Foundational Studies, Selected Works

+ P29.L + ..- + P$,<

n'.

I t follows immediately that for an i < s we have p,
...,

--+

The following table gives values of p(n) for lowest n: m 1 2 3 4 5 6 7 8 9 10 11 p ( n ) 1 2 3 2 5 3 7 3 3 5 1 1

12 13 14 15 16 17 5 1 3 7 5 5 1 7

n I 18 19 20 21 22 23 24 25 26 27 28 29 30 3 1 32 7 19 7 7 11 23 11 7 13 7 11 29 13 31 13

p(m)

I

We note the following properties of p ( n ) : &<"(n) 1 ;

(2) (3)

(4)

p(m) is always prime;

if

n92


(follows from theorem V);

p(m)=n if and only if m i s prime;

a d ~ 4 4 ,then p(n)>2.

Proof of (4): for n G 3 2 the values of p(n) are given in the table.For n>32 the left aide of (1)exceeds 2. From (4) and (1)we obtain immediately $6) The proof of eufficiency hae been given by Mre. W. Ssmielew; comp. foot-note 10).

[lo], 164

AXIOM

OF CHOICE FOR FINITE SETS

119

Theorem VII. Implication [2]+[n] holds if and only if n = 2

or n=4.

Another consequence of (1)and (3) is Theorem VIII. [m]! + [ n ] ! if and only if there is ru) prime p

between m and n. Proof. If mp by (2) and (3)and consequently even the implication [m]! + [ p ] cannot be true. If, on the other side, there is no prime between m and m, we have p(s)<m for o
28. We shb11 now prove Theorem IX. Condition (Y)i s necessary and s u f f i d e ~ ~ t for the implication [Z]+[n] in the following m e s : (i) 'n is prime;

(ii) n<15;

(iii) n=16, 18.

The proof is based on some lemmas. Lemma 13. [ r c k ] d [ k ] for every positive integers a and k. An easy proof will be omitted here. Lemma 14. If A has m elemeats and B n elements, A.B=O, a d if we know to realige the proposition [km+ln] where k a d 1 are

%on-negative integers not both 0 , then we may Ch008e an element from A+B. Proof. Consider the set A* of ordered pairs where a c A

...,

and i=1,2, k, and the set B* of ordered pairs <j,b> where b o B and j=l,2, 1. The sum A*+B* has k m + k elements, and we can by the hypothesis select a particular element p of A*+B*. p is an ordered pair whose second member belongs to A+B and may be taken as selected element of A+B. Lemma la 27). If p is a prime, A has n p elements (n= 2,3,4, ...), and if we kmw to redim ths propoSath [p], thm we can &fimeifedively a decomposition A=Al+Al into a sum of two disjoint non-empty sds. Proof. Let 2 be the class of subsets of A having exaotly p elements. From every X e d we can by supposition choose an element X*. For a e A denote by n, the number of X E A such that X*=a. Hence, the sum of all na is equal to the number of ele-

...,

-

*') This lemma and its proof are due to A. Tarski.

120

FOUNDATIONAL STUDIES

(101, 165

. Since this number is not divisible by p i P ! and the number n . p of all ?la is divisible by p , we follow that not all n, can he identical. Hence, denoting by A , the set of those a E A for which nu has the lowest possible value and putting A 2 = A - A , , we obtain the desired decomposition. Lemma 16. If p is a prime, A has nap elements ( n = 2 , 3 , ...), and if we know to realize the proposition [ n . p -11, then we can define A , into a Sam of a effectively a decomposition A = A , + A , + finite nnumber of disjoint sets of the power >1. Proof. Accordingly to the hypothesis, l o every a E A corresponds an element f ( a ) choosen from the set A - { a } . We have thus a function / ( a ) defined for u E A and such that f ( a ) + a . If the set of values of f coincides with A, then f is a permutation of A and can be decomposed into cycles. I n every cycle there is more than one element, because f(a)+a. If the number of cycles is greater than 1, they define a decomposition of A of the desired type. I f f is a single cycle, we consider f p instead of f and obtain a permutation for which the number of cycles is p > l , and in every cycle there is n > l elements. If f is not a permutation, we denote by A, the set of values of f. Sets A, and A - A , are both non-empty and we have a decomposition A = A , + ( A- A , ) . It is already of desired type, if A - A , has more than 1element ( A ,has never one element, because f(a)+ a ) . I n this exceptional case we have A - A , = { a } and may put A = ( & - Ma)))+ {a,f(a))merits of

2,i. e.,

$0

...+

24. We pass now t o the proof of theorem IX. Suppose that n and 2 satisfy the condition ( d l ) and that [Z] is true. If n is prime, 2 must contain a number of the form n-k and we follow by lemma 13 that [ n ] is true. I f n=4, 2 must contain a t least one number of the form 2i. Using lemma 13 we get the proposition [2] and, by theorem VII, the proposition 141. I f n=6, 2 must contain a t least one number of the form 2i and at least one number of the form S j . Lemma 13 yields propositions [2] and 131, i. e., the proposition [3]! from which we obtain[6] by theorem VI.

[lo], 166

AXIOM OF CtlOlCF FOR FINITE SETS

121

Let us suppose that n =8 and that A has 8 elements. 2 contains in this case numbers of the form 2k and 31+5m; we have thus propositions [2] and [31+5m] a t our disposal. Accordingly to lemma 15 we decompose A into a sum A=Al+A, of two non-empty disjoint sets. The notation can be arranged so that A, has a t least as much elements as A,. A, can therefore have 1, 2 , 3 or 4 elements. I n the first case we take the unique element of A, as the distinguishedelement of -4. I n the second case we con select an element from A, in virtue of [2]. I n the third case we may choose an element from A=A,+Aa using lemma 14. I n the last case we choose a n element a from A, and an element b from A, using the proposition [4], which is, as we already know, the consequence of [2]. We obtain thus decomposition A = { a , b)+ (A-{a,b)) and we may apply the same reasoning as in the first or second case. Hence we can always choose an element from A. Cases ?%=lo,n = 1 2 and 12x18 may be treated in similar manner as n=8. For n=10 2 must contain numbers of the form 24 5j, 3k+71, for n=12 numbers of the form 2i, 5 j , 5k+71 and for %=18 numbers of the form 24 3j, 5k+131, 7 p + l l q . Treating the case n=18, it is well to remember that [6], [8] and [9] are consequences of [2] and [3] (see theorem VI). A little more complex are cases n=9, 1 4 and 16. Consider first the case n=9. 2 contains then numbers of the form 3k and 2Z+7?n; we have thus propositions [3] and [21+7m] at our disposal. Let A be a set with 9 elements. Using lemma 15 we decompose A into a sum A = A , + A , of two disjoint non-empty aets and suppose the notation' to be arranged so that As has more elements than A,. A, may therefore have 1, 2, 3 or 4 elements. I n the first and third case we can immediately choose an element from A,. I n the second case we choose an element from the sum A = A , + A , using lemma 14. I n the last case we apply lemma 16 t o A, and obtain a decomposition of A, into a sum of a finite number of disjoint sets of the power >1. Since A, has 4 elements, only the decomposition A,=A'+A" into a sum two of sets of the power 2 is possible. Accordingly t o lemma 14 decompositions A = A ' + (A"+ A,) and A = A" (A'+ A,) define two elements a,b of A. We have thus

+

+

A = {a$) (A-{a,b)) and may proceed further as in the first or second caae.

122

[lo], 167

FOUNDATIONAL STUDIES

For a=9 the theorem is thus proved. If n=14, Z must contain numbers of t h e form 24 7 j and 3k+111. Remarking that 3k+111=3(k+21)+51, we follow that [8] is a consequence of [Z].Hence we have at our disposal propositions [2], [7], [8] and [3k+llZ]. Suppose that A has 14 elements. As in the foregoing cases, we decompose A into a sum A = A , + A , of two non-empty disjoint sets and suppose again that A, has at least as many elements as A,. The number v of elements of A, may therefore be equal to 1, 2,3, 4, -5, 6 or 7. In cases v=l,2,3,4,7 we can choose an element from A

without difficulty. If v = 6 we decompoee A, into a sum A, = A'+ A" of two disjoint non-empty sets using lemma 15 and proposition [2]. If one of the sets A',"' has 1 or 2 elements, the choice of an element from this set is already possible. If A' and AIf have both 3 elements, we consider the decompositions A = A'

+ (A"+ A,),

A = A"+ (A'

+ A,).

to which correspond two well-defined elements a,& of A in view of lemma 14 and of proposition [3k+111]. Hence A = W}+( A - ( W )

and we are in the same situation as for v=2. It remains the case v = 5 . A, has then 9 elements and we may apply lemma 16 to the set A, obtaining a decomposition (1)

A, = B,

+ B; + ...+ B,

into a sum of disjoint sets of the power >1. Let us denote by b, the number of elements of Bi (i=l,2,...,q) and by b the least of these numbers. If not all br are equal to b, we may decompose A, into a sum of disjoint non-empty sets A, = A;+A;, taking as A; the sum of those B, for which bi=b and as Xi the sum of the remaining Bi.Arranging now the notation so that Ak has less elements than A& we obtain the decomposition A=A;+(Ai+A{)

[lo], 168

AXIOM OF

CHOICE FOR FINITE SETS

123

in which A; has 2 , 3 or 4 elements and return so t o cwa v=2,3 or 4 discussed previously. If all bt are equal to b, then b=3, q=3 and (1) takes the form A,,=B1+Ba+Bs where Bl,B, and Bs have 3 elements. Consider the decompositions: A = Bi (Ba+Bs+ 4, A = Ba+ (&+ Bi Ai), A = Bs (Bi Ba Ad, to whch correspond 3 elements a, b,c of A in virtue of lemma 14 and of proposition [3k+111]. We write now down the decomposition

+ + + + +

A = {a,b,c)+ ( a - { a , b , c ) )

and return so to cases v=l,2,3 in which we can already accomplish the choice. Case *=14 is thus discussed in full. I n case n=16 the reasoning is the same &B for n=14.Z contains in thia case numbers of the form 2i, 3j-t 13k and 5E+llZ. Theorem IX.is thus proved completely. Results of this section suggest a supposition that condition (bd) is in every cwe sufficient for the implication [Z] +[*I. I was not able to solve this quation even in the case n=15 and 2 ={3,6,13).

ON ABSOLUTE PROPERTIES OF RELATIONS ADIDRZEJ MOBTOWSKI

1. We shall be concerned in this paper with properties of relations. For simplicity we shall consider only two relations, a binary relation R and a ternary relation 8,but the generalization to the case of my number of any relations presenta no difficulty. We shall denote by A0 the field of the relations R and S and aasume that A0 is infinite (not inductive). The properties to be dealt with are expreseible in a symbolic language L whose principal features will be sketched below. The variables of L are of Merent types: (1) variables “z”,“y”, “z”, * . . representing elements of A . , (2) variables “X”, “Y”,“Z”, . representing subsets of A,,, (3) variables “K”, “p’, . representing sets of suhtm of Aa , and so on. We shall consider variables of only these three lowest types, in order to simplify our notations, but it will easily be Been that this restriction is nonessential. The simplest u*ell-forrnedformulas are

“a”, - -

R(z, Y),

Sb,Y, d,

%ex, X&

where the letters 6 ( z 9 9 , ( 6 1 1 , c r Z ” , iCX97, 1 <E 2 9 stand for any variables of their type. Y Other w. f. formulas are built up out of these by means of the stroke ‘ I / ” and of the existential quantifier “3”applied to variables of any type. It is well known that other connectives of the propositional and functional and also that any variable occalculi may be defined in terms of “I)! and “I”, curring in a w. f. formula is either free or bound. A w.f . formula is said to be elementary if all of its variables (free and bound) are of lowest type, otherwise non-elementary. 2. It is clear that any w. f. formula P without free variablesdefines a property of R and S . This property will be denoted by the italic letter P. It is elementary or non-elementary according to the nature of P. We write P(R, S) t o indicate that R and S have the property P. A property of relations R and S is said to be definable if it is identical with P for a suitable w. f. formula P.’ We shall consider in the sequel a property @ of relations R and S expressible as the conjunction of a finite or denumerably infinite number of definable properties. G(R, S) means that R and S have the property @. Let A be a subset of A0 , % a set of subsets of A and a a set of subsets of 8. The ordered triplet (A, %, a) will be denoted by a single letter “M”and called a model. If A contains all elements of A . , 9l all subsets of A0 ,and a all subsets of %, then M is called an absolute model and denoted by Mo = ( A @ 9, 1 0 , m). Let P be a w. f. formula. If we replace in P the quantifiers “(Elz)”,“(ElX)”, h i v e d Decamber 18, 1946. P(R& could be read: R and S sstisfy the w. f. formula P (“erfiillen die Aummgefunktion P”). See A. Taraki, Dw Wahrheiibbcgridi in dcn formali&rtm Spruchm, S W Q j&i&sophica,vol. 1 (1936). pp. 281-406. Several notiom defined -hat vsgnely in fiw, 8,4 aan be expreeaed exactly with the help of the I L O ~ ~ O Mof sstkfaction and of truth llis dehed by Tmld.

