s e 2.
be
r(s)
the greatest
is s a t i s f i e d , o t h e r w i s e
m
f o r which
V y (...)
,
In C a s e 1 l e t
T h e n we define :
H. H e r m e s
In C a s e 2 w e put
s S F = H =0,
(5.12)
g(s)=O.
In both c a s e s l e t b e (5.13)
T s = Ts- l , of Ps V Q s V H s s - 1 ,
(5.14)
Ts = Ts,l
Here
P
and Q
a r e introduced
F~ o t h e r w i s e .
U
b y (4. I), ( 4 . 2 ) , ( 4 . 3 ) , w h e r e F', F, H, D
a n d E a r e defined i n (5. 3), ( 5 . 4 ) , ( 5 . 5 ) . Using (5.13), (5.14) a n d (5.2) we g e t i m m e t i a t e l y
In t h e definition quantifiers i n (*))
.
( e . g. V n ( n
of
5
s
F',
HS, T
s
a n d g ( s ) w e h a v e only r e s t r i c t e d
~. .) i n (5.10) and V m V y ( m 5 s A y
In t h e t h e o r y of r e c u r s i v e functions i t is p r o v e d
5 s ...
)
that this fact
g u a r a n t e e s t h a t e v e r y t h i n g is r e c u r s i v e .
By i n s p e c t i n g (5.9), ( 5 . 1 0 ) a n d (5.12), (5. 13), (5.14) a n d (5. 2) we s e e that (5. 16)
S
S
0
F h a s a t m o s t o n e e l e m e n t . H , T s, A S a r e finite. 3. Connection with L e c t u r e 4. F i r s t we find t h a t
S
T h i s is t r i v i a l e x c e p t f o r C a s e 1. H e r e a c o m m o n e l e m e n t of F a n d m u s t b e both of t h e f o r m
HS
2 p + l - & ( s ) and 2n+ & ( s ) which i s i m p o s s i b l e .
-
H. H e r m e s
Using
(5.4)
H'
a n d t h e d e f i n ~ t i o nof
it is e a s y t o v e r i f y t h a t
-
(5.18)
~ H S S 1.
Now w e m a y c h e c k ( u s i n g 5. 17) t h a t H e n c e a l s o l e m m a t a 10 a n d
Axioms
1 to
7 ( L e c t u r e 4) a r e t r u e .
1 1 of L e c t u r e 4 hold.
~'1
.
4. We want t o s h o w t h a t A' (In a s i m i l a r way it c a n b e shown 1 1 ' t h a t A' A ) -We a s s u m e t h a t A < A a n d look f o r a c o n t r a d i c t i o n . 1 1 L e t b e fO, f 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 s of A' r e s p . A According to
j .
.
K l e e n e ( L e c t u r e 3 ) t h e r e is ( u n d e r o u r a s s u m p t i o n ) a n u m b e r e , s . t . 1
(5: 19)
r
f ( x ) = U(
yT
1
exy)
P
.
We now f o r m u l a t e a condition (5.20). In no. 5 we s h o w t h a t we g e t a c o n t r a d i c t i o n a s s u m i n g t h i s condition, a n d i n no. 6 the negation is a s
(5.20). This
finishes t h e proof
we show t h e s a m e a s s u m i n g . 1 0 for A < A . T h e condition
follows:
(5.20)
F o r each finite s e t
5 . We n e r e a s s u m e tc L e m m a
L t h e r e is a n u m b e r s, s . t .
(5.19) a n d ( 5 . 2 0 )
10 ( L e c t u r e 4) t h e r e is a n u m b e r
. Let
b e k = 2e+l. According
u s.t.
L e t be
L is f i n i t e
. By
(5.20) we h a v e a n u m b e r
s
s. t .
(*+j
.
We now c h e c k t h e
- 29
H. H e r m e s
a s s u m p t i o n s of L e m m a 1 1 ( L e c t u r e 4). ( 1 ) g ( s ) = 2 e + l = k. ( 2 ) Hss
-1 /
H e n c e F' re
u
by (5. 18). (3) S i n c e g ( s ) = 0 we h a v e s S
If s < u then
0.
L fI F 3 F
< s. (4) is i d e n t i c a l w i t h
D r s , i. e .
H
r
fI F
s
(++). H e n c e 7 D r s . q r k and that
#
0, we would h a v e
- NOW L e m m a
s)
> 0 and --C a s e 1.
= 0 corltradicting
(5.21). ( 5 )
Let
L fI F
S
be
#
w). T h e r e f o -
r 5 u. 1:' we a s s u m e
0 in c o n t r a d i c t i o n t o
11 shows that t h e r e i s a number
r s. t.
~ H r m for each m.
F o r e a s i e r reference we rename former
S
F's by (4X),
r by
s (and f o r g e t about
the
.
