Representability of Sets in Formal Systems

REPRESENTABILITY OF SETS IN FORMAL SYSTEMS BY

ANDRZEJ MOSTOWSKI The aim of this paper is to advocate a method (due in principle to Gijdel

[2], though never elaborated by him in details) to present in a uniform way

the theories of recursive, hyperarithmetical and related families of sets. The gist of the method is to defim these families using the notion of representability in suitable formalized theories. The techniques worked out by Kleene and other writers yield probably better results when one wants to discuss properties of a single family; the writer believes however that the method developed below is very helpful when one wants to discuss common properties of these families and to detect reasons of their affinities. The method will be presented for families consisting of sets of integers and sets of functions. An extension to higher types has not yet been tried, but seems to present no essential difficulties. The writer had planned to entitle the paper “ Kleene’s theories as I see them.” Although the final title is more conservative, the influence of Kleene’s work on the present paper should be obvious to every reader even moderately acquainted with the literature.

I. GENERAL THEORY CHAPTER

OF REPRESENTABILITY

1.1. Formal systems. In Chapters I and I1 we shall deal with formal systems having a common language and differing from each other by the notion of consequence. The common language of these systems is that of .second order arithmetic 131 with constants for both types of objects (integers and functions). Latin 1.c. letters will be used for variables and constants of type 0 (integers) and Greek 1.c. letters for variables and constants of type 1 (functions). We use the first letters of the alphabet for constants and the last ones for variables. Numerals are denoted by a,, a,, ‘ . . We denote by A x the set of axioms consisting of the usual axioms for the propositional and functional calculus with identity, of Peano’s axioms for arithmetic, of the so-called pseudo-definitions, i.e., axioms of the form (&)(x)[&) = 0 = 01, where 0 is a formula in which the variable is not free, cf. [3], and finally of the special form of the axiom of choice which allows one to permute the functional and the numerical quantifiers, cf. [ll, p.2171. If X is a set of formulas then Cn,(X) denotes the set of formulas which can be obtained from A x u X by the usual rules of proof. We assume that in each formal system which will be considered in our theory there is defined a function of consequence Cns acting on sets of formulas and yielding such sets. Further we assume that this function satisfies the

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Received by the editor April 6, 1961

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REPRESENTABILITY OF SETS IN FORMAL SYSTEMS

469

following axioms:'

(A) X c Cns(X). (B) X E Y implies Cns(X) E C n d Y ) . ( C ) Cno(Cn8(X))E Cn8(X) and Cn8(Cno(X))E C n , ( X ) . (D) If P' is a closed formula and I E CndX U {P'}), then ?F=I I E C n d X ) . (E) There are infinitely many inessential constants of both types. A constant a or a is inessential for S if for every set X of formulas none of which contains a (or a) the condition I E Cn8(X)implies I' E CnXX) where I' results from I by a substitution for a (or a) of a free variable of the appropriate type which does not occur in I. Note that Cno(Cn8(X)) = Cns(Cno(X))= CnAX) by (B), (C), and the obvious properties of Cno. In some theorems we shall assume that Cns is an idempotent operation. These theorems are marked by an asterisk. If 2 is a set of formulas then ExtdZ) denotes the system S' with the function of consequence defined thus: Cn,,(X) = Cns(X u 2). 1.1.1. Zf there are infinitely many constants (of both types) which are inessential for S and do not occur in formulas of 2, then the system Exts(Z) satisfies (AHE). Let 1 be a functional variable or a functional constant and (p a function from integers to integers. We denote by D,(A) the set of formulas A(&) = & P ( ~ ) and call this set the diagram of (p. To maintain the symmetry between both types we denote by &(I) the formula l = 8,; here l is a numerical constant or a numerical variable. 1.2. Representable sets and functions. Let o be the set of integers 2 0,o" the set of all mappings from o into o. Elements of o will be briefly called numbers and elements of om functions. We denote functions by the letters (p, 4, r?, * We denote by Rk,1the Cartesian product omX . . . x om X o X . . * X x w ' ; elements of Rk,, are denoted by German letters p, q, . . . . Thus o= p is a sequence ((p,, . . , ( p k , nl, . . . , nl) consisting of k functions and 1 numbers. Let the German 1.c. letters a, 6, . . . , 0, IU, . . . denote sequences consisting of k inessential constants (or variables) of type 1 and I inessential constants (or variables) of type 0. Such sequences are called briefly k , l sequences and we shall write a = ( a I ,. . ., a k , a,, . . -,at) and similarly for other letters. We Put +

a .

-

D,(a) = DV1(al) u

. . . u Dq,(ak) u Dn1(al)u . . . u D,,(al) .

A set A C Rk,Iis weakly represented in S by a formula I if 0 has k free variables of type 1, I free variables of type 0 and I These are esentially Tarski's axioms [17];we did not include all of Tarski's axioms in order to have a wider range of applications.

410 (1)

FOUNDATIONAL STUDIES

q EA

= @(a) E Cn@&a));

here @(a) is the formula obtained from @ by a substitution of the constants a for the free variables of @. The family of weakly representable subsets of Ri,[is denoted by s k , l ( S ) or briefly by 9 k . l when S is fixed. If besides (1) the equivalence q non E A

=

-

@(a) f Cns(Dq(a))

holds, then we say that A is strongly represented in S by @. T h e family of ) briefly by strongly representable subsets of Rk.[ is denoted by ~ ; , I ( . S or 9”: .1 . It is worth while to remark that families 9 k . l and 9:,[ may well coincide, e.g., if S is a complete system. R,,,,,, is represented in S by a formula @ with k m A mapping f : free variables of type 1 and 1 n free variables of type 0 if for every q in

+

+

Rk.1

here D and a are k , l sequences of variables and of inessential constants and II, and b are m ,n sequences. T h e family of representable mappings of Rk.1 into R,,,,,, will be denoted by &,t;m,,,(S) or briefly by K I : ~ . ~ . If m = 0, i.e., if f maps Rk,i into w or into a Cartesian product of finitely many copies of w , we can replace (3) by (3’)

@(a, h) E CndDQ(du D/(&))

and obtain an equivalent condition. Since (2) and (3‘) imply (for m = 0) (3”)

-

@(a, b) E Cns(DQ(a)u D,,(b))

for n # f h )

we infer that 1.2.1. Iff E A < I ,then ~ ,the ~ relation f(q) = n is strongly representable. I f f : Rk.1 + R0,% and the relation f(q) = n is weakly representable, then f E f i , t ; o , n . No similar theorem holds if m # 0. 1.3. Properties of strongly representable sets and of representable functions. We list below a series of elementary theorems whose proofs can be obtained immediately from the definitions. 1.3.1. S‘,*.[ is a Boolean algebra of sets. 1.32. If A E S ? ~then , I , A x (I) E and A x w’ E 9Z’lL,,~. Every set which arises from A by a “permutation of axes” belongs to s k : along icith A . 1.3.3. If A E g k : + 1 , then the set A” = ((1: ((1, n ) f A } belongs to %?. A theorem similar to 3.3 would be false for the operation A V = (11: (rp, (1) f A}; we only have a weaker result 1.3.4. If A E s k T 1 , l ( s ) and a is a constant inessential f o r S then A’ E .@k?Ext@,(a))). I.3.5*. If f E - % , I ; ~ , % and A E .%:.,‘ then !-‘(A)E .%*L.