,

[14], 34

ON ABSOLUTE PROPERTIES OF RELATIONS

125

“(3X)”respectively by “(3z)(zul).”, “ ( 3 X )(XcPI).”, “(3X)(Xea).”, we obtain a new w. f. formula P, and a corresponding property P M depending on M. The passage from P to P M is described aa a relativizstion of P to the model M. It is plain that P,, = P. If we relativise to M the properties P whose conjunction is 9,we obtain a new property (P, , called the prpperty relativized to M.

3. As Skolem pointed out: the meaning of any non-elementary property P depends on the meaning attached to the word “set.” In other words we cannot expect in general that the properties and a M are equivalent. This is of course not surprising as long as we leave M quite arbitrary. But suppose that M fulfills a finite or denumerably infinite system A of set-theoretic axioms, i.e., w. f. formulas without free variables and containing neither “R” nor “8”. One might conjecture that it is possible to choose these axioms so that they will characterize completely the notion of set, or at least that for a suitable A any model M fulfilling A will be indistinguishable from the absolute model within . our formal language L. 9, would then be equivalent to Let us write A, to indicate that M fulfills A, i.e., that all the w. f. formulas of the system A yield true sentences when we relativize them to M. We may then express the above conjecture by the formula (1)

AM I M

[*Af,(R, fl

*df(R,

S)l*

We shall say that 9 is an absolute property (with respect to A), if the formula

(1) holds for 9.

The purpose of this paper is to give a necessary condition for the absoluteness of 9 in the above sense. As a corollary we shall establish the existence of nonabsolute properties. 4. Let us suppose that the absolute model Mo fulfills all axioms of A? (2)

It follows that A contains the axiom of extensionality, (3)

( X , Y,X)

(5)[Z&X

= ZEYI =, [XEX = YEXI)

1 Th. Skolem, Uber einigeSatzjunktionenin der An‘thmetik. Skrifter utgitt m D e t Norrke Videnskaps-Akademi i Oslo, I. Mat.-naturv. kl. 1930, no.7 (1931),p. 7: “Die eben bewiesenen %tee werde ich jetet anwenden um etwaa Licht auf die Schwierigkeiten zu werfen, auf die man stiisst, wenn man versuchen will, den Begriff der g m e n Zahl vollstiindig zu charakteriaieren. Bekanntlich muss man hiersu u. a. daa Priuzip der vollstsndigen Induktion benuteen, oder wenn man will das Wohlordnungsdom: In jeder Menge gamer poeitiver Z d e n mit mindestens einem Elemente gibt es eine kleinste Zahl. K i j ~ t eman nun dem Begriff“Menge” einen absoluten Sinn beilegen, so wSre die Charakterisieruq volletandig und fruher hat man sich die Sache immer 80 vorgestellt. Seitdem man aber damber klar geworden ist dam such der Begriff “Menge” selbst einer Begrtindung-etwa eher axiomatischen-notig hat, und dass man dadurch zu einem allgemeinen Relativkmua gefflhrt wird, 80 liisst sich jene Auffaasung nicht mehr aufrecht erhalten; die Moglichkeit eimr vollstiindigen Charakterisierung der Zahlenreihe ist jedenfalle eweifelhaft.” * A weaker assumption of self-consistency of the system A* (see below) would be su5cient.

126

WI, 35

FOUNDATIONAL STUDIES

(if there were variables of higher types in L,we would have this axiom alrso for higher types), and the axiom of infinity, which we do not need to write down explicitly. Instead of A we shall now consider a system A* of first-order axioms with constant terms “E”,“T,,”,“T,”, “T2”. A* contains the axioms (z>[To(z) v TAz) v Tz(41,

(4) (5)

( z ) [ T d d 3 4’t(z)l,

i # j, (6) (2,Y>[Ti+l(Z).YEZ 3 Ti(Y)I, i = 0, 1, a d further all the w. f. formulas (sentences) P* which arise from axioms P of A if we perform on them the following operations: the quantifiers “(3z)”, replaced reape~ti~dy by “(~z)T~(z).”,“(ZlX)Ti(X).”, “(3X)”, “(33E)” “(3?3T2(X).”,and “.? by “E”; the variables of any type are allowed to mge over the whole domain of individuals. Note that if P is the axiom (3), then P* is equivalent to

(7)

( X , Y , 33 { T ~ ( X ) . T ~ ( Y ) . T ~ ( ~ ) . Q3 [ T(zEX O ( ~ ) zEY)I 3

Ixn = Y r n ] )

The new system A* is self-consistent. This follows by the c l d c a l interpretation method. Define the predicates TO, TI, T2,E for z, y E A0 $a aa a3 follows: To(z)= z is an element of A0 , Tl(z) = z is a subset of A. , T2(z) = x is a set of subseta of Ao , zEy = z is an element of y. Then take A . pl0 a. as the range of quantifiers in the axioms of A*. It is TI,T2 , E easily seen that the assuniption (2) implies that the predicates TO, fulfill all axioms of A*. It follows now by the theorem of Skolem-Lowenheim‘ that there are predicates T o , q ,T,, E with natural numbers as arguments which fulfill the axioms of A*. Hence E is a binary relation between natural numbers, and T o , T I , Tz are sets of natural numbers. Note that TOcannot be finite, since the axiom corresponding to the axiom of infinity cannot be fulfilled in a finite set. Put a. = z for z e T o , a. = (&u)(yEz)for zeT1 and a, = (dJ(yEz)for zeT2 . In view of (4)-(6), a, is defined univocally for every integer x; it is aa integer if %&To,a set of integers if n T 1 , and a set of sets of integers if zeT2. Let A be the set of all a, for z&To; 91, the set of all a, for zeTl ; and a,the set of all a, for zeT2. And put M = (A, 8, a). We shall show that

+ +

+ +

(8)

a,ea,

= zEy

for zeTj and yeTj+l,

j = 0, 1.

This follows immediately from the definition of a, if j = 0. If j = 1, yeT2, and xEy, then a,&% by the definition of a, . If a=&%,then a,e(d,)(uEy) and 4 Th. Skolem, Uber einige Grundlagenjragen dsr Mathemcrtik, Skiifter ufgitf av Det Norske Videnrkaps-Akademi i Oslo, I. Mat.-naturv. kl. 1929, no. 4 (19!29), pp. 23-29.

11-41, 36

127

ON ABSOLUTE PROPERTIES OF RELATIONS

.

we infer that there is an integer u such that uEy and a, = a,, By (6) ueT1 . If VEX, then aocasand therefore ay&au; (8) being true for j = 0, we obtain VEU. We show in similar manner that if VEU,then VEX. Hence (u>[ueTo3 (VEX3 vEu)]. By (7) we obtain now XETZ3 (xE3E E uEX) and therefore xEy = uEy. Since uEy, we obtain finally xEy, and (8) is proved. From (8) we get the following more general result: Let P ( z , y, . . , X , Y , ,X , D , . .) be a w. f. formula built up from the elementary w.f. formulas

.- -

-

-

zEX, XEX

(9)

by means of the stroke “I” and the quantifiers “(9x)To(x).”,“ ( 3 X )Tl(X).”, “(X)T&).”. Let arise from P* by replacing “E” by “E” and the quantifiers “(9z)To(x).”,“(3X)T1(X).”,“ ( 3 X ) T&€).” respectively by “(32)(z~A).”, “(aX)(XeQt).”,“(9X)((xca).”. Then

X E T , , ~ E T. ..XET~.YETI.. ~, . ..TeTo.’&Tt.. . . 3

(10)

P*(x,y,

. , . , x , Y , . . . , X , 9, . . .) = P O ( a za,, , . . . ,a x , a p , . . . ,a t , ab, . . .)

This is shown by induction on P*. If P* is one of the w. f. formulas (9), then (10)follows from (8); and if we awume (10) for two w. f. formulas P* and &*, we obtain by easy logical calculus the same equivalence for the w. f. f o ~ u P* l I~&*, (3%) TO(X).P*, TI(X).P*, (3%) T*(X) .P*. If P* has no free variables, then (10) yields:

(ax) P*

(11)

= pa.

Now let P be any axiom of A. Pass from P to P*; P* is true for predicates E, T o , Tl , Ta, and hence by (11) Po is true. But Pois exactly the m e as the Pu of $2. Hence M fulfills all axioms of A. We have thus proved the following theorem : Ti) There i s a model M = ( A , a, a ) in which A i s an infinite set of positive integers, 8 a denumerable set of subsets of A , and a a denumerable set of subsets of % SU& that A M . Without loss of generality we may assume that A is the set of all positive integers. 5. We shall denote by No the set of positive integers, by CJt the set of all subsets of No, and by v the set of gll subsets of %, and we put N = (NO,%, v ) . C will denote the discontinuous set of Cantor, i.e., the pet of realnumbers x = ~ ~ 1 c i - 3where - i ei = 0 or c; = 2 for i = 1, 2, 3, . . . . We put c;(x) = ci(n x ) = ci (the ith digit of 5 ) . The function y = ci(z) is continuous. The functions +(m,n ) = 2m-1(2n - 1) and #(m, n, p) = +(+(m, a),p) establish a one-one correspondence between positive integers and ordered pairs or ordered triples of such integers. If q = g(m, n), we put m = dq), n = &); if q = +(m,n, p ) , we put m = ~l(n),n = A d ,P = m(d. We now let correspond to every binary relation R who* field is NOthe real

+

128

FOUNDATIONAL STUDIES

+

number xu = 2 C&ci-3-', where ci = 0 if -R(s&), m(i)) and ci = 2 if R(rl(i),rz(i)). The set of all possible x R will be denoted by a: ZED2 = x-4eC. For a ternary relation s we put sixnilarly 38 = 2* C7k-3-',where ci = 0 if -S(pl(i), &, p s ( i ) ) and ci = 2 if S(pl(i),~ ( i m(i)), ) , and we denote by DI the set of all possible xa: XED,= x - 8 e C.' The relation R for which xu = t will be denoted by R, . Rt is defined only for t e D2 D8 and is binary or ternary according as tcDt or teD8. From the equivalences (tcD&Ra(m, n) = (1 - 4 c C).(C+,.)(t) = 2) and (te&).Rt(m, n, p ) = ( t - 8 c C).(c~~,,..,,,,,~(t) = 2) we obtain immediately:'

+

+

(ii) The sets' D 2 . E t [ R t ( mn) , ] and D*.Et[Rt(m,n , p ) ] are dosed. his result can be generalized as follows?

(iii) If

+ is an elementary property of relations, then the set

(12)

ErJrC&.sCLbQid%

R31

i s Bmelian and its class is at most o. Prooj. Let P(R, S, x, y , . . . , u) be an elementary w. f. formula. shall show that if m, n , . . . , q are positive integers, then the set

(13)

We

E n [ ~ ~ D a s e l ) J . P d RR,,,, m,n, . . . , q)l

is Borelian and its class is finite. This follows immediately from (ii) if P is one of the w. f. fOrmUleR R(z, y), S(z, 31, 2). We proceed now by induction on P . If the theorem holds true for two w. f. formulas P'(R, S, 2, y, . . . , z, u, . . . , w) and P(R, S, x, g, . . . , z, s, . . , , t), it holds ale0 for the w. f. formulas Q'(R, S, y , , . , , z , u, . . . ,w ) = (3lz)P'(R,S, Z,y, . . . ,z,u, . . . ,w)and Q?R, S, Z, . . . , z , u, . . . ,w,8, . . . ,t ) = P'(R, 8,Z, . . . , w)~P'(R,S , x, . . . , t ) , since

Era[TdhcDaQk4Rr,R., n, . . . ,dl = Er.[~cD2.seDJ.(3~)(~eNo).P:(R,, R,,Z,n,

. . . , q)I

=

C : l ~ r , [ ~ ~ ~ . ~ ~&, ~ dm,% 12, .R ..r ,dl ,

and

E,,[reLseDaQ:(R,, R , , m , . . . , p , k, . . . ,I , h, . . . ,i) D~ x D~ - E,,[P%,

R. ,m, . . . , P, k, . . . I

01 +

=

Ds X Da - EJPe,(R,, Rapm, . . . P , h, . . . ,i)I

+

8 In order to discuss the properties of sets, we denote by D I the set of all numbers 2 E where E E C and put for r E D I : R, = E k ( r ) = 21. a We use here several theorems concerning Bore1 sets. They can he found, e.g., in the monograph: K. Kuratowski, Topologie I , Monografje matematyczne, Warsaw 1938. 7 E .,[. ..I denotes the set of (2, v ) which satisfy the condition [...I. * K. Kuratowski, Fundmenfa mathemuticue, vol. 29 (1937), p. 99.

~41~38

129

ON ABSOLUTE PROPERTIES OF RELATIONS

Hence the set (13) is Borelian of finite clam for any P. The set (12) being the common part of a finite or denumerably infinite number of sets like (13), we infer a t once that it is Borelian and its claw is a t most o. (iii) is thus proved.