H e n c e w e have : (5. 23)
a n d ~ H s mfor each
@sk
B y (5.23) we h a v e c e by (5. 11)
g(s) = k
g(s) = 2e(s)
= 2e+l
+ &
rn.
{ 0. We t h e r e f o r e h a v e
(s)+l. That gives
E ( s ) = 0,
C a s e 1. Hene ( s ) = e and
by ( 5 . 9 )
where ber y
r ( s ) i s determin2d a s
in
( 5 . 9 ) . A c c o r d i n g t o (*)
we h a v e a num-
s. t.
(5.25)
r(s)
5
1 s , T f0
s , y L
e per(S)y, U(y) = 1
S
H e r e we m a y r e p l a c e -------
f0
s
by -
that in o r d e r to check wh?ther
fO: In L e c t u r e 3 , no. 2 we h a v e m e n t i o n e d
T
1
zxy holds o r n ~ t ,we h a v e l o a s k t h e g o r a c l e f o r g only f o r a r g u m e n t s which a r e l e s s t h a n y . In o u r c a s e we
y < s . H e n c e i t i s sufficient t o show t h a t m e values for each x <s. ------have
(a)
0
If f ( x ) = 0 t h e n we h a v e x
e AUs'
f0
S
and
-
f0
x e A O , cO(x) = 0
h a-v e -
.
the sa-
H. H e r m e s (b)
If
fO(x) = 0
n, 2x B T n wise
let
by be
we would h a v e
0
then
xe A , x
A0 n ( 5 . 2 ) . We want t o show t h a t
0
fs(x) = 1. H e n c e
H' nTn-
4 0,
X ~ A ; . Then
i. e . Hsn-
for a certain number 0
fs(x) 2x
H
= 0, if x S
-< s.
Other-
b y ( 5 . 1 0 ) . But t h e n
c o n t r a d i c t i n g (5. 23).
-
Hence we
have: (5.26)
r(s)
5
s.
Now i n (a) w e d e r i v e fl(p:(s))
y
5 s,
T
1
f0
e p:")
y, ~ ( y =) 1
= 0, a n d i n (b) fl(pr(s))
( a ) We h a v e Fs ( f r o m $sk.
(5.23)), i. r . F'
.
= 1.
C Ts,
hence
2 ~ ' ( ~ ) +1C T s by (5.24). pe e '(') A's+l (5.21, p:(S) A', (b) F r o m ( 5 . 2 6 ) we get u(J.LyT1 e p;(s) y ) = 1 , h e n c e f0 f ( p S ) = 1 b y (5. 19)
fl(~~~(~))=0.
.
6. We no!-assume
L
a finite s e t (*it+)
Let be
m
( 5 . 1 9 andetle_-n_etaJ&n_ of ( 5 . 2 0 ) . s. t . s(s
> o ~ g ( s )= 2 e + l 4 ~ 1 s+ L ~ F ' # 0)
.
m 2p e + l
is g r e a t e r t h e n e v e r y e l e m e n t of L. m ( c ) t h a t t h e a s s u m p t i o n p e~ A' l e a d s t o a c o n t r a d i c t i o n a n d
a number s . t .
We d e r i v e i n
In t h i s c a s e we h a v e
m pe E A 1 a l s o l e a d s t o a c o n t r a d i c t i o n . m ( c ) We a s s u m e t h a t p e e n 1 . H e n c e t h e r e i s a n u m b e r s 1 s . t .
i n (d) t h a t t h e a s s u m p t i o n
H. Hermes
p
m
1
.
6 As
s > 0, s . t .
number
Since T = 0. t h e r e is a 0 1 S m ~ E T ~ - T ~ - THence ~ - ~ . F ={zpe + l ? This
F r o m this we get
1
2p
m
shows that we have Case Now we have
s
>
+
2pe +i,sTs
.
l(no. 2),
and that
.€
( s ) = 0, e ( s ) = 0, g(s) = 2e+l. s 0, g(s) = 2e+l, 7 F ' s (because-F C Ts,,). Applying
w)
m
2 p . +1, the only element we get L Il FS f 0. But this cannot be since e s of F , i s not an element of L according. to the choice of m. d)
We a s s u m e that
pyl.
Using (5.19) we get
e
s. t.
1
e pmy and U(y) = 1. There i s a sefO quence of configurations Co, , Cn) Going through , Cn, s.f. y = G ( Co, this sequence, the T M with Gbdel number e has only a finite number Hence we have a number y
T
.. .
of question
about the value of
a r e also given by
0
.
f
0
. For
.
sufficiently l a r g e
Hence in (5.27) we may replace f s ficiently large s. We may a s s u m e in addition that (5.28)
f
. ..
6(s)=0, e(s)=e,
Now we get from
m
5
0
-s these values 0
by fs
with suf-
s , y L s.