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REPRESENTABILITY O F SETS IN F O R M A L SYSTEMS

471

The notion of a recursive mapping fi Rk,! + Rm,* will be defined formally in $11.1. It will be seen there that for k = m = 0, n = 1, this notion coincides with the usual notion of a recursive function with 1 arguments. then 1.3.6. r f f is a recursive mapping of R k . 1 into Rm,,, and A E f-'(A) E S?~,L. Note that no assumption of idempotency of Cns was needed in Theorem 3.6. contains all recursive mappings; permutations and identijca1.3.7. tions of variables do not lead outside the family of representable functions. 1.3.8. Superposition of two fzmctions one of which is representable and the other recursive ieads to representable functions. I.3.9*. The family of representable fulrctions is closed with respect to superpositions. 1.3.10. A E g:,,, if and only if the characteristic function of A belongs to

gc,,

9 m , n : O . 1.

1.3.11. r f A E G?:,I+~ and if for every q in Rk.1 there is an n such that ( 9 , n ) E A, then the function min, [(q,n)E A ] 6eZongs to .$%,f:o,l.

As the last theorem we note that the family of strongly representablesets is not affected by extensions of S. More exactly 1.3.12. I f X is'a consistent set of formulas, then .%';J(S) = S:L(Exts(X)).

1.4. Properties of weakly representable sets. 1.4.1. If A, B E L%?~.I, then A n B E 9 k . l . 1.4.2. I f A E g k , 1 and B E LZ.&,then A u B E .9Pk.I. It is not known whether the union of two weakly representable sets is always weakly representable. Most probably this is not the case, but no counter-example is known a t present. Theorems 1.3.2, 1.3.3 and 1.3.4 remain true for weakly representable sets. The Theorems I.3.5*, 1.3.6, are probably false if &'* is replaced by 9. following weak form of 3.5* survives: 1.4.3. If A E so.^ and f E S5,1:a,~, then f - ' ( A ) E @ a , l . The following example shows that Theorem 1.3.12 does not hold when g* is replaced by @. Let Cns be the function Cne defined in [lo] and IZ a formula such that the set Cn8({n})be consistent and complete (cf. [lO,'p. 1661). In this case the family ,%?a,l(Exts({IZ})) is a Boolean algebra because it coincides with .GPo*.I(Exts({IZ})). In [lo] it has been shown that the family gO.'(s) coincides with the family IZ: and hence is not closed under complementation. We note a weaker theorem (which we may note in passing, holds not only for , ! % ? kbut , l for .GP'l as well). 1.4.4. If 17 is a closed formukc, then ak,l(EXts({R}))S SZPdS). Indeed, if B represents A in Exts({IZ}), then IZ 3 B represents A in S. It is remarkable that under special, but not too narrow assumptions the analogue of Theorem 1.3.12 can be proved for the family S P k , i . We shall discuss this phenomenon (discoved for recursive sets by Ehrenfeucht and Feferman) in later sections. 1.5. Universal functions. Let e,, el, . . . (or more exactly e:"", e:".", . . .) be an enumeration of the Giidel numbers of formulas with k free functional

472

m1, 33

FOUNDATIONAL STUDIES

variables and 1 free number variables. T h e formula with the G a e l number e. is denoted by en. The sequence e. is primitive recursive and logical operations on formulas (including substitutions) correspond to primitive recursive operations on integers en. Put

U ( n ) = U "'"(n)= {q : Z,(a) E Cns(D,(a))}.

1.5.1. g k . 1 coincides with the family of sets U ' k , " ( n )n, = 0,I, . . . . Thus U"." is a universal function for the family 5 P k . I . Theorem 1.5.1 provides us in the usual way with examples of nonrepresentable sets. 1.5.2. The set {(q, n) : (q, n) @ U'k"+"(n)}is not weakly representable. If one wants an example of a subset of &&%+l.O which is not weakly representable one can take the set {(p, PI,

.'

' ,(Pk)

: (p, +'I',

' ' '

,pk)

U'k" o ' ( d o ) ) }.

1.5.3. The set {(q, p) : q E Ui.'"'(p)} is not strongly representable. Let T be the set of Gijdel numbers of closed formulas which are provable in S, i.e., which belong to Cns(0). 1.5.4. There is a recursive function of two variables g ( n , m ) = g.(m) such that U'"'(n) = g;'(T). 1.5.5. [email protected]%':I, W T#.GPo,1. Theorems 1.4.1 - 1.4.3 and the analogues of Theorems 1.3.2 - 1.3.4 for weakly representable sets have strengthened versions showing that if the operations mentioned in these theorems are performed on sets U(p), U(q), . . . then the result is a set U ( x ( p , q , . .)) where the function x is recursive. E.g. the strengthened version of Theorem 1.4.1 reads: +

U(P) n U(q)= U M P , q)), where x ( p , d = min, [e, = rZp 8z &'I Let A R k . l . s 5 1, m = (n,+!,. . . , nl) and let Am,ebe the set

.

{ ( ~ ~ ~ , ~ ~ ~ , p k , n ~ , ~ ~ ~ , n , ) : ( p ~ , ~ ~ ~ , ~ k , n ~ , ~ ~ ~ ,

If A = U " ~ l A 1 ' ( then r ) , Am, e= lJ'"''(r'),where r' depends on m,e, and Y . Using the strengthened versions of Theorems 1.4.1-1.4.3 and of the analogues of Theorems 1.3.2 - 1.3.4 for the family .A? and repeating the proof of Kleene [4] we obtain the recursion theorem (or, in Myhill's terminology, the fixed point theorem). then 1.5.6. There is a recursive function E(m, r ) such that if A = Utk.l+"(r), A m , n ( m , v i = U'k,8'(E(m Y ),) . I.6*. Degree6 of representability. In the whole 56 we assume that Cn,JCns(X))= Cns(X). Let y , + be functions, i.e. elements of R l , o . We say that the degree of representability of p (in S) is not hiqher than that of 4 (symbolically cp 5 ++"I, if p is representable in Exts(D+(a)), where a is an inessential constant for S:

co s s 4 = c o ~ ~ , ~ , ~ . ~ ( E x t s ! D ~ ( a ) ) . This definition is an obvious adaptation of the .definition due to Kleene-