If Q, is not eIementary,the set (12) is not necessarily Borelian. As a matter of fact, it may even not be projective. However, we have the following: h m a . If Q, i s absolute (mathrespect to any A), then the set (12) is Boreliun and its clcrss is at most o. 6. We shall define a correspondence between real numbers and subsets and sets of subsets of NO. For z = 5 c7-1ci.3-i (where ci = 0 or ci = 2 for i = 1, 2, . . .) we put S(z) = E,[c. = 21. For z = 5' T-lcr.3-' (where ci = 0 or cs = 2 for i = 1, 2, . . .) we put em(%)= 5 ~ + ( , , , s ~ - 3 - iand c7-1ci-3-i 6(z) = ( S ( e l ( z ) ) ,S(Q(Z)), . . . , S(e.(z)), . . . ). For z = 5 (where ci = 0 or ci = 2 for i = 1, 2, . . .) we put e.(z) = 5' ~b1cg(n,i)-3-' and u(z) = (6(el(z)), E;(cl(z)), . . . , e(e,,(z)), . . .). (Had we more logid types in the language L,we would continue these definitions in an obvious manner). If x = 5' E: and g = 5' q where E : dand ~ E Cwe , put M ( z , y ) = (No,

+

+ +

+

e(4, .b)).

+

+

+

(iv) The functions y = e.(z) and y = ~ ( z are ) cuntinws. (v) The sels & = E.[z - 5 E C],L%= E& - 5' are Borelian (closed).

E

C], and

68

=

E,[z - 5'

E

C]

(vi) Theseb

&

=I

E,[zENo. ~

g1 = E,[ze&. y d l . S(Z) = S(y)],

~ 6 1 E . S(y)], 2

g1 = E , [ Z E ~ ~y&eZ. . S(Z) E e ( y ) i , = E,[ze&. ye&. S(z) E

&)I,

4

=

E,LZEB~. y € e 2 . ~ ( 2 = )

9 s = E,[z&.

ye8a. u(z) =

sb>i, &)I

are Borelian and their classes are finite. Proof. (5, y) E G, = zEN*ye%[c,(y) = 21, hence Go is closed. (z,y) E 91 = ze61.yeOl.(n)[(n,z) E 60 5 (n, y) E &,,I, hence 9, is a G I . We have further (z,y) E El I zEel.y& . (3n)[n."S(z) = S(en(y))]= zet%.ye&.(3n)[nEN0. (2, en(y))E &I. The function en(y)being continuous, we infer that & is a Gi, . From equivalences (z, y) E 9 2 = (s&e2.y&e2.(n)[S(e,(2)) E a(y)l.[S(en(y))E S(z)]) = (ze6?.yee,(n)[(en(z), y ) E &1].[(e.(y), z) E GI]} we infer further that 4;is a G- . The proof for GZ and 9 8 is &mil&. (vii)'IfA C N o , there is an zcel 4 that S(z) = A . If % C % and % i s finite or denumerable,there i s a ye02 such that S(y) = 8. If cu C v and cu i s finite or humerable and wery element of a i s finite ur denuwable, there 2s a z& such flMtU(2)

= a.

Proof. Let nl < Q < . . . be a finite or infinite sequence built up from all rW2-3-"', we obtain zeOl and S(z) = A. elements of A. If we put = 5 Let A , ,A2 , . . . be a finite or infinite sequence built up from all elements of 8. From the first part of the theorem we infer that for every n there is an znE&

+

130

~ 4 ~ 3 9

FOUNDATIONAL STUDIES

such that A,, = S(z.). Obviouely and Cr,(+(,,Q,(Zrl(+(n,0,)'3-i

9(zm) = Am and

+

Let US put' 1 = 5' ~ ~ C r ~ ( ~ ( Z r l (-3". * > ) We have 5 xL~+(n.~b).3-' = 6 =: 5 CLIC~(G)*~-' f : x n a d therefore s(e,,b)) =

+

em(@) =

+

+r a

6 0 = ( S ( e M ) , S ( e W , .. ) = ( A , , AS, - - - I = P[. Let P[,, a[, , . . . be a finite or infinite sequence built up from all elements of a. As we have just proved there is for any n a y.f& such that a(y,,)= %, . Putting z = 6' ~ ~ l c r , & r l ( ~ ) . 3we - i prove exactly tu in the foregoing 0

+

case that Z&, e,,(z) = y. and therefore u(z) = { W, , %* , . . .1 = a. The propsition (vii) is thus proved.

(viii) The a t H = E+r[~f&.~~&&(t~~] is not cnrpty. ) is a model M = (No,PI, a) such that W ie a denumerable Proof. By (ithere set of s u b t s of No and a a denumerable set of denumerable subsets of 8 and A,, . By (vii) there are red numbers z& and @ such i that a(%) = P[ and which proves ~(y)= a. Hence M ( z , y) = (NO,3,a) = M ,and therefore that (2,y) e H. Hence H is not empty. (ix) The set K = E , [ z r ~ y ~ ~ r f ~ s f ~is ~Borelian , , ( ~ of . ~dcuM ~ ] o d

most.

...

Proof. Let P(R, S , z, y, - . * , z , X, Y , , 2, 3, B, ... , 8)be a w. f. formula with the indicated free variables. We shall show that for any integersh,i,...,j,k,l,...,nr,n,p,..-,qtheset ( 14)

Jp = d p ( h , i ,

... ,j , k , l , - - ., m , n , p ,

E , I z f ~ y f ~ r r ~ s E P Y ( z . y )R. ( R,A, r , i,

- - ,j ,

6 . -

,q) =

*

S(e&)), SMz)), . . , S ( e d z ) ) , G(G(Y)),@(dd), * * , @(e,(v))l is Borelian and its class is finite. Let US proceed by induction on P. If P is the w. f. formula %EX,then a

Jp(h, k) = Elyr.[zet&.Y E & . ream 8fD1. hfS(ek(z))] =

E , [ z f 6 * . y f L . r f ~ . 8 E D 1(h, . e&)

E

601,

and hence Jp(h, k) is Borelian by (iv), (v), and (vi) and its class is finite. If P is the w. f. formula X f 3 , then J P ( k ,n) = EWr,[zE&. y&.

@(eh)>l SEDS.(eke), e,(y)) f &I.

re&. s e a . S ( 4 d )

= EW,[ze&.ye&. re&.

E

Hence Jp(k, n) is Borelian and its class is finite by (iv), (v), and (vi). is the w. f. formula R(z, y), then J P ( h ,i) = Eryrs[z~e2. geeS.rt&. seDI. R,(h, i)] = E,,lze02. yet%. rfDZ.scDt. (c,(t.,a(r) = 211. 9

Use is made here of the axiom of choice.

If P

132

[141,41

FOUNDATIONAL STUDIES

CPN(R~ , RJ = *~u(q,.ro)(Rr , R3 3 (20 9 vo 3 r, 8) E K. Hence the set E,[%N(R,, R,) z = = YO] is an intersection of the fourdimensional Borel set K whose class is at most F, and of the plane z = % , y = yo . The set E,&(Rr, R,)] is therefore Borelian and its class is F ,

.

.

.

7. We shall apply the theorem to mme particular properties of relations.

(a). Let BO(R) = R is a well ordering relation. The set

0 = Er[rdh.BON(R~)] is known to be an analytic complement and not Borelian." It follows.that there is a model M such that A M and -(R)[BON(R) -= BOM(R)],i.e. (3R)[BOw(R). -BON(R) v -BOM(R) Box@)]. But it is easy to ~ e that e BON(R) 3BOU(R) and hence (3R)[-BON(R) BOM(R)].This means that there is a model M which fulfills all axioms and a relation R with field NOsuch that there are decreasing sequences" . - . n&nzRnl but no such sequence is in M. (b). Let R smor S be defined aa in Principia muthemutica, *151.01. The set

.

.

E&&&

(15)

.

SE&

.R, smor R.1

is analytic and not Borelian. In fact

R, smor R,

E

-

(*)[ZEDS R,

.

E

.

1 -+1 R, = &Z,IRrIRz].

The set E,,[zEDI. R. c 1 -+ 1 R. = &R,IR,] being Borelian, we conclude that the set (15) is anslytie. If this set were Borelian of class a, the set At = E,[r& R, is of type €1 would be Borelian of class a for any ordinal number [ (sincethere are s such that R. is of type [). Hence the set 0 would be expressible as a sum C r < o A c ,the At being Borel sets of class a. This is impossible, since 0 is not a Bore1 set." It results now from our theorem that there is a model M such that A M and two relations R, S such that -(R smorN S E R smoru s>. Since R BmOrM S 3 R morNS , it must be that R smorN S and -R smoru S, i.e., there are oneone relations T such that R = SIT but no such relation" is in M.

.

(c). Let F(R) mean that the relation R is well-founded, i.e., that there ia no decreasing sequence n&*Rn:. If the set E,[~&&FN(R)] were h d h , the set 0 would be also. Hence F(R) is not an absolute property. (d). It is easy to prove that the property of being inductive is an atwolute property of a set. Writing Fin(A) to indicate that A is inductive, we have n elements]. Hence the set E[reDl. E,[mD1. Fin(R,)] = xzJCr[r&D1.R, Fin(R,)] is Borelian of finite class. See K. Kuratowaki, Fundamento methematicor, vol. 29 (1937), p. 58. is defined 88 the set {#(a1, 11, +(na , 2 ) , +(n, 31,. sequence nl , n, , n~ , 1) K. Kuratowski in the paper cited in footnote 10, page 66.

10

11 A 11

-.

I.e., the set E+~,.,JmTnlis not in M.

,

*

1.

[141,42

133

ON ABSOLUTE PROPERTIES OF RELATIONS

Consider now the eimilrvity (equality of power) of two sets. The set E&&. x D 1 . R, sm R,] is Borelian of finite class, since it can be expressed aa a sum

.

Cmd{E,[r&Dl.R, haa n.ekmmts] X E.[s&D1 R, h a n elenenfa])

-

+

E,[r&Dl.Fin(R,)] X E*[S&l. Fin(R,)]. DI X This example shows that the theorem converse to that proved above is pmb&ly false, Bince the relation R sm S e m s not to be absolute. UIPTPEBSITX

or wAB8Aw

On the principle of dependent choices. BY

Andrzej

M 0 s t o w s ki (Warszawa).

Let us consider the following weakened form of the axiom of choice:

(X)

I

if 11'is a binary wlation and B a set +O aml if for every s e l l there is a y c I3 s w h that xBy,then. thfre is asequence x1,x2,...,xlI,... o j elptjitnts of H such that x,KxlI+~ for n = 1 , 2 , . . . I ) .

It will be proved here that the yencral asiow of choice (which we shall denote by (2))i s independmt of ( X ) , 1. e . , cannot be proved from (T) and the usual axioms of set-theory. An independence-proof has sense only with resppeet to a well defined formal system whose consistency is either proved or assumed as an hypothesis. Our proof applics only t o such systems of settheory as remain self-consistent after adjunction of the following axiom (AT)

there i s a %now,-dtmtnierable set of e l m f a t s which are .not sets.

It can be shown without difficulty that the system G described in one of my former papers 2, satisfies t'his condition. Hence we shall ta,ke 6 as a basis for our proof. I n order to prove that (2)is independent of (T)we have to coiistruct in a self-consistent theory GI a model in which all the axioms of 6 as well as the axiom (T) are fulfilled and in which the axiom of choice is false. *) This axiom ha5 been considered

by A . Tarski

iti

his recent paper

i l zionzatic and algebraic aspects of two theoren48 ow sums oj cardinals, this voliiiiie, p. 79-104. T:irski calls (2') the priiicipls of drpedent choices. L~ Fiiiit1;inwiita illntlieniatirae 32 (1939),pp. 201-253.

[15], 128

135

ON THE PRINCIPLE OF DEPENDENT CHOICES

We shall take as GIthe system 6 enriched by the axioms (N) and (2)but we shall make free use of many notions known from intuitive set-theory without defining them meticulously with the help of primitive notions of 6. Since it is known that the eonour result can be stated as follows: sistency of 6 implies that of GI3)), If G is self-consisttnt, then the iinplicatiola (!P)+(Z) i s not procable in 6. Let A be a non-denumtrable set of pairs {as,bs}

s c

S

whose elements are not sets. Lct A', be the set of all u,'s and a.11 b,'s:

..>

KO={. ..,as,b,, . and l e t Kt be defined by induction on

6 a s follows

(!$(X)=set of subsets of X). If f is a one-to-one mapping of h ' , onto itself and tji E h-,), then f ( m ) is defined as the value of f for the argument m . Suppose that f(n) is already defined for n ' E , and that rti K c - z K , . 4<5

We have then ? , ~ C ~ a,nd K , can define f ( m ) as '1<1

rl
E [n c e i ] . Thus f(w)

fin)

is defined inductively for every element f n of any Kc. A mapping f is called adinissible if for evcry

8 6

S

f ( { a , b, d ) ) ={as, bdl.