(5.27) :
This shows together with (5.28) that we nave Case 1 (no.2). Therefore thes + 1 = {2pr's)+1j, g(s)= r e exists a number r ( s ) 2 m, s. t . F e = 12pe(s) = 2e+l. We will show in
a moment that - I F ' S . Now applying (ff*Y) we get
2pr(s)+l, only element of F', cannot be an element of e F~ , cannot be a n element of L according to the choice of m , since
L~F'
0.
But
r ( s ) 1 m;
It remains to show that F ' s . Otherwise F
S
C
T
s- 1'
hence
H. H e r m e s
.
'('I + I , g T s - l Hence t h e r e is a n u m b e r q s. t. Z ~ ' ( ~ ) + ~ E - TT '"e e 9 q-1' T h i s Leads t o .rq= )2p:(s)+l] Helice w e h a v e C a s e 1. w h ~ r e F q =
.
-
.
( q ) + ~ &(q)] B y c o m p a r i s o n we g e t 8 (q) = 1. r ( q ) = r ( s ) w h e n c e i 2 p e (q) (i. e . - - t P q c T g(q) = 2 e + l . Now we h a v e q .> 0 , g(q) = 2 e + l:lF1q ). q- 1 Applying (*k4 we get L n F~ # 0. But 2 ~ : ( ~ ) + 1 , f h e o n l y e l e m e n t of F~ , i s not arl
e l e m e n t of
L.
R E F E R -----E N C E-S : S e e L e c t u r e 4. ------
n c-t i-on? -L -e c-t u- r-e- -\'i;--.-P-r i-m-i-t i-v-e -R-e-c-u-r s-i -v e- -F-u1. In L e c t u r e 2 ,
no. I
the c l a s s
-P
of p v i m i t i v e r e c u r s i \ : r t'crn-
c t i o n s h a s been i n t r o d u c e d a s t h e s m a l l e s t c l a s s of ( t o t a l ) functions t o which 1 belong S a n d e v e r y vn and w h i c h is c l o s e d u n d e r s u b s t i t u t i o n a n d C: p r i m i t i v e r e c u r s i o n . T h i s c l a s s i s a p r o p e r s u b c l a s s of t h e c l a s s o f a l l
.
r e c u r s i v e t o t a l f u n c t i o n s , a s h a s b e e n shown by A c k e r m a n n : t h e function d e fined by
H. Hermes
i s total and recursive, but not primitive recursive. The c l a s s of recursive functions i s not an a r b i t r a r y mathematical construction but has an llabsolutenmeaning, because it coincides with the c l a s s of computable functions. It i s not known whether the class of primitive recursive functions i s also *absolute11 in a s i m i l a r sense. The definition of P
depends
on the c l a s s of natural numbers a s the underlying do-
D of manipulable
main
ning of primitive
objects.
We may a s k ourselves: what i s the mea-
recursivity for other c l a s s e s
-P
2. We may consider
C
of manipulable objects?
a s being generated by two different kinds
of operations: (1)
independent of the pecularities of N: To these open n rations belong the substitution, and every U. (a function U . may be considered a s a
Operations
0 - a r y operation )
.
(2)
Operations depending on the pecularities of N: Here we have thf 0 1 0-ary operations C and S and the primitive recursion, which i s a bi0
nary
operation leading from functions
function
f
(cf. Lecture 2, no. 1)
g, h to a a functions g, h
to a
.
N may be considered a s arl absolute f r e e algebra of type < 0, 1 >, 1 i s the 0-ary operation and S the unary operation. This inwhere C: dicates immediately that
0
Co and S'
a r e dependent on
N; the s a m e holds
for the
primitive recursion where in the f i r s t equation occurs 1 in the second equation S
.
C:
. and
3 . How can we transfer: the notion of primitive recursivity of another
domain D
D of manipulable objects ? Let be g a
onto N.
longsn
to
~ s d e l i z a t i o nwhich maps
Then we can associate with each (total) function N a s its
g-transform-
a function
fg
which
f which Itbenbelongstt to D,
H. Hermes
by the following definition :
.
We define
P a s the c l a s s of the g-transforms of the elements of P_ If -g we choose different GBdelizations g l , g2, the c l a s s e s P_gl and P may -g2 differ from each other.
On the other hand we may use the processes (1) and imitate the p r o c e s s e s (2) (cf. no. 2) in a natural way, in order to get a class
P-D
of
functions which henceforth may be considered a s genuine primitive recursive functions. We want to s a y that we have a satisfactor cept
of primitive recursivity for
y
D if it i s possible to introduce
a natural way and if there is a Godelization g s . t .