WI, 34

R E P R E S E N T A B I L I T Y OF SETS IN F O R M A L SYSTEMS

473

Post [S] to the more general situation considered here. Similarly as for recursive degrees we define-cp z S(I, as (cp S s + ) & (4 s.9cp). I.6.1*. The relation S s is reflexive and transitive; the relation z S is un equiva lence relation. The equivalence classes under z S we call degrees of representability. We show similarly as in [6] that I.6.2*. Degrees form an upper semi-lattice whose minimal element is fi,l.Q,l. I.6.3*. If V E 26,1,0,, and 4(n)= 9(cp(n)),C w ) = ds(n)), then 4 Sat? and C S s9 , but in general non = s C . In the next two theorems we denote by %A the characteristic function of a set A and put r = X T (cf. 1.5.4). I.6.4*. If A € then Ssr . Indeed, x,,(n) can be represented as r(V(n)),where cp is recursive. I.6.5*. r f T‘ is the set of Godel numbers of formulas provable in Exts(D,(a)), where a is an inessential constant of S , then r < s x T , and X A < s f o r every

+

A in &Po.l.

The theorem is proved by showing that r = s ~ T , would imply that TI is strongly representable in S’ = Exts(D,((a)) which contradicts 1.5.5 (with S replaced by S’). The formula r S s x T , is a consequence of the following result: for arbitrary cp, if T * is the set of Gijdel numbers of formulas provable in Exts(Dq(a)), then 2 T*S;Scp . Theorems I.6.4* and I.6.5* generalize the basic properties of the “jump operation” of [6].

1.7. Separability and decidability. We call two subsets A, B of Rk.1 separable in S if there is a C in .P:, such that A C C and B n C = 0 . 1.7.1. T and the set Tn of Godel numbers of formulas which are refutable in S are not separable in S. The proof is identical with the proof of 2.5.B in [3]. S is called S-undecidable if T$.G%?’:l;it is called essentially S-undecidable if no consistent system Exts(X) is S-decidable. From 1.7.1 we obtain the result that,under the assumptions made in 51.1, 1.7.2. S is essentially S-undecidable. 1.8. Properties ( A ) ,(C), and (S). Most of the formal systems encountered in practice enjoy one or more of the following fundamental properties: (A) I f - A , BE s k , i , then A U BE 9 k . 1 , k , I = 0 , 1 , 2 , . . . . ( C ) If X is a consistent weakly representable set of formulas then the set

is weakly representable, i.e., Mk.I(X)E s k , l + l ( s ) , k , I = 0 , 1 , 2 , . . . . The phrase “weakly representable set of formulas” means of course a set such that the set of the Gijdel numbers of its elements is weakly representable. A special case of (C) in which we assume x = 0 is called (co). Sets k f k , I ( o ) are denoted simply by kfk.1. (s)For every A, B in. &Pk,lthere is a formula 0 with k f r e e functional variables and l free number variables such that

474

FOUNDATIONAL STUDIES

- -

@(a)f Cna(Dq(a)), if q E A - B @(a) E Cns(Dq(a)),i f q E B - A

@(a) E Cns(Dq(a))or

, ,

@(a) E Cns(Dq(a)),i f q f A

uB.

@ is called a separating formula for A and B.

Condition (S) was formulated for the first time (in a slightly weaker form and for a special system S) by Shepherdson [15]. Condition (C) was used implicitly by several authors. 1.8.1. Each of the properties (A),(C), ( S ) is preserved when one passes from S to a system Exts(D,(a)), where a is an inessential constant and cp a function. 1.8.2. (c,) implies that {(q, p ) : q E u“ “ ( p ) )E g k , l + l ( s ) and T E ~ o . I-( s ) %.:(S, . PROOF.The first part is identical with (Co). In view of 1.5.5 it is sufficient Let 9 be a recursive function such that e$& = to prove that T Ego.l. & ( x =x)’ for every closed formula @. Since’ 101

E

T = Zv‘:&(a)

E Cns(Do(a))= (0, &@l)) f

Mo,l

we infer by 3.6 that T E S ~ . ~ ( S ) . 1.8.3. (S) implies that S k * [ coincides with the family of sets A S Rk.1 such that A E @ , I and Rk.1 - A f gk,l . Obviously every A in g k : belongs to g k . 1 together with its complement (cf. 1.3.1). If A and Rk,[- A are in 9 k . i then every separating formula for these sets strongly represents A in S. 1.8.4. (C) and (S) imply that S is essentially incomplete, i.e., that for every weakly representable consistent set X of formulas the system S’ = Exts(X) is incomplete. PRoof. Condition (C) implies that the set T’ of Godel numbers of formulas provable in S’ is weakly representable in S along with X . If S’ were comwould imply w - T’ E @o,l(S),whence by 1.8.3 we plete, then T’ f .@o,l(S) whence by 1.3.12 T’ f &%:(S’). Since this contrawould obtain T’E go:@), dicts 1.5.5 we infer that S’ is incomplete. Theorem 1.8.4 gives an abstract form of the incompleteness theorem of Rosser [14]. 1.8.5. (C,) and (S) imply that the theorem of reduction [7] holds in %,i(S), k,l= 0,1, . ’ ’ . Hence the second separation principle holds for weakly representable sets and the first separation principle holds f o r complements of weakly representable sets. PROOF.If A , B E Hk,&S) and @ is a sepal‘ating formula for these sets then the reduction of A u B is effected by taking A l = A n {q : @(a) f Cns(Dq(a))), BI = B n { q : @(a) E Cns(Dq(a))l 1.8.6. (C,), (A), and ( S ) imply that f o r every pair A , B of disjoint sets in L%b,i(S) there is a formula 8 such that 8 weakly represents A and -8 weakly represents B.

-

*

‘0’ denotes the Giidel number of the formula Q.