We shall say that a set Q -Kt is symmetrical if there is a denumerable set S,CX such that m and 8, satisfy the following condition @(X,,nz): Q(Xm,m)

if f i s an admissible mapping m d f ( a , )=as for ( I n ) =I n .

( then f

8

E flm,

An ana.1ogous definition applies to classes a.11 of whose elements belong to one of the sets Kg. This follows from results of B. Giiclel. See hie book The Consistency of the Continuum Hypothesis. Arinals of Mathematics Studies, Number 3, Prin'ceton 1940.

136

[lS], 129

FOUNDATIONAL STUDIES

Let M p be the set of those ryc rh'd which are hereditarily symmetrical, i. e., such that from /ilk c

E

...c

/ltl c ' /it

follows that w,ml,wa2,...,wtk are symmetrical (k=1,2, ...). A set m is called remarkable 4, if it belongs to one of the sets M E . A class X is called remarkable if it is symmetrical and all its elements are remarkable. S o w replace in every axiom of 6 the words iltdioiaual,

bY

set,

da88

atentent of KO, Tenlarkable set, remarkable class.

It can be shown without difficulty that every axiom of 6 becomes then a provable proposition of GIs). The axiom of choice becomes a false proposition since the set A is remarkable and there is no remarkable set W which has exactly one element in common with every pair { a s ,b#}. It remains to show that (T)becomes a provable proposition of the system G,. For this purpose l e t us consider :I remarkable set B and a remafkable binary relation R (i. e., a remarkable set of ordered pairs) and suppose that for every x E B there is a y Q B such that gRy. It follows from the axiom of choice (which is valid in GI)that there is a sequence x1,x2,...,x,,,... (i. e. a set of ordered pairs , n=1,2 ,...) such that xnRxn+l for la=l,2 ,... We shall show that this sequence is remarkable. It is clearly sufficient to show that this sequence is symmetrical. Since x,, is symmetrical, there is for any integer IZ a, denu00

mc~ableset S,CA'~ such that @(S,,s,). The set So=zC,, is a denun=l merable subset of S and obviously satisfies the condition @(So,xn) for n = l , 2 , ... This proves that the sequence x,,.r2,...,x,,,... is remarkable and our proof is finished. We can apply the same method to establish another result of this kind: 4) Following R. Doss, Journal of Symbolic Logic! 10 (1945), pp. 13-15, we use this word as a translation of the German word ,,ausgeeeichnet" which I have used in my paper referred to in the footnote*). 6 ) For details see Fundamenta Mathematicae 32, pp. 220-235.

[15], 130

ON THE PRINCIPLE OF DEPENDENT CHOICES

137

Let .Z(m,n) and Z*(m) he the following iixiomrs:

Z(m, n)

Z*(m)

I I

if A is a set with the cwdinal nzntzbes nt ulzd if every element of A is a non-roid set with curdilzal lznzmber
if A is a set with cardinal nwazber t m ulzd if ecery element .of A is a non-coid set, tkcn thew is a ftcnction f s w h that f ( X )z 2- for X

B

A.

Modifying a little the foregoing proof, we can show that if m is tt cardinal number definable in the system Go due to Bernays 6), then We implication Z*(m)+Z(m, 2 ) is not provable in G (provided that 6 is self consistent). 6 ) P. Bern ays, Joiiriial of Syiiibolic. Logic. 2 (193i), pi). G.ii niitl 6 (1941), pp. 1-17.

PROOFS OF NON-DEDUCIBILITY IN INTUITIONISTIC MJNCTIOIYAL CALCULUS

ANDRZEJ MOSTOWSKI

I t has been proved by 8.C. Kleene and David Nelson that the formula is intuitionistically nondeducible, i.e., nondeducible within the intuitionistic functional calculus.' The aim of thii note is to outline a general method which permits US to at a b u the intuitionistic nondeducibility of many formub and in particular of the formula (1). First let us recall some notions due to Birkhoff? A lattice L is called complete if for every non-void subset A = (a,) of L there exists in L a least element 8 = &a, (tbe sum of A) such that a, 5 s for every a+ from A and a greatest element p = Pa, (the product of A ) such that p 4 a, for every a, from A. The sum of L is denoted by 1 and the product of L by 0. A complete lattice L is called Brouwerian if it satisfies the infinite distributive law

b

+ PA

=

P,(b

+ a,).

Let a and b be two elements of a complete Brouwerian lattice L and let US a as the product of the set A of all z such that b S a 2. Then define b b 2 a is the least element c such that b S a c. For brevity we write +a instead of 1 a. An important example of a complete Brouwerian lattice is furnished by the set of all closed subsets of a topological space 1. It is evident that in this lattice the operations PA,,and b A a have the following meaning:

+

+

-

SA, = EA,,

(2)

Pa, =

na,,

b

A

-a = b - a;

here Z,n, and - denote the set-theoretic sum, product, and subtraction and the bar denotes closure. Let L be a complete Brouwerian lattice and I any non-void set. A function F'(z1, - - ,zb) called a k-argument ( I , L)function if its arguments run through Z and its valuei belong to L. The set of all these functions is denoted by %(I, L). We shall say that

-

+

Received November 28, 1947. 1 5. C.. Kleene, On the interpretation of intuitionistic number theory, this JOURNAL, vol. 10 (1945), pp. 109-124; see especially the last sentence of 010, p. 117. 1Garrett Birkhoff, Lafftcetheory, New York 1940. See a180 J. C. C. McKinsey and A. Taraki, On closed element8 in closure algebras, Annalr of mefhmtaficr,vol. 47 (1946), pp. 122-162.

[la], 205

NON-DEDUCIBILITY IN lNTUlTlONlSTlC FUNCTIONAL CALCULUS

139

is an ( I , L)functional of characteristic (n,no ,nl,ni , . - if @ is a function whose values belong to L and which has n arguments running through I , n, arguments running through %(I, L),nl arguments running through Z ( Z , L),and so on. The integers w, no, nl , m , . . are all non-negative and all but a finite number of them are 0. An (I,L)functional@is called a zero-functional or is said to vanish identically, if its value is 0 (the sero of L) for every choice of arguments. We write then @ = 0. Consider now a formula a )

A = A(zi, ,x., Fi, * * . , F.,,Gi, ,G,,, Hi, H,*, *.*) of the functional calculus and suppose that it has n individual free variables, no functional variables with 0 arguments (propositional variables), ni functional variables with 1 argument, nt functional variables with 2 arguments,and so on. The formula A csn be conceived as an ( I , L) functional of characterii3tic n, no ,nl , nt , . . * , if we give the foliowing meaning to the signs occurring in A: (i) the individual variables are interpreted aa Variables running through Z; (ii) the functional variables with k arguments are interpreted as variables running through the set %b(Z, L);(iii) the logical operations M v N and M & N are interpreted aa the lattice multiplication and lattice addition; (iv) the logical operM ;(v) the logical operation M 3 N is interpreted aa the lattice operation N ation -,Mis interpreted aa the lattice operation A M ;(vi) the logical operations ( x ) and (32)are interpreted as the lattice operations & and P . . The functioml which we obtain by this interpretation from the formula A will be called the functional corresponding to A and denoted by 9,. Exsmple: If A is the formula (l), then the corresponding functional is *

(3)

@A(G)=

f

{[A

-.

S&(Z)]

a

*

,

[Ss- -G(z)]}.

The characterii3tic is (0, 0, 1, 0, 0, -). THEOREM. Ij A is intuitionistically dedwibb, lhen the ewresponding junctiolurl vanishes i&ntkdg for m - dset Z and every cmnplete B r m d iattiec L. Proof. If A arises by substitution from one of the axioms of the intuitionistic propositional calculus, then, as shown by B d o f f , McKinsey and Tarski,* *A = 0. S&, = Oandif Ahas the form If Ahas theform (z)B 3 B, then@, = B 3 @a!)& then @A = P, A % = 0. Hence the theorem is true if A an &om of the intuitionistic functional calculus. It remains to show that if A and B are formulas such that the corresponding functionals Or and 4 vanish identically for every non-void set Z and every complete Brouwerian lattice L and if C wises from A and B or from 'A alone by one of the rules of proof admitted in the intuitionistic functional calculus, then 4 vanishe identically for every Z and L. There are four rules of proof in the intuitionistic functional calculus: 1" the modus ponens or the rule of detachment; 2' the rule of substitution for indi'See. Birkhoff, loo. cit. p. 128, and McKineey and Tareki, loc. cit.. Theorem 1.3.

140

1161, 206

FOUNDATIONAL STUDIES

vidual variables; 3” the rule for (z);4” the rule for (3s). Let us consider these rules separately. Rule lo. In this case B is the implication A 3 C. Since *A = 0 and = 0, we infer from %, = b A *A that aC = 0. RuEe 2”. If C arises from A by substitution, then +c is either equal to *A or arises from + A by an identification of some variables. Since @A = 0, we obtain = 0. Rule 3”. In this case A has the form D 3 E and C the form D 3 fz)E where the variable “z” is not free in D. Hence *A

=%A

h,

@c =

Ss%

&S.

Since = 0 we infer that & 5 eD, The variable “z” being not present in b , we have also S. h ,i.e., aC = 0. Rule 4’. In this case A has the form D 3 E and C the form (3z)D 3 E where the variable “2” is not free in E. The proof in this case is Similar to that in caw 3”. The theorem thus proved shows that in order to establish the intuitionistic nondeducibility of a formula A it is sufficient to define a set I and a complete Brouwerian lattice L in such a way that the (I,L)functional *A does not vanish identically. Let us apply this method to the formula (1). Take as Z the set of positive integers and as L the lattice of all cloeed subsets of the Euclidean plane 1. Let Q, (z = 1,2, -) be a sequence of points which is dense on the boundary of a cloeed circle K and let G(z) be the set consisting of all points of K and of all points of the infinite straight l i e joining Q . with the center of K. By (2) we have then S&(z) = 1 and therefore

--

zzS&i(z)

(4)

x

LA1

P

&(I

Furthermore --G(z) = (1 G(z)) = - 1 1 K = K = K and it followa

- -

(5)

S,

1) = A

1

= A =

-L

0

I :

1.

- G(z) =

1

-K

a(z) = K .

From (4), (5), (3), and (2) it follow that *A((%)

-t-

(Iz s&l(z)] [SS A ( L ~ K ) = ~ ‘ 1- K K = 1 - K Z 0.

A E

a(%)]) A

1

- 1- K

The formula A is therefore intuitionistically nondeducible. Aa another example let us coneider the formula B (6) ( Z ) P I v G(4l 3 IF1 v (z)G(z)l, where “ K ” is a propoeitional variable. The corresponding functional is

@E.(FI, G ) FI*SA(Z) &[F~.Gl(z)l; it has the characteristic (0, 1, 1, 0, 0, .).

.

=

[16], 207

NON-DEDUCIBILITY IN INTUITIONISTIC FUNCTIONAL CALCULUS

'

141

In order to ahow the intuitionistic non-deducibility of the formula B, take aa Z the set of positive integers and as L the lattice of closed subsets of the denumerable apace 1

h i , + ,.**,,,

...,o

with ordinary topology. Let Fa be the set consisting of points 1, f, $, 1

2n+ 1'

+

*

, 0 and G(z) the set consisting of

the single point 1 1

1 z.

.- - ,

We have

1

then A.G(z) = 0, 8#".G(z)l = 0, SAQ) = {%, ;, - . -,%, * - - ,0) and F~.S,cl(z)= ( 0 ) which provea that %(Fa, Gi) = ( 0 ) # 0. Hence the formula (6)is intuitionistically non-deducible. It is interesting to remark that its double negation is intuitionistically deducible.

We conclude with 80- problem euggwsted by the foregoing investigations: Are there, for every intuitionistically non-deducible formula A, a complete . Brouwerian lattice L and a non-void set I such that the (I,L)functional 9 daea not vanish identically? Do there exist a lattice Loand a set lasuch that the ( l o , &) functional d m not vanish identically for any intuitioniatically nondeducible formuh A? Can the lattice of closed plane sets and the set of positive integere be taken for Lo and Zo ? mOIVBBBlTI OF W A S M W

ON A SET OF INTEGERS NOT DEFINABLE BY MEANS OF ONE-QUANTIFIER PREDICATES bY

ANDRZEJMOSTOWSKI(Warszawa) 1. This note is a sequel to my paper ,,On definable sets of positive infegers"') quoted below as D. For explana-

tion of symbols and notations the reader should refer to section 1 of D. Quotations like D 1.23 mean: theorem 1.23 of D. We shall assume that conditions R, and R2 given in D 6.1 are satisfied. As it was mentioned in D 6.23 and D 6.31 this assumption implies the fact that Pt)is the class of general recursive n-adic relations and is the class of general recursive functions. Let Rf"' be the smallest finitely additive field of sets 3 PF) a:'. It follows from D 2.18 and such that D 2.31 that C Pr'. @I. # T I . Qf), i. e., that there It will be proved that exists a set A which belongs to @' but does not belong to @). For a clearer explanation of the meaning of this result, we have to remember that for every set of Pr' there is a definition of the form En[z, nq ,(n, PI d e R11 (1) there is, with a general recursive R,. For a set of likewise, a definition of the form (2) E, [DP zq(n, P, 4e &I with a general recursive R,. For the sets of the general form of definition is

+

el.