4.
class
As a f i r s t example for a satisfactory
of words over
solution for the con-
a finite alphabet
ED in
P = P -g -D'
solution we mention the
{ ao, . . . ,a N l
(including the empty
. This < 0,1,. . .,
word E)
c l a s s may be considered a s an absolute free algebra of 1 1 type 1>, where E i s the 0-ary operation, and So ' ' . . ' s N 1 a r e unary operations, defined by S .(w) = wa. (concatenation). In o r d e r to J E, So , , S; and define priimitate (0) we now select the functions
1'
...
mitive recursion by the following N+2 equations:
A s s e r has proved that. we have a satisfactory solution by indicating an appropriate GBdelization.
H. H e r m e s 5. T h e e x p r e s s i o n s of
m a y be considered a s an have t h e
expression
defined by
Fichts
m i n i m a l l o g i c (cf. L e c t u r e 1, no. 6)
absolute f r e e algebra as
of type , w h e r e we
t h e 0 - a r y o p e r a t i o n a n d a b i n a r y o p e r a t i o n J,
J(x, y ) = (xy). H e r e w e h a v e t h e following p r i m i t i v e r e c u r s i o n :
In a s i m i l a r way w e m a y p r o c e e d i n the c a s e of a n a r b i t r a r y a b s o l u t e f r e e Mahn h a s 6.
.
n >, w h e r e n a r e n a t u r a l n u m b e r s ( j = 1,. . ,r ) . 1'"" r j shown t h a t we have a s a t i s f a c t o r y solution in a l l t h e s e c a s e s .
a l g e b r a of type
o In
l1D1une facon
is
Therefore one might a s s u m e that in particular
ordinal number 3.6.1.
N(a)
'5
and any D n (I1 P e r ogni numero cardinale infin nito n, 2 6 il primo numero' inaccessibile >nn)
.
The function (3.5.1. ) was introduced in 1937 (cf. Kurepa C157 , formula (2))
.
The existence of the function (3.5.1.)
is implied by the axiom of choi-
ce; probabljr ,the converse holds too. 3.6.4.
Here is a s e r i e s of nice continuum hypotheses.
Let n
be a
natural number; '.then f o r every ordinal q
(the case n = l yields the Cantor's 3'. 6.5.
A
general continuum
hypothesis)
one has
.
,great variety of continuum hypotheses is obtained in the
D. Kurepa
following way:
F o r every ordinal number
3.6.6.
a
l e t m(q)
be a positive integer; then
Hypothesis.
F o r el erg-
diadic
gp(G{O,.i'Jone h a s
Ord -+
ordinal mapping
of the preceding continuum hypotheses in acceptable.
Each
3 . 7 . On factorials.
--
Fo" any ordinal (cardinal) number c>--'inal (cardinal) numbers m the
,ar-dinal number of
dinallty
11
all
such
n
that
let m
< n.
.. . =
2n
.
F o r natursal Inirlgers w e have
a s well a s ( 3 . 7 . 3 . ') where
such
P ( n ) denoiea the s e t of a l l uniform mappings
that
be ? h e s e t of a l l the If
the permutations of
, then f o r every transfinite cardinal n
( 3 . 7 - 1.)
In
.
n
is defined a s
any s e t
S 3f
car-
D. Kurepa
(3.7.5)
o<S
(T,
holding for e v e r y kA satisfies ( k ~ )< ~ 2
kA=1, 5, 6 , .
this relation is satisfied for
in
every chain
has
( k ~ ))I kT ~
A
. It
fi] p.
130 (Prof. P2); equivalence
is interesting that one h a s the following
PPP2
D. Kurepa Theorem
4.4.
. The
kPD (T,Q = ( k ~ ) +for every infinite t r e e
(4.1) conjuction
of
hypothesis
(Kurepa [92] )
the general continuum
In particular, infinite
tree;
on
incompatible with of
reals
preceds b
in
the
and
well
5.
2kW0 =
Without
if
akwl
continuum
(R;
if
+
2
contradicting
the
some ordinal o( satisfying
every infinite t r e e
As a matter
of
for every
and
well
Q(X)
ordering of
a c t b ea s b
of
and
the relation Q(W) would
.
(T,q+l , then for
we
have
b T, kT a r e either equal o r consecu-
(T,,< ) s u c h that
kX
>
; consequently,
kw
d-subset
PX C P D ( O , 6 ) and
therefore kP
D
(0, ,kPX >,
prove the relation
(6.1. )
Notations.
kw,-
> kw
'
= kw,,
means means
O(
or
X
= kwd
, we a r e not
.
.
the imediate predecessor
the immediate follower
of
O(
of O(
.
kT =
6+1
f
kX means cardinality of
do(+
2
. If a = q , kT
and this proves the relation (6.1. ) able t o
a
then
tive (D. Kurepa [6]p. 105, theor. 1) ; in either case t h e r e ,is some
K
is
hypothesis we a r e not able to prove
+
If for
any
R ,
C H
kPD(T,