[811, 36

REPRESENTABILITY OF SETS IN F O R M A L SYSTEMS

PROOF.Let Y and u be recursive functions such that v(n) = r.-ZLk."1 Sbki:; = Sb(x/B.) ZAk*'"'. Sets

475 and

A* = ((4, n ) : q A v (q,u(v(n)))E k f k . 1 1 , B* = {(q, n ) :q E B v (q, 0)) E kfk.11 are weakly representable in S according to (A) and (G).Let Zik.'"' be a separating formula for A* and B*; we can assume that the last of its number variables is x. If 8 arises from Z:k"+l' by the substitution of Bq for x, then 8 has the desired properties. Theorem 1.8.6 gives an abstract form of a theorem due to Putnam and and Smullyan [13]; the idea of the proof given here is due to Shepherdson [U]. 1.8.7. (C), (A), and (S) imply that if X is a weakly representable consistent set of formulas, then f o r every pair of disjoint sets A , B in .B'k,I(S) there is a formula 8 such that 8 weakly represents A and -8 weakly represents B in ExtdX). The proof is similar to that of 1.8.6;the only difference is that we replace M k . 1 by kfk,i(X). 1.8.8. (C), (A), and (S) imply that %,i(S) = a.I(Exts(X))for every consistent weakly representable set X of formulas. PROOF. The inclusion S . l ( S ) C 9h.I(Exts(X)) follows from 1.8.7. If A E Hk.[(Exts(X))and Z, represents A in Exts(X), then q f A = ( 4 , p ) f Mk,r(X) whence A EM k , t ( S ) ,since, by (C), Mk,I(X) belongs to S . l ( S ) . Theorem 1.8.8 represents an abstract form of a theorem discovered by Ehrenfeucht and Feferman [l]. It is an open question whether the operation Ext preserves any of the properties (A), (C). For the property (S) the answer is negative as is obvious from the observation that 1.8.3 is false for the system Sp discussed in [lo]; cf. the remark following Theorem 1.4.4.However, the property (S) is preserved under finite extensions: 1.8.9. If S has the property ( S ) then so does the system S' = Exts({17)), where 17 is an arbitrary closed formula. S ' )A,, B E g k , l ( S ) ,cf. 1.4.4. Let b be a separatPROOF.If A , B E ~ ~ , ~ (then ing formula for A and B in S. We easily show that it is a separating formula for A and B in S'.

CHAPTER 11. APPLICATIONS OF THE

GENERAL THEORY

11.1. System So. The function of consequence for this system is simply the function Cno. Thus, apart from the existence of inessential constants, So is identical with the system A of [3]. 11.1.1. So satisfies axioms (A) - (E) and the condition Cno(Cno(X)) = Cno(X). The notions of recursive and recursively enumerable subsets of Ro,lare known. Subsets of R l , oare called recursive or recursively enumerable if they are unions of recursive (recursively enumerable) sets of neighbourhoods in the Baire space R,,o under the usual numbering of neighbourhoods. These definitions can be generalized in an obvious way to subsets of Rk.l for arbitrary k , 1. With these definitions we have:

476

1811, 37

FOUNDATIONAL STUDIES

11.1.2. S ? ~ . I ( S ~and ) gk:(S0)coincide with the families of recursively enumerable and of recursive subsets of Rk.1. 11.1.3. S%.t(Exts,(D+,(a))) and GP~,(Extg,(D&))) coincide with the families of sets which are recursively enumerable (recursive) in bp . 11.1.4. The family ~ , o : o , l (consists S o ) of the muppings f: Rl,o-+ o such that f(p) is a recursive functional in the sense of [4]; the family .9?&.o(So) consists of the mappings f : Rl.o4Rl.owith the following property: there is a recursive functional F with one functional and one numerical variable such that f ( v ) = 9 ( n ) [ W= ) Rep, n)l . The characterization given in 11.1.4 can easily be extended to functions in .9&m,a for arbitrary K, I, m, n. Functions of this family will be called recursive. 11.1.5. So satisfies conditions (A),(C), and (S). PROOF.( A ) is obvious from the properties of recursively enumerable sets. (C) follows from the possibility of expressing the relation of provability in So by an existential statement whose initial quantifier binds a numerical variable and has as its scope a formula which defines a recursive relation. (S) is proved as follows. Let A , B be recursively enumerable sets, let r ,A be formulas which represent them in So and let IT be a formula with the free variables x , y , b such that IT strongly represents in So the following relation P:

-

m is the Godel number of a formal proof of

5 from

Dq(a).

Hence IT(&, a,, a) E Cno(D,(a))if P ( m ,p, q), and n(s,,a,, a) E Cno(Dq(a)) if non -P(m,p , 9). Repeating the proof of Rosser [14] we show that the formula ( E X ) ( ~ (8X q-7 , ,b) & (x’)[z’< x

3

~ L ’ ( x &di, ’ , b)])

is separating for A and B. 11.1.6. Systems Extao(D&)), where a is an inessential constant of So, satisfy conditions (A),(C), (S). This theorem which results from 1.8.1 and 11.1.5 explains why sets recursive or recursively enumerable in arbitrarily given functions have properties similar to absolutely recursive and recursively enumerable sets. System So is known to possess various properties which do not follow from the general theory of Chapter I (e.g. the existence of effectively inseparable recursively enumerable sets). We shall not deal here with these properties since our aim is to show how much can be already obtained from the general assumptions made in Chapter I and not to develop the theory of recursive sets and their generalizations.

11.2. Systems S., These are systems obtained from So by the repeated use of the rule o. The precise definition runs as follows: For a set Xof formulas we put

Here R and

7c

are ordinals and R is a limit ordinal. We define S. as the system

(811, 38

471

REPRESENTABILITY OF SETS IN F O R M A L SYSTEMS

whose function of consequence is Cn.. By an easy induction on 11.2.1. S, satisfies conditions (A) - (E).

IT

we obtain

11.3. Constructive dehition of systems S,. Before we can discuss further properties of systems S. we must introduce some definitions. Let 9 be an arbitrary function and a a constant inessential for So. We denote by P a ternary relation which is recursive in 9 and universal for the family of relations (i.e. subsets of R0J primitive recursive in 9. Let W e be the set of e in o such that R,' is a nonempty well-ordering relation. The order type of R,' is denoted by lei'. From [9] it is known that

{lel':eE W'} = { I T IT: < of} = o:, where or is the first ordinal not constructible in cp. Along with the functions Cn, we shall consider auxiliary functions Cnf, where e is in W'. Whereas the definition of Cn, would be unacceptable to a constructivistically minded mathematician, the definition of 6:would be almost acceptable for him. I E a : ( X ) if and only if there is a function 5 whose domain coincides with the field of RP, whose range consists of sets of formulas and which satisfies the following conditions: (a) if no is the minimum of R.?',then 5(no)= C n o ( X ) , ; (b) if n is the successor of n, in the ordering R.?',then B(n) = F(B(n,)); (c) if n is a limit element of the field of RP, then F(n) = F ( j ) with summation extended over j such that RP(j, n ) and j # n ; (d) I E g ( j ) , where the summation is extended over the field of RI. If 'e E W', then there is exactly one function 5 satisfying (a) - (d) for each set X. 11.3.1. &:(X) = Cn,,,,(X)for e in W'. PROOF.By an easy induction on /el'. Let P be a formula which strongly represents R'" in Exts,(D'(a)); thus P has 3 free numerical variables and one inessential functional constant a. Let d(a, t) be the formula

u

u

= t , x , Y ) v P ( a , t , Y, 41) & [ p ( a ,t , x , Y)& P(a, t , Y, x ) = ( x = r)l&[P(a,t , x , y ) 8z P(a, t , Y , 4 = P(a, f, x , 4 1 & (P)(Ex)CP(a,f, P ( x + I), N x ) ) =

( x , Y , z)(CP(a,t , x , x ) & P(a, t , Y. Y)

[ B b ) = P(x + I)]))

.