QF)

e'

') Fundament. hbthematicae. vol.

34 (1947). pp. 81

- 112.

[17], 115

ON

A SET

143

OF INTEGERS

[(n, p ) Rl1t - - * * " p [(n, P) &I, Zq"n, ci, S,I,- . zq"n, 9)e S,])), where @ is a Boolean polynomial (i. e., a logical sum of logical products) of its arguments and Rl,..., Rk,Sl,..., S, are general recursive sets. Hence for the set A, whose existence will be proved below, a definition of the form ( I ) and a definition of the form (2) exist but no definition of the form (3) can be applied. We remember here that if a set B simultaneously admits (n, p ) 6 R1]and En[-ZP (n, p ) e R, J definitions of the forms En[n,, with general recursive R,, R,, then B is itself a general recursive set *). These theorems are quite analogous to the following well known results concerning projective sets: if a set B is an A - set as well as a CA - set, it must be Borelian; but a set which is a PCA-set as well as a CPCA-set does not necessarily belong to the smallest (denumerably additive) field of sets over A CA. 2. In the subsequent proofs we shall make much use of the primitive recursive function s(k,n) defined in the following manner: we first define an auxiliary function t(k,n): t(1.n) = n, t(k 1,n) = s, [t(k,n)] and then put s(k,n) = s1[t(k,n)], where s1( n ) and s2(n) are determined by the equation

(3)

E n {@ ("p

f

. 9

+

+

n

=

P ("I (2 s,(n) - 1 ) .

The following theorem exhibits the usefulness of this function : 2. 1. For any finite sequence of integers rl, r2,..., rh there is an integer n such that s(k,n) = r k for k = 1, 2,..., h. 3. In this section we shall prove two lemmas (3.3 and 3.4) complementary to the theorems given in section 2 of D. *) S. C. K I e e n e, Transactions of the American Mathematical Society,

vol. 53 (1943). p. 53; E. L. Post, Bulletin of the American Mathematical

Society, vol. 50 (1944). p. 290; -D 5.51.

144

[17],116

FOUNDATlONAL STUDIES

3.1. I f f ( m ) is a function of class pi*’), fhen the sef JV= Ei, m [i < f(m)l is of class P (2). ,

QC)

Proof. Evidently i 0 , that W is of class P,“ and Qf) (cf. D 2. 16). If k = 0 , we obtain W e P ~ ) * Qi.~ e., ) , WePF-Qf) in view of D 5.51. 3. 2. Every function flm) of class P?*” is general recursives). Proof.‘) By definition D 5.1 the set If=El, m [I =f (m)l belongs to P:”’. Since (1,m) e I f =lIh [(h=f (m)) + ( I =h)] and the set of (h, I,, rn) satisfying the condition in square D 2.16 entails I I e Q y ) . Hence by brackets is of class D 5.51 I, e =p’oz’, i. e., f is a general recursive function. 3.3. I f M e Pfc2’ and either (i) k 5 1 and f E P’’”)or (ii)k>l and f e @? :,: .then the set P=En,m [ “ i < f c m , (n, rn, i ) e MI b fl) is of class Pk . Proof. Let us first consider the case k = 0 . By definit b n s D 1.3 and D 1.5 there are decidable propositional ) y ( x , y ) such that functions p ( ~ , x , yand (4) (n, rn, i) e M 5 I- ~ ( n m, , i), i =f ( m )=I- y ( i , m). The propositional function y*(x, y ) defined as w (x, Y) cu c x w”u, Y)1 -f

QF),

- QY)

QY),

0”“

-f

’) A similar theorem is valid in the theory of projective sets: if the geometric image of a function f is an analytical set. this function is Borelian. See, e. g.. K. K u r a t o w s k i, Topologie I, Monografie Matematyczne. Warszawa 1933, p. 253. ‘1 I owe this proof to a remark of Professor K. Kuratowski.

rm,117

ON A SET OF INTEGERS

145

is decidable again and satisfies the conditions i =f (m)3I- y y j , m), (5) I- nxr.x*,y Y*(X 11 Y ) W T X 2 . Y ) + x, =XJ. (6) Now define 6 (g, y ) as -% { V * ( & Y ) * K [x 0. As the set M is of class P;"), there is a set N eQr?:) such that (n, m, i ) c'M = Z j [(n,m,i, j ) e N ] and consequently (n, rn) 6 P nil, we rewrite (9) as (10) (n, rn) e P = z r n i [i
-

146

FOUNDATIONAL STUDIES

[17], 118

and remembering that k>l, we conclude from (10) that

PEP;+,I.

Theorem 3.3 is thus proved. Applying de Morgan’s laws we obtain: 3.4. I f MEQ:””’ and either (i)k I 1 and f el‘:’) or (ii) k>l

and f ePF?:, then the set Q =E,,,[Z;
Qr).

We shall also need the following theorem due to Kleenes).

4.2. There is a function # ’: universal for the class P(i’” and such that the set (12) E,k[nE#:’(k)I is of class p(:+”. by the formula

Define now the function

(13)

-

(d

.Ci<,,(,,,,

We shall prove that 4.3. The function

the set (14) is of class

(s(2i- 1,s,(d) -PI“’(s(2 i,qh)))]. is universat for the class Rr)and

e+’)~(an+’).

Em,

E

+’: ( m ) ~

Proof. Since P ~ ( s ( j , m ) ) e Pfor ~ ) 1 - 1,2,...,2s2(m)-1, 4.1 and (13) entail e ’ ( r n ) e R : ) for every integer rn. Let us suppose that P ERy’,i.e., that (11) holds for some M,,..., Mk, N1, ...,N,ePT’. As the function is universal for pr”’, there are integres r,, rs, ...,r2, such that

c)

‘)

S. C. K l e e n e . 1.

c.9, p. 47.

[17], 119

147

ON A SET OF INTEGERS

c’(ru),-

(15) MI P!)(r+J, N,= for j = 1 ,2,...,k. By 2.1 there is an integer r such that s (1,r) r, for j - 1,2,...,2 k. Putting m=2*(2k 1) we obtain s,(m) = k 1, s,(m)=r and s(j,sl(m))=s(j,r)- r j for j = 1.2,...,2k. Therefore in view of (13) and (15) Tr)(m) = E i < k + 1 [FYI(?* 1-1) -#:)(ra 31 =2i<
+

+

neZy(m)= -Zi<8, (m) n q z p {[(n, s (2- 1,s, (m)),P) e BI * (17) [(n, s (2 i, s, (m)), q ) non e BI I. The formula (16) proves that the set (14) is of class l‘(:+’). Using (17), D 2.16 and 3.4 we obtain that this set is also of class Q:’” which proves our theorem.

4.4. Ry)# Qr’. Proof. Using 4.3 and Cantor diagonal theorem, we notice that the n dimensional set A = E,,,, [(m, PI non e T ~ ) ( P ) I is different from every (q) and therefore it does not belong to Rr’. By 4.3 and D 2.12.we further obtain that A e Pf) Qf’, q. e. d.

c’

9

7&. Andrzej Mostowski and Alfred Tarski: Arithmetical classes and types of well ordered systems. Preliminary report. For notations see Bull. Amer. Math. SOC. Abstract 54-11-74, 54-11-75. The aet A being ordered by the relation 4 , 7(A) denotea the order type of the system 8- (A, b ).Let L be the class of all (A, b) with R #O well ordered b y S. A decision procedure for Th(L) is established. Given any ordinal 7, let M(r), N(7),P(r), Q(r) be mpectively chases of all 8EL with: r(8)=r+r,for some ordinal qZ0; 7(8)=f+r+q for some I and r,, q < y * o ; ~ ( $ f ) - y ; r ( 8 ) = o " * E + y for some EZO. Theorems :AC(L) is the smallest family containingall the classes L,M(@),N(clrr*p), where 1,p=O,1, * * ,and closed under class addition and subtraction. AC(L)..as a Boolean algebra under class operations has a well ordered basis of the type # AT(L) AC(L) consists of a!l P(r)'s,and AT(L) -AC(L) of all Q(r)'s,with 7 <#. The families AC(L), AT(L), AT(L). AC(L), AT(L) -AC(L) are countably infinite. For 8, BEL, we have 8183 if, and only if, either ~ ( 8-r(B) )
-

-

%?@I

ON THE RULES OF PROOF IN THE PURE FUNCTIONAL CALCULUS OF THE FIRST ORDER ANDlZZEJ MOSTOWNU

We consider here the pure functional calculus of first order F: aa formulated by Church.]. Church, I.c., p. 79, gives the definition of the validity of a formula in a given set I of individuals and shows that a formula is provable in F: if and only if it is valid in every non-empty set I. The definition of validity is preceded by the definition of a value of a formula; the notion of a value is the basic “semantical” notion in terms of which all other semantical notions are definable. The notion of a value of a formula retains its meaning also in the case when the set I is empty. We have only to remember that if Z is empty, then an m-sry propositional function (i.e. a function from the m-th Cartesian power I“to the set (f,t ) ) is the empty set. It then follows easily that the value of each wellformed formula with free individual variables is the empty set. The values of WITS without free variables are on the contrary either f or t. Indeed, if B has the unique free variable c and 4 is the value of B, then the value of (c)B according to the definition given by Church is the propositional constant f or t according as 40’) is f for at leaat one j in Z or not. Since, however, there is no j in I, the condition +(j) = t for all j in I is vacuously satisfied and hence the value of (c)B is t. To see this still more clearly let us consistently treat functions of one variable as seta of ordered pairs. Thus 4 in the example above is the empty set of ordered pairs and KCJiave the equivalence

{the value of (c)B i s

t)

(j)b e I

3 (j,1) c 41.

It is evident that the right side is satisfied if I is empty. It follows from our argument #at each wlT bhginning with a non-vacuous universal quantifier: and also each wtT containing free individud variables, is valid in the empty set. A WIT beginning with an existential quantifier is therefore never valid in the empty set provided the quantifier is non-vacuous. It can easily be checked that the axioms of F: given by Church are valid in the d e s of the empty set and that the rule of generalization [111] as well substitution and of alphabetical change of bound variables [II], [11’], [11”], [II”’] preserve validity in the empty set. The rule of modus ponens does not preserve validity in the empty set. For instance the formulas F(z) v -F(z) and [F(z)v -F(z)] 3 (3z)[F(z)v -F(z)] are valid in the empty set but (&)[F(z) v -F(z)] is not. Received March 14, 1950. 1 Alonzo Church, Introduction to mathematical logic, Part I. Annals of mathematics studies Number 13, Princeton, 1944. * A quantifier is vacuous if it is followed by an expressionin which the variable bound by the quantifier is not free. Cf. Willard Van Oman Quine, MuthematicuZ fqpic. Cambridge 1947, p. 74.

,

150

FOUNDATIONAL STUDIES

1281, 108

The following weaker form of the rule of modus ponens can easily be shown to preserve validity in all sets (including the empty set): (I*]. If A and B are wffa such thut aU individual variables free in A are free in B, d if A and A 3 B are theorems, then 80 is B.

PROOF. It is su5cient to test whether the ruie [I*] preserves validity in the empty set. This is evident if B has at least one free individual variable. If B hss no free individual variables then neither does A, and since by hypothesis the values of A and A 3 B are t, it follows that the value of B is also t.’ Let F r be the eystem based on the same axioms 88 F: and on the rulea [I*], [II], DI7, PI“], [II“‘],and [III]. We shall prove the following:

T~EOREH. A formula is pnwcable in Ff if and on& if it is valid in each set (the empty set included) PROOF. We shall write “h A” instead of “A is.provable in F:” and “h A” instead of “A is provable in F:’. Let A be a wff without free individual variables valid in each I. Hence A is valid in each non empty I and h A. Let F be a functional variable with one argument which doea not occur in the formal proof of A in the system F‘, and let p be a propoeitional variable with the same property. We introduce the following abbreviations: T(a) = F(a) v -F(a), T = (ax)T(x),

T’ = (x) -T(x),

1 = pv-p.