This formula is obtained by expressing in the language of So the usual definition of well-ordering. 11.3.2. If e E W' and I e I' L 1 , then d(a, a,) E Cn,,,,(De(a)). The proof proceeds by induction on I e 1' and uses the fact that in Sowe can prove a formula expressing the well-known set-theoretical theorem sayin: that a relation well-orders its field if and only if every segment ,If the field is well-ordered. Kreisel and the author (independently of each other) have shown that there

418

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are integers e in W v such that d ( a , 6,) non E Cn.(Dv(a)) for z < I e 1". Kreisel calls such integers and the corresponding ordixials I e Iv "autonomous". Their existence shows that Theorem 11.3.3 cannot be strengthened by replacing I e 1' by a smaller ordinal. 11.3.3. There are formulas r(t,x , y , b, E ) and A(t, y , b, E ) such that for every formula 0 and every e in W v that following conditions are satisfied ( i ) if , c Iv 2 1 and R:(f,f ) , then @ E G X ( o , ( a ) ) = r(&, a/, are-, a, a) E Cnl,lcP(Dq(a)u &(a)) , (ii) if l e l v z 1, then @ E cnZ(Dq(u)) 3 A(&, arei, a, a ) E Cn,.lvp(D,!a) U &(a)) , (iii) if le iv 2 1, then 0 6 Tn:nlP(D,(a))= -A(&, b,a, a) E Cnlslvtl(Dy(a)u Dv(a))), (iv) A is satisfied in the standard model of So under the interpretation of t , y , b, E as e, [@I, q, (p if and only if 0 f &:(Dq(a)) . PROOF (IN OUTLINE). To construct the formula r we express in the language of So the arithmetized definition of the relation 0 E &Y(Dq(u)) . The formula which we obtain in this way has the form

(W)[ZI(t,E, 4) 82 Z*(t,E , 7, b) = 7/(2"(2Y + 1)) = 01 , where Zl and 2,can be described as follows: Zl(t,E, 8 ) is the formula

(u , V)[8(2"(2V + 1)) = 0 = P(E, t , u, u)l (i.e. 2, "says" that p(Z"(2v 1))= 0 implies that u is in the field of R:). &(t, E , 8 , D) is a conjunction of three formulas each of which gives necessary and sufficient conditions for the vanishing of p(2"(2v 1))in cases (a) when u is the minimum of R:; (b) when u is the successor of an element ul; (c) when. u is a limit element. The formulas describe (in the language of S) the three situations described in points (a), (b), (c) of the definition of G : ( X ) . Having constructed the formula r we show that (1") r is satisfied in the standard model of So witli t , x , y , E, n interpreted a s e, f , r01, (p, q if and only if 0 f z y ( D q ( u ) ). (2O; I f f is in the field of E , if r / is the order type of the segment of this field determined by f and if @ E ay(Dq(a)),then

+

+

r(&, 6f,are>, a, a ) E Cnmax ( 1 , r , ~@,(a) u Dda)) . The proof of (1") is straightforward and the proof of (2") proceeds by induction on r j . The implication from right to left in (i) results from (1") and the implication from left to right from (Z").'

* It is rather significant that we proved this theorem oy means of semantical considerations. Probably a syntactical proof would allow us to obtain much stronger results and in particular to characterize the family S T ~ z(Exts,(X)) , with an arbitrary X. We do not know, however, whether a purely syntactical proof exists.

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(ii) results from (i) by taking as A the formula ( E x ) r ( t ,x , y , D, E ) . (iii) is proved similarly as (ii), but uses the lemma A(E, t ) = (E!rl)[Zdt,€, 7)& 2 2 %

E, T , 0)l E Cno(0)

+

and Theorem 11.3.2. (To explain why the lower index in (iii) is I e 1 and not simply I e as in (ii) we remark that - A begins with a general quantifier and so we must apply once more the rule o in order to prove this formula.) (iv) is a direct corollary from (ii) and (iii). The theorems of this section have nonrelativised versions which we obtain by taking as (p e.g. the constant function 0. When referring to these nonrelativised versions we shall simply omit the index (p and the constant a (or the variable E) in the formulas A,r, and A. 11.4. Properties of S., n < ol. 11.4.1. Systems S. f o r 1 In < o1 satisfy conditions (A), (C,)and (S). PROOF.Let e be an integer such that I e I = n. (A) If 0, T weakly represent sets A, B C R k . 1 , then the formula A(&, 8rem3t a) v

-4%wail, a)

represents A u B in S,. (C,) From the nonrelativised version of 11.3.3 and from 11.3.1 (q, fi) E Mk.1 = $”(a) A(&,

4

E Cn,(Dq(a))=

a) E Cn.(Dq(a)),

where q(p) = r2ik “(a)1. These equivalences prove that if 8 represents g in So then the formula (Ex)[@(a, x ) & A(&, x , a)] represents Mk.1 in S,. (S) Let A, B c Rk,t,let 0 , T represent A, B in S, and let 8 be the formula

(Ex){P(&,x , x ) &k r(4, x, &wl, a) &k (x’)[P(&x ’ , x ) & (x’ + x ) 2 -r(&,x ’ , ha,-, a)]} . Using an argument similar to that of Rosser [14] we show that 8 is a separating formula; the difference between this proof and that of Rosser is that our proof uses the well-ordering Re instead of 5. 11.4.2. Systems S., 1 5 n < o1 do not satisfy conditon ( C ) . PROOF.Cn,(O) is representable (weakly) in S,. By 1.8.8 condition (C) would imply that LG3‘k,&S,) = 5%.1(Exts,(Cn,(0)) which is false, because Exts,(Cn.(0)) = Cn,.2(0). 11.4.3. The family .5%?o.l(Sr,)< , ol, consists of hyperarithmetic sets; every hyperarithmetic set of integers belongs to one of the families .5Zo,l(Sx), n
480

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11.5. Properties of the syskm Sp. 11.5.1. The family .G%.t(So) consists of IT: sets, i.e., of sets {q :(cp)R(cp,9)) ,

where R is arithmetic. PROOF. Evaluating the predicate %(a) E CnO(Dq(a)) we show easily that it is of type lIi, cf. [14]. Hence so are weakly representable sets. Weak representability of the lI: sets is an immediate consequence of Orey’s theorem 1121 (strictly speaking the proof given by Orey is applicable only to sets of integers, but a generalization to the more general case presents no difficulties). 11.5.2. So satisfies the condition CnQ(Cno(X)) = Cn,(X) for every set of formulas. PROOF. C n p ( X )is the smallest set containing X and closed with respect to the rules of proof of Se and to the rule o. In 11.5.4 we shall show that a cannot be replaced in $5.2 by any smaller 9 ordinal. The proof is based on the following important theorem discovered by Shoenfield (unpublished) and Spector [16]: 11.5.3. If X i s a set of formulas and x the characteristic function of the set of its Godel numbers, then Cnp(X) = Cn,;(X) . 11.5.4. Cno(D&a))= Cnq(D,(a)) # Cn,(D&)) for a < of. PROOF.The equation is an immediate consequence of 11.5.3. In order to prove the second part we consider formulas A(&, a ) defined in $11.3. The set of those of the formulas A(&,a) which are true in the standard model when a is interpreted as 9 is not hyperarithmetic in (p, since otherwise so would be the set W+‘which is known to be false [5]. Hence there is no z < of such that A(&, a)E Cnx(D,(a))for all e in Wq. On the other hand, A(&, a) E CnQ(Dq(a)) by 11.3.2. This shows that the sequence Cn,(D,(a)) is strictly increasing for n
11.5.5.

There is a formula E such that f o r each formula

@

I E Cno(Dq(a))= S(6rw, a) E CnO(Dq(a)) . cp.

PROOF.For simplicity’s sake we assume that q consists of but one function Take as E the formula

( W M E ,t )

Y9

E , E)1 .

If E(6r,i, a)ECno(D,(a)),then c” is true in the standard model when a is interpreted by 9 and hence there is an e such that A(&,a) and A ( & , 6 r 0 1 , a , a ) are true under the same interpretation of a. Hence e is in W p and (cf. 11.3.3 (iv)) I E CT!(D,(a)) E Cno(Dup(a)).3 If I E C ~ ~ ( D , (then ~ ) ) by , 11.5.4 there is a a < of such that @~ Ctz,(D ,(a)) and hence I E ar(D,(a)) for an e in W’. Using 11.3.2 and 11.3.3 (ii) we obtain B(Gro1, a ) E CtzO(D,(a)) . 11.5.6 SO satisfies conditions (A), ( C ) , and (S). (A) If I,?P weakly represent A , B in SO thrii the formula 3(8recail, a) V 8(6rc,,,i, a) weakly represents A u B in SO. (C) The predicate

i$?”(a)ECnQ(Dq(a)u

x))

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REPRESENTABILITY OF SETS IN F O R M A L S Y S T E M S

481

is of type II:, cf. [14]. Hence M,,,(X) is of type 17: and hence it is weakly representable in SO. (S) We limit ourselves to the case when A and B are subsets of Let I , Y weakly represent A , B in SO. Then the following formula 8 (W{A(E,t ) &i 4 t , 8 r e w i , f , 6)& (t')[P(E,t , t', t') 3 - 4 t ' , 8wwl, E , E)])

is a separating formula for A and B. To prove, e.g., that (p E A - B implies @(a)E Cno(&(a)) we choose according to 11.5.4 a n integer e in W' such that 0(a)E C&'(&a)) and then proceed similary as in Rosser's proof [14]replacing everywhere the less-than relation by Rr. 11.6. Shoenfield's rule. This rule' is analogous to the rule w. We say that I is provable from a set X of formulas by means of the Shoenfield rule if there is a formula W with exactly one free variable x such that

(x)[A(x)= PY(x)l= 0 E CndO) , W(8,) E X for every e in W . (W is here the (nonrelativised) set of integers such that R, is a we;:-ordering and A is a formal definition of W cf. 511.3.) Let Cn,' be a function analogous to Cn,, but based on the Shoenfield rule instead of the rule W . Let 8,be the system based on C d as the function of consequence. 11.6.1. Cn,'+,(X)= Exts,+,(X U Z0), where 2, is the set of formulas -A(&) with e nonf W . We obtain a proof of this theorem by showing that (a) every application of the rule o can be replaced by an application of the Shoenfield rule; (b) Zo E Cnf(0); (c) if P ( & ) E C ~ , ~ u , ( Zo) X for all e in W , then A(&) 3 W(8,)e Cns,(X u Z,) for all n. Theorem 11.6.1 reduces the Shoenfield rule to the ordinary w rule. As a corollary we obtain 11.6.2. 20 satisfies conditions (A) - (E) and the equation C n i ( C n i ( X ) )= Cni(X). 11.6.3. Sets weakly representable in 20 coincide with sets of type 17: relatively to W sets strongly representable in 8 0 coincide with sets hyperarithmetic relatively to W. The second part follows from the first. From the evaluation of the predicate @(a)~Cn~(D& u Z,) a ) which is of type 17: relatively to 2, it follows that sets weakly representable in zOare of type I7: relatively to W . I t remains to show that if A is of this type, then it is weakly representable in ZD. Let us assume that there is a relation Q recursive in W such that q e A = (+)(Ep)Q(W(p), q). If I strongly represents the relation Q ( Y ( p ) ,q) in Extso(Dxw(a)), then the formula Y : (E)(Ex)O(E,x , 0, a ) has the following property: q e A if and only if P is satisfied in an arbitrary w-model of So containing x , ~and q under the interpretation of I, as q and a as XW. It follows 4

Its use was suggested by Shoenfield in a conversation.

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FOUNDATIONAL STUDIES

WI, 43

that q E A = Y ( o ,a ) E Cno(D,(a) u a W ( a ) ) . Now let X ( a ) be the formula ( x ) { a ( x ) 5 1 & [ a ( x ) = 0 = d ( x ) ] } . In view of 11.3.2 Dx,(a) c - C n ~ ( 2 ,u {X(a)}) and hence qeA implies F(a, a ) E Cn,(D,(a) u 2, u { X ( a ) } ) , whence (E)[X(E)2 Y ( a ,t)] E Cn;(D,(a)). Conversely, if this condition is satisfied, then Y ( a , a ) is satisfied in the standard model under the interpretation of a a s xw and of a as q, whence q E A. Hence A is weakly representable in Zo. 11.6.4. 8,has properties (A), (C), and 6). (A) The union of two sets of type IT: in W is of the same type. (C) If X is a set of formulas such that the set of its G a e l numbers is a IT: set in W , then the predicate Zdk “(a) E Cno(X U 2, u D,(a)) is of the same type. (S) Since Extso(Dx,(a)) satisfies (S), the same is true of Exts,(Dx,&a) U { X ( a ) } ) ,where X ( a ) i s the formula used in the proof of 11.6.2 (cf. 1.8.9). Since, as can easily be shown, @(a,a ) E Cno(X u &,(a)

u { X (a)} )=

( t ) [ X ( E= ) @(a,El1 E C n d X u 2,) ,

we infer that [email protected](Exts,(X)) = 9k.r(Ext.yo(Dxw(a) u {X(a)}))which proves the theorem. CHAPTER 111. SYSTEMS WITH