We let correspond to each wff B which is either a part op A or a part of a formula which occurs in the proof of A a wff B* in the following manner: if B is quantifier-free, then B* = B; if B = -& , then B* = -B: ;if B * & 3 B,,then B* = B: 3 B! ;if B = (a)&and ais not free in &,then B*- B: ; if B = (a)& and a ie free in & ,then B* = 1. The capital letter B with or without subscripts will always denote a wff for which B* is deiined. L E M1.~ T’ 3 = By. PROOF. The c w of a quantSer-free B and the passsge from & and & to Bl 3 & and --& are evident. If a is not free in ]3r then h (a)Bl -& , and exactly the eame proof can be used to show that 1 0 (a)& E 4 . Using the inductive assumption we easily infer that lemma 1 is true for the formula (a)&. Finally, if a is free in Bl , we have to show that if the lemma holds for & , then

b T‘ 3 [(a)& 3 1) and h T’ 3 [l 3 (a)&]. 1 A different set of rules for the functional calculus, such that theorem obtained by these rules are valid in each set I whether empty or not, has been given by Jdkowski. Cf. Stanistaw Jabkowski, On the mlea of s u p p o & i m in fdlogic, Studia bgica, Number 1, Warszawa 1934, $5.

1281, 109

151

RULES OF PROOF I N T H E P U R E FUNCTIONAL CALCULUS

The first formula is evident, since the propositional calculus holds in the system F*, . In order to prove the second it is sufficient to show that

10T’ 3 (a)&. This will be done as follows. By the propositional calculus we have

h “ U a ) 3 BI , 6) h [T’ 3 -T(a)] 3 ([-T(a) 3 &I 3 (T’ 3 Bd). (ii) By axiom (8’) (Church, I.c., p. 66) and the rule [II’] we have further h T‘ 3 NT(a). (iii) Formulas (i) and (iii) show that the rule [I*] is applicable twice to (ii) (since the condition concerning the variables in the antecedent and in the consequent is satisfied). We thus obtain T‘ 3 B1 and consequently, by rule [III],

h ( a W 3 &I. Now substitute T’ for “p” and & for “F” in axiom (7’) of Church, I.c., p. 66. Rule [I*] is again applicable and we obtain h T’ 3 (a)Bl. Lemma 1 is thus proved. LEMMA 2. bo T’ 3 A. PROOF.T’ and A are valid in the empty set. Since the rules of Ff preserve validity in the empty set, we infer from lemma 1 that A* is valid in the empty set. Since however A is a WEwithout free individual variables, the formula A* is either 1 or a quantifier-free combination of 1’s and of propositional variables. Since 1 was defined as p v ~ p we, see that A* is a formula of the propositional calculus. Since it is valid in the empty set, it has the value t for each system of values for the propositional variables and is therefore a tautology of the propositional calculus. It follows that h A*. Combining this with lemma 1 (for B = A), we obtain h T‘ 3 A. LEMMA 3. Ijh B, tha h T 3 B. PROOF. If B is an axiom, the lemma is evident. Assume that the lemma holds for two formulas Bl and & and B is the formula resulting from Bl or from BI and & by one of the rules of proof admitted in F: . We have to show that the lemma holds for the formula B. The c t l ~ e of s the rules [II], [11’], [II”], [II’”], and [111] do not cause difficulties. In the case of the rule [I*]the proof rum as follows: By the inductive assumption we have (9 hT3[&3BI and bT3B1. (ii) Let al , . . * , ak be all of the variables which are free in BI and which are not free in B. Using tautologies of the propositional calculus we obtain

t-0 { T 3 & 3 B]) 3 ((T 3 Bd 3 ( T 3 W a d and hence, by the rule [I*],

-

0

.

T(ad

J Z

Bl1)

152

(iii)

1281, 110

FOUNDATIONAL STUDIES

h ( T 3 B 1 ) 3 ( T 3 [ T ( a l ) ... T(a,+)I B B ] ) .

Rule [I*] is applicable to (ii) and (iii) and yields

FOT 3 [T(ar) ... Tfak) 3 B]. Proceeding as in the proof of Church’s theorem XII, p. 47, we obtain

h T 3 [(3adT(al) . . . (XadT(aJ

3 BI

and hence, by the rule [II’”],

T 3 [T.T

- * *

T 3 B].

Ey the propositional calculus we now obtain bo T 3 B. Lemma 3 is thus proved. LEMMA4. koT 3 A. This follows immediately from lemma 3. LEMMA5. koT v T’. The proof is the same as in F j . Proof of the theorem: By lemmas 2 and 4, (T v T’) 3 A, and hence by lemma 5 and rule [I*], A. This proves the theorem for formulas without free individual variables. To accomplish the proof i t is sufficient to remark that (1) A wf€ is valid in all sets (including the null set) if and only if the same is true of its closure.‘ (2) If the closure of a wff is provable in F: then so is the wfl itself.

It is not difficult to adapt the method explained above to other standard formulations of the functional calculus of first order. Thus, e.g., in the formulation due to Quines one has to replace the axiom-schema *lo2 by the following one : If aU free variables of 4 are free in $, then k ‘(a)(& 3 4) 3 [(a143 (a)+]’. The resulting system has the property that a foimula is provable in it if and only if it is valid in every set.‘ We remark finally that rule [I*] can be replaced by the following somewhat stronger rule: If A and A 3 B are theorems then so is B, provided that A and B satisfy the additional condition: A contains no free variables unless B has at least one free variable. To illustrate the difference between the systems F j and F*, we give here the changes which must be introduced into 25 theorem schemata listed by Church, I.c., pp. 4 7 4 8 and 53-56, in order to obtain schemata valid in F*, . [XI is valid only under the assumption that a is not free in A ; the schema, of course, becomes utterly uninteresting. [XII’is valid only if a is free in B. [XIV] is valid only if a is free either in both A and B, or in neither of them. [XXI]:as in [XI]. In the sense defined by Quine, I.c. p. 79. sQuine, I.c., p. 88. 6 I am indebted for this remark to Mr. Grzegorcayk, Dr. H. Hit, and Mr. Janicsak.

4

(281, 111

RULES OF PROOF IN T H E PURE FUNCTIONAL CALCULUS

153

[XXII]: $s in [XIV]. [XXV]: the sign of equivalence has to be replaced by the sign of implication in the direction from right to left. [XXVI]: as in [XXV]. [=I] is valid only if a is free in A . [XXXIII]: as in [XXXI]. The distributivity laws for quantifiers ( l a ) A v B := : ( 3 a ) A v ( 3 u ) B (a)A B :E : (a)A (u)B, hold in Flf: only under the additional assumption-that a is free either in both A and B, or in neither of them. When one examines the proofs of the modified schemata one sees very clearly that the changes were necessitated by the lack of transitivity of material implication in FE . This is the most burdensome feature of the system F ; . We conclude by stating that this system is by far not as elegant as the usual functional calculus.

.

UNIVERSITY OF WARSAW

.

.

. .

ANDRZEJ MOSTOWSKI

A CLASSIFICATlON OF LOGICAL SYSTEMS 1 propose myself in this paper to discuss a classification of logical systems (or. languages as philosophers are used to say)

in which certain portions of mathematics can be formalized. This classification will be based upon a classification of sets of positive integers which was firsf defined by S..C. Kleene in his paper [7)* and which was afterwards considered by me in a paper published in 1947*. The most important facts related to this classification will be recalled in section 1 and the classification itself extended to transfinite indices. In section 2 we apply the results of section 1 to obtain a classification of logical systems. In sections 3 and 4 we discuss some well-known systems and establish their position in the classification described in section 2. The subject itself as well as the method of its presentation will be of a mathematical rather than philosophical character. The author believes, however, that the results established in the paper may be of some importance for philosophers interested in the study of structure of various logical and mathematkal languages. 1. Throughout the whole paper we shall use the word ,,func-

tion” as synonymous with ,,function whose values and arguments tun over the set of integers”. We shall describe here some classes of such functions which are of special value in logical investigations 1.1. Prjmitive cccutdvc functions. In order to define this class of functions we consider the following three operations yielding new functions from given functions:

a

Numbers in brackets refer to bibliography at the end of this paoer. M o e l o w s k i 191.

[29], 238

155

A CLASSIFICATION OF LOGICAL SYSTEMS

I. Substitution. This operation leads from R + 1 given functions f, fl, .., fk the first of which has k arguments and the others have nl,, , arguments to a function F of nl+n,+ ...+nk arguments. The values of F are given by the formula

.

..

F (x:,..,,xi,,...,xf,...,xk"k )=f (f,(xi,...,xf,),...,fk (xt,...,xknk)). 11. identification of variables. This operation leads from a function f of k + 1 arguments to a function F of k arguments. The values of F &re given by the formula

III. inductive schema. This operation leads from two functions f and g the first of which has k and the second k + 2 arguments to a new function F of k + 1 arguments satisfying the equations

P(o,xl,...,x,)=.f(x,,.. .,Xk), P (n+ 1,xi,* * x,)=g(P (n,XI, f

9

.

9

xk), xi,* *

9

Xk).

if a function f can be obtained from other given functions

f,, f,,

...,fk, .. *

by a successive application of the operations I, 11, 111, then f is saki to be primitive recursive in the functions (1). Functions which are primitive recursive in the functions (2), (3), (4) listed below are called primitive recursive without further specification: (2) The constant function 0 (without argument$), (3) the successor function S (with one argument) such that S(x) = x + 1, (4) the identity functions 2; (with k arguments) such that 1: (XI,. ..,xc)=x?. G 6d e 1 [6],p. 179 or K 1 e e n e [7], p. 42. For the definition see R o b i n s o n [ill.

8

simplification of

156

FOUNDATIONAL STUDIES

[29], 239

It is evident that every primitive recursive function can be defined in a finite number of words: to define such a function it is in fact sufficient to fix the succession of the operations I, 11, 111 which leads from the functions (2), (3), (4) to the given function. (Of course each operation may be performed several times and each of the functions (2), (3), (4) may occur on many places in the sequence defining the considered function). Another important property of primitive recursive functions is the following: if a primitive recursive function f is defined in the manner just described, then the value of f for any effectively given set of arguments can be effectively calculated in a finite number of steps. This is evident for the functions (2), (3), (4) and in order to prove that all primitive recursive functions have this property it is sufficient to observe that the property in question is preserved by application of any of the operations

I, 11, 111.

The primitive recursive functions satisfy therefore the fundamental requirements of intuition'sm: finite definability and calculability for arbitrarily given arguments. 1.2. Primitive recursive sets and &tiom. A set A of integers is called primitive recursive if there is a primitive recursive function f of one argument such that

An n termed relation R between integers is called primitive recursive if there is a primitive recursive function f with n arguments such that

.

..,xn)=0.

R(x,, .. ,xn)= f(x1,.

Since we identify n-termed relations with sets of ordered n-tuples (xl,. ., xn), the preceding definitions fix at the same time the notion of primitive recursive sets of n-tuples of integers. From the finite calculability of primitive recursive functions it follows that if a set A is primitive recursive, then there exists a method which allows us to decide in a finite number of steps every particular problem of the form: does a given integer n

.

1291, 240

A CLASSIFICATION OF LOGICAL

SYSTEMS

157

belong to A o r not? We say briefly that the decision probfem for A is solvable 4. 1.3. Recunively enumerable oetr. A set A of integers or, more, generally, of n-tuples of integers is called recursjvely emmefable if there is a primitive recursive function f which establishes a oneto-one correspondence between the elements of A and the integers. The decision proMem for recursively enumerable sets is, in general, unsalvable s. Recursively enumerable sets possess however the following property (which is weaker than the solvability of the decision problem): if a set A is rec-ursively enumerable, then there exists a sequence built up from all the elements of A such that for any given n we may calculate the n-th term of the sequence in a finite number of step. A more careful analysis shows that every set with this property is recursively enumerablee. It can also be shown that if the decision problem for the set A is solvable, then A is recutsively enumerable'. This remark will often enable us to simplify considerably proofs of recursive enumerabiiity of certain sets which we shall encounter in the course of our discussion. It shows in fact that we can save us the actual construction of primitive recursive functions which enumerate the elements of the given sets and limit ourselves to an indication of a finitary method solving the decision problems for these sets. We remark still that the class of primitive recursive sets is narrower than the class of sets with a solvable decision problem s. 1.4. A classification of definable sets. We denote by F":) the class of k-termed recursively enumerable relations and by the class of those k-termed relations whose complements are recursively enumerable.

' P o s t [lo], p. 287. ' P o s t [lo], P. 282.

* This follows from B theorem proved by R o s s e r [la], p. 88 if me identify the notion of a finitarg method with that of general recursiviw. 7 In other words every general recurdve set is recursively enumerable. See e. 8. P o e t [lo], p. 290. * An example of a set which is not primitive recuasive hut which has a solvable decision problem will be given in section 4. 3.

158

[29], 241

FOUNDATIONAL STUDIES

The commgn part of f(f)and Q(i) is called fg). We further define as follows the classes Wi and Qf using induction on n. A k termed relation R belongs to the class Pj:* if there .is a relation S in Q@,+')such that

A &-termedrelation R belongs to the class Q ment belongs to the class P:Tl.

if its cornple-

In this way we obtain a classification of k-termed relations and in particular, for k l , of sets of integers. On first sight this classification may perhaps seem artificial. We come to it however in a very natural way if we consider the so called definable sets of integers. A set A is called definable (or more accurately definable within arithmetic) if there exists a formula E ( x ) with one free variable x built up from the elementary formulas of the fQm (5)

u-v+w,

u=v w,

u=v

Y

...

with tbe help of logical connectives and quantifiers ranging over the set of integers and such that the following equivalence holds:

x EA

jti

B (x).'