NONDENUMERABLE SETS OF CONSTANTS

In Chapter 111 we shall try to obtain some parts of the “classical” theory of projective sets within the frame of the theory of representability. Systems described previously are not suitable for this purpose because the classical theory of projective sets treats each function (i.e. each point in the Baire zero space) as a n individually given very simple entity, whereas in the theory set forth previously a function can only be described by an infinite set of formulas. Following an idea of Kreisel [8] we shall consider systems whose language contains 2no constants serving to denote individual functions. We call these systems infinitistic. The language of an infinitistic system contains the same variables, constants, functors, and predicates as the language of So. In addition it contains 2*O Constants rp, where cp runs over the set of all functions. Logical symbols available in the infinitistic systems are the same as in the finitistic ones. The rules of forming expressions are similar in both systems, the expressions rp(x) and ~ ~ ( 8 being .) treated as number expressions. Similarly as in the finitistic case each infinitistic system is characterized by a function of consequence satisfying conditions (A) - (E) and the additional condition ~ ~ (=8 ~ E) Cns(0) for arbitrary cp and n . In the present paper we shall not discuss consequences of these assumptions in the general case, but shall limit ourselves to some particular cases. 111.1. Arithmetization. Since formulas of the infinitistic systems may contain constants 7P, we cannot use integers to arithmetize these systems. Instead we shall use a mapping of formulas on functions. A mapping of this kind can be obtained as follows.

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REPRESENTABILITY O F SETS IN F O R M A L S Y S T E M S

483

Let us add to the language of So a n infinite number of functional variables * . which cannot be bound by quantifiers. A formula 0 of an infinitistic system can be obtained from a formula tB0 of the system So (extended by the symbols v i ) by substituting functional constants rv for all the symbols p which occur in 0,. Let qj,, v,~, . . , v j r - , be all these symbols and assume that cb is obtained from O0 by substituting rvi for vji, i = 0 , 1, . . . , k - 1. Put 70,p I ,

+ +

d o ) = r@ol , d k n s 1) = vj,(n) , 0 5 s < k , n = 0,1, 2, . . . We take (p as a “functional Gijdel number” of 0 . Since 0, is not uniquely determined by 0 , every formula of the infinitistic system may have (and in general does have) many functional Godel numbers. Our definition correlates Gijdel numbers only with formulas which contain a t least one functional constant. To simplify our exposition we shall exclude other formulas from further consideration, replacing, if necessary, 0 by 0 8~(re = re). 111.1.1.

There is an arithmetically defined function f: R l , Q - ,R l . osuch that

if v is a Godel nuwber of a formula 0 , then f ( q ) is a Godel number of

-0.

Similarly f o r other connectives of the propositional calculus, f o r quantifiers and for the operation Sb(En/re)0,. 111.1.2. Godel numbers of the axioms of the propositional and predicate calculus, of the axioms of-So and of the axioms re(&)= constitute a set which is open in the space R l , o . 111.1.3. For every rule of proof the relation: the formula with the Code1 number bpI arises f r o m a formula with the Godel number cp2 (or f r o m formulas with the Godel numbers cp2, (pS) is arithmetically definable. 111.2. System Sa. Let Cn,”(X) be the smallest set containing X , all the axioms of the propositional and functional calculus and of the second order arithmetic, all the axioms which have the form re(&,)= &,,, , and closed with respect to the usual rules of proof. In the axiom schemata (e.g., of the prowsitional calculus) the schematic letters @, P,. . . are to be replaced by arbitrary formulas of the infinitistic system. Let S,” be the system in which Cnr is the function of consequence. A proof in S,” is a finite sequence of formulas. Via arithmetization we can enumerate proofs in S,” in such a way that a single function be the Gijdel number of a proof. The set of GMel numbers of proofs is arithmetically definable. From this remark it follows that: 111.2.1 Sets weakly representable in Exta,”(X) are analytic in the set of the Godel numbers of formulas which belong to X. In case X = 0 we have a stronger result: 111.2.2. Every set which is weakly representable in Sr and contained i n Rk.1 is open in the usual topology of Rk,,.The same is true f o r systems Extsr (X) undcr the assumption that the set of Godel numbers of formulas which belong to X is open in R,,o. The proof follows from the observation that small changes of the functions q for which occurs in a formal proof change the given proof again in a formal proof.

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111.2.3. Every open set A E R,., is weakly representable in Sr. If X is aconsistent, then A is weakly representable in Exts,”(X). The proof is obtained by expressing in Sr the definition ot an open set as a union of denumerably many neighbourhoods. In the formula obtained in this way a constant rV occurs, where cp is a function enumerating the neighbourhoods in question. It is an open problem whether S,” satisfies conditions (C) and (S); condition (A) is obviously satisfied. 111.3. Systems S,”.5 Using the same o-rule as in Chapter I1 we define as in $11.2 functions Cn;; these functions yield sets of formulas of the infinitistic system when applied to such sets. Let S be the system with Cn; as the function of consequence and let T,“ be the set of Gijdel numbers of the formulas provable in S7. Every proof in S: can be represented as a graph. Let G be a directed denumerable graph with one initial vertex V , and such that every vertex is connected with denumerably many vertices. We assume that G is well-founded. It is then easy to correlate with G an ordinal called the height of G. A normal covering of G is a mapping of its vertices onto pairs (4,cp) such that the following conditions are satisfied: (1)4 is the Godel number of a closed formula of the form (xk)O and rp is the Gijdel number of a proof in Exts;({(x&3}); (2) if V’ immediately succeeds V in G and if these vertices are mapped onto where 4 is the Gijdel number of (x@ and 4’ the G a e l pairs (+’,rp’),(+,rp), number of (xm)@’, then there is an integer n such that q’ is the Gijdel number of a proof of Sb(x,/G,)O; (3) under the same assumptions as in (2) there is for every integer n a vertex V” which immediately succeeds V in G and is such that the pair (+”, cp”) onto which V” is mapped has as its second member the Gijdel number of a proof of Sb(z,/6.)0; (4) in the pair (go,cp,) which is the image of the initial vertex the first member is the G d e l number of a formula provable in S,”. If V, is mapped onto the pair (4,,rp0),where cpo is the Gijdel number of a proof of 0,then we say that G together with its normal covering is agraph of a proof of 0. From these definitions we easily obtain 111.3.1 If K < 9 then 0 is provable in S7 if and only if there is a graph G of the height K such that G is a graph of a proof of 0 . This theorem together with well-known facts concerning analytic sets yields the following three corollaries: 111.3.2. The set T,” is analytic for each n < 9. 111.3.3. Sets which are weakly representable in S7 are analytic f o r K < 9. 111.3.4. Sets which are strongly representable in S;” are borelian f o r n < 9. 111.3.5. Every analytic set is weakly representable in S;”. We shall sketch the proof for analytic subsets of R,,,. If A is such a set then there is an open relation Q such that (P f

A

= (E+)(n,P ) Q&), P , cp) .