It can be shown that if the elementary formulas (5) represent primitive recursive functions, then the class of definable sets coincides with the class of sets which belong to one of the classes P'," or Q!!) for an integral value of ,lo. The value of n for which a definable set A belongs to P!] or to is determined by the formula E ( x ) and it depends in particular on the number of quantifiers occurring in tbe nomal formll of E(x). If the normal form of E(x) is

Qr)

.,gJ...(xp-1, ...

(xi,. ..,xi1)(ax;,..

...

* For a mom exact definition of the notion of defimbiliv see T a r s k i [UJ.P. s a and section 4.4 below. ** M o s t o w s k i [9], p. 85'.

[29], 242

159

A CLASSIFICATION OF LOGICAL SYSTEMS

.then A belongs to QC);if the normal form of E(x) begins with an existential quantifier, then A belongs to PF). For instance the set of integers h such that

(i. e. the set of exponents for which the Fermat's last theorem is true) is of class Q(i). It is not known whether this set is of class Pi) 1.5. Universal functions. The problem of existence of sets which belong to P(!,) or Q $ but do not belong to classes with lower indices can be solved with the help of the following important notion. Let X be a denumerable class of sets of integers o r of k-termed relations between integers (k=L,2, . . .) and let F be a function of one argument such that F(n) is always a set. We shall say that the function F is universal for the class X if

.

A function F will be called a normal universal function for the class P ( i ) or Q(2) if it is universal for this class and in additioti the set

G,, ...,xk,m)[(xi,.. ,xk)E F ( m ) ~ h

A

belongs to the class P:+l) or QLk+l). It can be provead that if FF) and CF) are normal universal functions for the classes PLk) and QLk) (k=1,2, .), then the functions defineld by the equations 1.

h

A

x,,Y)'G:+''(m)I F?kt(m)=(Xl, * . * x k ) ((3y) (XI, * Glfi: I (m) =(x,, ...,xk)[- (x,,...,xt)c Fti (m)] 9

3

(6)

I\

A

are normal universal functions for classes PL'$

and Q3t13.

See e. g. C h u r c h [ 2 ] ,P. 60. Thie definition is taken over Irom the theory of borelian and plrojecfive sets. See K u r a t o w s k i [&I, p. 2'75 and M o a t o w l s k i [!?I, P. 93. il M o s t o w e k i [9],p. 95. ii

la

160

[29], 243

FOUNDATIONAL STUDIES

A normal universal function for the class

Pi&)can be cm-

structed as follows l4. We have already remarked in 1.1 that every primitive recursive function can be defined in a finite manner, essentially by a finite sequence of equations. If we enumerate the finite systems of equations, we obtain an enumeration of all primitive recursive functions and it is evident that putting

Ft) (m)=set of values of the m-th prim-tive recursive function with k arguments

we obtain a function which is universal for the class 3)It.can be shown that this universal function is at the same time normal.

Thus we have established the existence of normal universal functions for each of the classes f $ j and 0‘:). Using the normality property of the functions F(i) and G(kh and Cantor’s diagonal theorem15 (or - what is essentially the the reasoning known as Richard’s paradox)le we same prove that the set

-

belongs to the class Q(;)but not to the classes f(f) with This entails the inequalities

S<,
which prove that no two classes of our classification are equal 18. Equations (6) imply the following important equivalences which we shall use in section 4.3 K 1e e n e [7], p. 47K u r a t o w s k i [ S ] , p. 258. See e. g . R u s s e l l and W h i t e h e a d [ 4 ] , P. 61. l7 A detailed proof is given in M o 8 t o w s k i [9], p. 96-97, [9], p. 96 for the details of the proof. The inla See M o s t o w s k i clusion signs in the above formulas denote the relation of a proper Part 01 a set to the whole set.

[29], 244

A CLASSIFICATION OF LOGICAL SYSTEMS

161

(xl,...,xk)E fl;i2 (m) =(3a) [(xl, . . xk,a) Gtks,')(m)] =(aa) [(x,,...,xk,a)non € FhR++1')(m)]

~(3a)["(3b)(x ~,..., ~~,a,b)€G,~+*)(m)]

=(Xa)(b)f(x,,...,x,,a,b)non€GF+2t(rn)] =(3a)(b)[(xi,.. ,xk,a,b)E Fik+2)(m)].

.

We conclude this section with the following lemma:

Lemma 1.51. The universal functions FL*) arld GIk) are definable within arithmetic (in the sense explained at the end of section 1.4). It Is evident from the formulas (6) that if this lemma holds for an ilnteger n, *italso holds for the next higher ankger n + l , Hence it is sufficient to prove the lemma only for n=l. We shall not .reproduce this proof here and refer the interested reader to the paper Kleene [7] whene the lemma is proved in detail. 1.6. Classes PE' and 0 :' for transfinite values of alS. The problem how to extend the classification descrribed in the section 1.5 to transfinibe values of the index n is ,noit yet commplety solved and like every problem of a new definibion is not free firom a e r t a h arbitrariness. We shall indicate in this section a definition of the classes PE' and for values of a lower than .the SQ called first non-construct&le ordinal 014~. We define PE) as the class of all sums (unions) (9)

where A is any set of ,one of the classes PF) with n e w . Starting with this dass we construct classes QE', P&, Q:il,. .., PELn,, The definitions given and results mentioned in this section seem to be not published anywhere. Nevertheless they are rather widely known to specialists in the field of recursive funtiona. Especially I understood fro= conversations with Professors P o s t and K 1o e n e that they found equivalent definitions and results in approximatively the same time. za See C h u I c h [ 5 ] .

162

[29], 245

FOUNDATIONAL STUDlES

.

f k) Qm+n,. .in exactly the same way as we constructed the classes considered in section 1.4 starting with the class Pi”’. It is not difficult to show that all these classes possess normal universal functions. In order to obtain such a function for the class PE) we represent every integer n in the form

-

R =24J (2ss (n) I)

and put

P!!)(ml= Z Fj4 (j)where A=&,,) (ss(m)). (i.i ) E A

QtLq

Normal universal functions for the classes PZkq and are then defined inductively by the formulas (6) in which we put IE=O +q. After all these classes and functions have been defined we can pass to indices of the form 0.2+p. W e define PEi as the class of all sums (9) where A is any set of cne of the classes P : i q wiht a finite p. Further we apply to P the ?ame procedure as in section 1.4 and reach the classes Pf!,+qand QS+q with 9<0. Using arguments similar to that used above we prove that all these classes possess normal univemil functions. We can therefore extend our classification first to the index 0.3 and further to indices 0.3+q with q
z).

Lemma 1.61.

The set

does not belong to p“‘

-

[29], 246

163

A CLASSIFICATION OF LOGICAL SYSTEMS

This .lemma implies the -inequalities (7) for all transfinite values of n em,. Lemma 1.62. If f is a primitive recursive function with one argument and ACPg’ , then the set of integers n such that f ( n ) € Abelongs also to the class PI). The p m f of this lemma even in the case a=w (the only transfinite case we need far what follows) requires a rather de-. veloped technique whioh we do not wish to presuppose here. For finite a’s the p r o d is giveam in Mostowski [9], pp. 10L-105. We make stiM the following canark in order to justify OUT definitions. The classes P f ) and 9:) as we have defined them depend a priori on the universal function I$) and G F with p
-.-

ek)

2. b this section we apply the classification of sets of integers described in the section I to obtain a classification of lo= gical systems. First we explain to which systems oar theory will be applicable. 2.1. Logical systems. Notion oftruth. Although our investi gations will be purely fonmal we shaii nevertheless accept a de-

164

FOUNDATIONAL STUDIES

[29], 247

finite philosophical p o h t d view with respect to logical systems. We shall not consilcier logical systems as void schemata de-

prived of any interpretation. On the contrary we shall assdme the objective existence of a khd of ,,mathebmaticalreality” (e. g, of the set of all integers or of the set of all real numbers). By objective existence we mean existence indepandently of all‘ 1irtguistic const r w tions. Tlhe role of logical and mathmatical systems is to describe this realilty. Every logical sentence has thus a meaning: it says that the mathematical reality has thlh,is or some other property. If the mathematical reality does in fact possess the given property, then the sentence expressing this praperty is tme, otherwise it is false. The intuitions unlderlying this distinction can be made precise and an exact Idefinition of truth can be furmulated ‘by Using the well known methods of Tarski [15]. The point of view characterized above allows us to make plawilble the existence of undecidable sentences in almost all, logical systems. Such sentences evidently exist if it is not possible to prove all the true sentences expressible in the considered system. Now every proof in a logical system consists in a succession of some simple 0,perations (called the rules of proof) which tan be mechanilcally performed m one o r two expressions. The unprovability of certain true sentences can be explained by the fact that the propertries of the ,,mathemat,ical reality” are more complicated than the properties which can be established by successive applications of hhe rules of proof to the axiom. We do not intend to defend the philosophical correctness or even the philmophicabl acceptability of the point of view here described. I t is evident that it is entirely opposite to the point of view of nominalism and related trends. 2.2. Expressions and numbers. Sinoe the publiicahion of the famous paper Godel [6] it is well known that the study of expressions is equivalent to the study of integers. It is namely possilble to put the exgressions d any logicail systemiinto aone-to-one correspondence with integers. In this way eaclh set of or operatton% on expressi*onscorresponds to a well defined set of inbegers or to a function 04 integers to integers. To certain purposes it is

[29], 248

A CLASSIFICATION OF LOGICAL SYSTEMS

165

even useful to identify expressions with the c o r r q m d i n g integers. Eadh-logical system detmmines in a unique manner certtatilin sets of eqressions (e. g. the set of meaninigful expressions, the set of provable sentences, the set $oftrue serntelnces) and centain operations an expressions (e. g. mles of proof). Since we can identify ex,pressimq with integers, it foldlows that every classification of sets of integers or of f u n c l h s determines a classifkatiaa of 10gical systems. Examples of such classifications are well known in the literature. Some of them are given in next section. 2.3. Constructive and decidable systems. A iogical system is called constructive if the set of its meaningful expressions and the set of its axioms are primitive recursive and its rules of proof are finite in number and are primitive recursive functions 21. For each constructive system there exists a finitary method enabling us to (decide in each particular case whether any given expression is meaningful in the s y s t m and if so whether it is or is not an axiom of the system. Furthermore we can always .determine in a finite number of steps .an expression which we obtain from obher effectively given exlpressions by any of the rules of proof. The set of expressions provable in a 'constructible system is recursively emmerable. Conversely if the set of expressions provable in a system S is recursively enumerable, it is always possible to choose axioms and rules of proof so that they be both primitive recursive *e. A system S is callled decidable if tbe seit of its Iprovable sentences and ,the set of its ungrovable sentences are both recursively enumerable. For each decidable system Ithere exists a finitary method whkh allows us to decide whether any given senlli TLk defin'ition corresponds approximatively to the intuitione underlying R o s s e 1"s distinction between constructive and non-constructive logics. See R o s e e r [13], Compare also the remarks made by T a r 8 k i [15], P. 3% footnote 79. Some logicians believe that non-constructive systems do not deserve the name of logical systems at all and have no other value than that of a purely theoretical construction.

166

FOUNDATIONAL STUDIES

[29], 249

tence is provable in the system or nut. Conversely i f such

a method exists the system is deciddblea.

It is well to observe that tyte mtims of construetivity and of decidability are applbable t o all logical systems i-Mly whether they possess or rrot an intapzetatiin in the ,,mathematical reality” which we have discwed in section 2.1. 2.4. Systems Pa and Q,. Let ‘us now consider only such logical systems for which it is possible to define the notion of tnuth. Since the set of true sentences of a system is evidently the most important set connected with the system we prolpose to cldssify the Jbogicd systems according to the structure of !this set. We define therefore: a logical system S belongs to the class Pa or 0, if the set of (numbers which correspond to) true sentences of the system belongs to the class P!’ or 9:’. In the ,next section we shall give exalnples of systems of different classes.

3. The fdlowing two exmptes are well known from the literature. 3.1. The propositional calculus. Because of the presence of free variables in meaningful expressions of the propositional calculus it is wual to conceive them not as sentences but as matrices (or sentential functions). We prefer however a different interpretation according to which meaningful expressions of the propositional calculus are sentences, i. e. have rw) free variables. We imagine to this end each expression as preceded by a string of general quantifiers which bind all the variables occurring in the expression. It is generally known that there exists a finitary method (called the truth-table method) which allows us to decide the truth or falsity of each sentence of the propositional calcult~s. In view of the remarks made at the end of section 1.3 the existence of this method proves athat the set of ,true sentences of the propositional calculus as well as its o m p l m e n t (to the set 20 Decidable systems are usually defined as systems which posgess general recursive sets of provable sentences. Bee e. g. T a r s k i [16], p. 47. In view of a theorem of K 1e e n e [?I, p. 56 this defhition is esuivalent to the one given in the text.