5 Results stated in this and the next section were obtained by the author in collaboration with Mr. L. Szczerba

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REPRESENTABILITY OF SETS IN F O R M A L SYSTEMS

Let 1y weakly represent Q in So" and let X(E,C) be a formula such that X ( r + ,73) f Cnf(0) and (E!E)X(r+,E) E Cn;(O) (cf. [lo], where this formula is denoted by "C is p ) . With these notations we easily prove that the formula (with one free variable r )

(EE,C ) ( X , Y)[X(E,C)8z V'(C(x),Y , r)l

weakly represents A in S. Let B, be the set of functions cp such that the relation cp(2'"(212 1)) = 0 is a well-ordering of type n and put BQ= U.OB,. Let Bord(F) be a formula obtained by expressing in the language of So the definition of the set Bo (cf.

+

1101). - _.

111.3.6. lf cp E B,, then Bord(r,) f Cnf..i.,l,(0)for n < 0 . The proof proceeds by induction on IT and uses the same remark which we mentioned in connection with the proof of 11.3.2. 111.3.7. TT # Tr+ifor n < 0 . PROOF.Otherwise all formulas Bord(r,), where cp E BQ would be provable in an Sr which would prove that the set BQ is analytic contrary to the wellknown theorem stating that this set is exactly C A . Cf. [7]. 111.3.8. Every Bore1 set is strongly representable in a suitable system S; with n
In the proof of this theorem we use notions of Lush's theory of sieves. Every Bore1 seta A is determined by a closed sieve W, (cf. [7, p. 3911) and the number of constituents of the complement of A is a t most denumerable. We can assume W, to be the whole space so that every cp determines a function 9 such that $(n) 5 1 for n = 0, 1,2, . . and the relation t9(2"(2n 1 ) ) = 0 is nonempty and orders the set of indices n satisfying cpE W,, . Writing the definition of rp in the language of So" we obtain a formula 8(E,C) satisfying the conditions 1

(E!C)@(r9, C)f C n X 9 ;

+

@(r,, 76)E Cn,"(O). Using these properties of 8 we can show that the formula (EC)[@(E,C) & Boyd (C)] strongly represents A in ST,where R is any infinite ordinal which exceeds the indices of all the constituents of the complement of A . Theorems 111.3.5 and 111.3.8 reveal an essential difference between the systems S, (a < 0,)and S," (n < a). Whereas the families SP~.&S,)strictly increase with a for x < ol, the families @,i(S,") are constant for 1 5 n < 0 . On the and are both strictly increasing. other hand, the families .%:(S.) An obvious corollary from this state of affairs is that Theorem 1.8.3 fails for the systems S," (1 5 n < 9)and hence that these systems do not satisfy condition (S). 111.3.9. Systems S," satisfy (A) and (C) f o r 1 I R < Q . PROOF.(A) is obvious and (C) results from the evaluation of the predicate @ c C n ; ( X ) which is analytic if X is a set of formulas whose Gijdel numbers form an analytic set. if p determines 9 in the above sense, then

-

6

We assume for simplicity that A

c Rl.o.

486

FOUNDATIONAL STUDIES

MI, 47

111.4. System SF. The properties of this system are very similar to those of SP. 111.4.1. S," is closed with respect to the rule o. PROOF.If G , is a graph of a proof of @(a,) in S,,, where K, < Q, then joining these graphs together we obtain a graph of a proof of (r)@(n)in S,", where K = sup K. 1. 111.4.2. S k , t ( S F )coincides with the family PCA. PROOF.Evaluation of the predicate cp E Tr reveals that Tr E PCA. Hence . S k . r ( S zE ) PCA. Now let A E PCA and assume for simplicity that A 2 It follows that there is an arithmetically definable relation Q such that cp E A E (E+)(e)Q(cp, e) . The right hand side of this equivalence can be transformed to (E+)(f(p, 4) E Bo), where f is an arithmetically definable function. Hence cp G A = (E+)[Bord ( T / , ~ . + ,E) CnF(O)] . If Ql strongly defines ill Sr the set {(cp, 4,s) i3 = f(cp, +I? , then

+

+,

which proves that A E L%?,o(S) . 11.4.3. Sz satisfies conditions (A) and (C). This is an obvious corollary from 111.4.2. 111.4.4. So" satisfies condition (S). The proof is obtained mutatis mutandis from the proof of 11.5.6. From 111.4.3. and 111.4.4. we can obtain various corollaries using the general theory of Chapter I. As an instance of such theorems we can quote the following: two disjoint CPCA sets are separable by means of sets which are simultaneously PCA and CPCA sets. BIBLIOGRAPHY 1. A. Ehrenfeucht and S. Feferman, Representability of recursively enumerable sets i n formal theories, Arch. Math. Logik Grundlagenforsch. vol. 5 (1959)pp. 38-41. 2. K . Godel, Uber die Ldnge von Beweisen, Ergebnisse eines mathematischen Kolloquiurns vol. 7 (1936)pp. 23-24. 3. A. Grzegorczyk, A. Mostowski and Cz. Ryll-Nardzewski, The classioal and the o-complete arithmetic, J. Symb. Logic vol. 23 (1958)pp. 188-206. 4. S. C . Kleene, Introduction to metamathematics, Amsterdam, North-Holland Publishing Co.;Groningen, P. Noordhoff N.V.; 1952. 5. , Hierarchies of number-theoretic predicates, Bull. Amer. Math. SOC. vol. 61 (1955)pp. 193-213. 6. S. C. Kleene and E. L. Post. T h upper semi-lattice of degrees of recur&ive unsolvability, Ann. of Math. vol. 59 (1954)pp. 379-407. 7. K. Kuratowski, Topdogie. I. Monografie Matematyczne XX, 2d ed. WarszawaWrociaw, 1948. 8. G . Kreisel, Set theoretic problems suggested by the notions of potential totalities, Infinitistic Methods (Proceedings of the Symposium on Foundations of Mathematics), Warszawa, 1961. 9. W. Markwald, Zur Themie der h s t r u k t i v e n Wohlordnungen, Math. Ann. vol. 127 (1954)pp. 135-149. 10. A. Mostowski, Formal system of analysis based on an infinitistic rule of proof, Infinitistic Methods (Proceedings of the Symposium on Foundations of Mathematics),

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Warszawa, 1961. 11.

-, A generalimtion of the incompleteness theorem, Fund. Math. vol. 49 (1961)

pp. 205-232.

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