[29], 250

A CLASSIFICATION OF LOGICAL SYSTEMS

167

of all sentences) are both re~~ursively enumerable. Hence tihe set of true sentences (or, more accurately, of the numbers corresponding to true sernrtences) is simultaneously of dass Pi1)and @;I, i. e. it is of class P!). Hence the propmitima1 calculus is a Tystem of class Po. 3.2. Functional calculus of the fitst order. Similarly as in the case of propositional calculus we do not conceive the expressions of the functional calculus as matrices but as sentences without free variables. W e achive this by adding a sufficient number of genehl quantifiers (binding the individual and the functional variables) at the beginning of each expression. The ,definition of ‘truth for Chis system has been sketched by Tarski [15], p. 352-357. It is known that the set of all true sentences of the f(unctiona1 calculus coincides with the set of its provable sentences 24. Since the functional calculus is a constructive system it follows that the set of its true sentences is recursively enumerable and hence the system belongs to the class P I . On thepther hand the system is not of class Po. Otherwise the system would be decidable which is known to be not the case 25. Hence the functional calculus of the first order is precisely of class PI.

3.3. Elementary arithmetic Se. The rest of this section will be devoted to a rather detailed ,discussion of an elementary system of arithmetic S, which has been defined and considered by Carnap in his bock 111. Carnap calls this system the ,,language I”. The system contains an infinite number of variaibles XI, x2, X3r...

and an inifinite number of constants denoting functions and relations. To each primitive recursive function f with k arguments corresponds a name-forming constant F f with k arguments. Similarly to each primitive recursive n-ary relation r corresponds a sentence-forming constant R , with n arguments. 24

Glide1 [5]. C h u r c h [a].

168

FOUNDATIONAL STUDIES

[29], 251

We further define the notion of a number-bhearetic expres. sion d the m t b n of a sentence. A number-theoretk expression of rank 0 consists of a nameforming constant and a sequence of variables in the same m m ber as the number of arguments of the constant. A numbertheoretic expression of rank 0 has thus the form.

Number-theoreric expressions of rank p

+ 1 are expressions

where f is any primitive recursive function of R arguments and are number-theoretic expressions of rank p at most not all of which afle of rank lower than p . I f f has no arguments and n is the (constant) value of f, then the (number-theoretic) expressim Ff is called the n-th mmeral and is denoted by n. Numlber-theoretic expressions will be denoted usua'lly (by the letter I' with subscripts. A sentence olf rank 0 is an expression consisting of a sentence-forming constant and a string of number-theoretic expressions whose number is equal to the inumber of arguments of the constant. Thus each sentence of rank 0 has the form

rl,.. . , T n

where r i s a primitive recurswe re~lati~oin with n arguments and the :?I s are nuimber-theoretic ,expressions. Sentences of rank p 1 have one of the forms

+

I

El E2

where E3 is a sentence of rank p , I' a number-theoretic expression in which the variable xk does not occur, and El and E2 are sentences of Fank p at most not both of ranks lower than p. Sen-

[29], 252

A CLASSIFICATION OF LOGICAL SYSTEMS

E eventually with subscripts. Note that in the system Sesentenws are allowed to contain fmc te.nces will be demoted by the letter

variables. The meaning of sentences (13) and (14) will be explained in 4th mxt section, where we shall define the nutim of truth f o r the system S.. For a better understanding we remark already here that we read (13) as not Elor mt E, and (14) as there is a niurnba not greater thm l? such that F 3 .

3.4. Definition of truth for the system S.. In order to define the notion of truth we must first introduce same auxiliary definitions. We shall consi'der infinite sequences whose adomains consist of all the variables x k and emnterdomah of integers. Se quences of this sort will be called simply sequences and will be -denoted by a letter Q, with or without subscripts. By Q, (k)we denote the value of Q, far the value xk of the argument, Let Q, be any sequence and I' any number-theoretic lexpnession. We shall 'define an (integer I' (Q,) by induction on the rank of I'; this integer will be saild to denote the value of 'I for the argument (-sequence) Q,, If I' has the rank 0, i. e. is of the form (lo), we put I' (a)= f (a (kl),.. . ,c9 ( k J ) . If l? has the rank p 1 i. e. 3s of the form (1 1) where rl, . .,, l?, have at most the sank p, then we put 'I (a) = f (Q, (rJ,..., Q, (Fa)). The definition of (a) is thus complete. It follows from this definition that if I' has no free variables, then I' (Q,) has a constant value independent of @. In particular r(@) = n if I' is the n-th numeral. It should further be remarked that the value of I' (Q,) can be calculated effectively if Q, and the values Q, ( k ) f o r those x k which occur in I' are effectively given. Next we define by induction on the rank of E 'the following notion: a sequence Q, satisfies a sentence E. If the rank of E is 0, ii. e. if E has the form (12), theln 4) satisfies E if and only if the elation r holds between the m.mbers r 1 (a),..., r, (a):

+

170

FOUNDATIONAL STUDIES

[29], 253

Let us suppose that we have already defined the notion of satisfaction f a r sentences of ranks 4 p and let E b e a sentence of rank p + 1. If E has the form (13), then we say that 9 satisfies E if it either does not satisfy El or does not satisfy E,. If E has ithe from (14), tlhen we say that 9 satisfies E if there exists a sequence 9,such that

a,satkfks Es, 9, (h)= (h) for h 9, (k)< r (9).

*k,

We come now to the definition of tmth and falsity: a sentence E is true i f every sequence @ satisfies E; it is false if no sequence 9 satisfies E . It should be remarked here that the law of estcluded middfe: every sentence is either h e or false does not hold in the considered system. This is evidently emsed by the presence of free variables in the sentences of our systeqm. 3.5. Auxiliary lemmas. We prove in this sectian a series of lemmas wrtJich will be nt3eded p a d y to an evaluation of the class of S. atbd partly to same consider&htns of section 4. Not all proofs will be given: those whioh are left out can be supplied easily by a straightforward iIkdndion on 4he ra& of expressions occurring in them.

Lemma 351. If I' is a number-theoretic expression and iP,(k) = 9,(k) for all k such fhat x k occurs in I', then I' (@%) = (*a).

Lemma 3.52. I f E is a sentence and 9,(k) = @,(k) for all k such that xk is free in. E, then

{aasatisfies E } = {aasatisfies E } . In the next lemmas we shall use the symbol Si Elto d p o t e tbe result of s&stit
SEPa...

&*

[29], 254

171

A CLASSIFICATION OF LOGICAL SYSTEMS

Lemma 3.53. Let I? be a number-theoretic expression with the free variables x k l , . ..,xkn, i, j two integers less than n. Let us further denote by I" the expression S."' I' and by @' a sekj quence Such t b t w ( k ) s @ ( k for ) k+ki and @'(ki)=@(kj)'*. Then

r ( q=r(q.

Proof. We use induction o n the. rank of I'. If I' is the expression (lo), then I' is the cxpressim. Y

..

.9

Xki-1

9

Xkj

3

at+,

and we have the equations

--

r'(@)=f(@(kl), * q k i -11, cp(ki), r(@')mf(*'(kJ,. - 7 9 Q'(kn)) =

---

f(@(kJ,

9

p

@(ki-A W j ) , @(ki+ d r

9

-*

9

XkJ

W i + A * - , qk)), *

-

9

'WJ)

which prove the demma for expressions of 'rank 0. Let us a s m e now that the lemma holds foc expressions of ranks < p and let 'I be the expressioh (11) -ofrank p I. We have then

+

Since the equations rj(@')+ rj'(@)hold for j = 1, 2,. .., n acqording to the inductive assumption, we obtain from the above equations r(@')=I"(@). Lemma 3.53 is thus proved.

Lemma 3.54. If E is a sentence with the free variables .., d= Ski E and CP' has the s m meaning as in lemma 'k j 3.53, t k n {@' satisfies E } = {asatisfies E' }. &,#.

xkn,

Proof. We use induction on the rank of E. If E is the expression (12), them E' = R,(I'{, . . . , r;) where the accen6 have !he W can be defined a8 a sequence which we o b i n from identifioation of the k,th term with t4e kj-th.

@

by

811

172

FOUNDATIONAL STUDIES

[29], 255

same meaning as in 3.53. From the definition of satisfaction we obtain the e(quiva1ences

{a' satisfies E } = r(F1(@'),. .., I?,(@'), {a satisfies E'} = r(I';(@), . .. , TL(@)); since Ti(@') = ri' (a) by lemma 3.53 these equivalences entail the assertion of the laemma for sentences of rank 0. Assume now that the lemma h d d s fmor sentences of ranks < p and let E be a sentence of rank p 1. We have to consider two cases: (13) and (14). In case (13) we have E' = El' 1 E; and the definition of satisfaction gives the equivalences:

+

{a'safisflesE}=((@'does not satisfyE1}or (@'does not satisfy I&})

{asatisfiesE') =((@doesnot satisfy El)or {Udoes not satisfyE,']) which immediately imply the colntenti,on of the !em.ma. were In case (14) we have E'=(Sxk)p E l since xki and x kj assumed to be free in E. Let us assume *thatW satisfies E, i. e. that there is a sequence G1such that Q1 satisfies E,, aL(h)=@'(h) for h*k, and al(k)' satisfies E.

Lemma 3.55. Let k , i be two integers, a1 and CP two sequences such that al@) =@ ( h ) for h i k , @.,(k)=i. Let 1' be a number-theoretic expression, E a sentence and put T ' = S y , E' =S f E . Then I''(@)=r(@J and {@ satisfies E') satisfies E } .

[29], 256

A CLASSIFICATION OF LOGICAL SYSTEMS

173

Proof. We apply induction on the ranks of I’ and of,€. If r is the expression (lo), where e. g . kl=k then I”=Ff(i, x k 2 ,...,xkn) and r‘(@)=f ((9(i), @(k,), ,,, (9(kn))=f(i,@(k.J,. ., @(kn)). On the other hand r(@J=i(@1(k),@l(kJ,* a&)) =f(i, @(kZ),..., @(kn)) and therefore I’((91)=l?’((9). Suppose now that this equation holds for all expressions of ranks\< p and let (1 1) be an expression of rank p + 1. We have thmI”=~~(I’,’, ..., );?I where l’i=Sfrj and from the definition we obtain

.

.

-

9

rya)=f(rl‘p(,. .., r,,ya)). r(@1)=f&pJ,. .., r n P

Since rj’ ((9) = r j (G1) by the inductive assumpti, , tbe colatention of the lemma follows. We prove now the second part of the lcmrfia. If E is the sentence (12), then E ‘ = R , ( r i , . .., rn’),

.

{ (9 satisfies E’) = r (I?((@), .. , I?:(@)),

{@, satisfies E } t r(r1(al) ,. .., rn(al))

and since l?i((9)=rj((91) we obtain the equivalence asserted in the lemma. Let us now suppose that this equivalence holds for sentences of r a n k s 4 p and consider a sentence E of rank p + 1. In case (13) E’ = El’ I Ei and the inductive assumption together with the definition of satisfaction entail the equivalences {@satisfiesE ‘ } q { Q )does not satisfyE,’) or {adoes not satisfyE i ) ) =({ a1does not satisfy El)or {a1does not satisfy E2}) ={Q1satisfies E ) . Hmence the lemlma is proved for this case. It pemains to considler tbe case of an expression E of the form

(14). Let us therefore assume that

E=(3

xdr El

174

FOUNDATIONAL STUDIES

[29], 257

where x h does not occur i n r . If b = k , then E' = E and the lemm8a follows at once from lemma 3.52. Assume now t h a t h*k. We have then E ' = ( a x h ) p El'., Suppose first that a sequence @ satisfies E', i. e. that )there is a sequence @* such that @*(j)=@(j) for j + h , @*(h)
*

@)I(" =W j ) for j h, @-(h) =@*(h). (15) We shall show tthat @**=@,-. In fact
Lemma 3.55 is thus proved.

Lemma 3.56. A sequence @ satisfies a senfence (3xk)r E if and only if it satisfies at least one of the sentences E i = S t E for

i< @(I-).

Proof. Let us first suppose that i
Suppose now that @ satisfies ( 3xc)rE. Hence fcAlows tahe existence of a sequence Ql such that Qlsatisfies E, @,(h)=@(h)for hfk, and @,(k)
Lemma 3.57. I f I' bas no free variables and I'(@)=n for any E is true (false) if and only i f at 9, then the sentence

[29], 258

175

A CLASSIFICATION OF LOGICAL SYSTEMS

least one (all) of the sentences S'; E (i = i , 2,.. ., n) is true (are false). This lemma is a direct corollary to the lemma3.56.

Lemma 3.58. I f I' is a number-theoretic expression whose free variables are some of the variables x k l , ..., Xk, and if Q(kl)=hl, -.. ,@(k,)=hn, then for each sequence
Proof. If r is the expression (lo), then S kh ll.,. *....h,. . k n r = ~ ~ h l , ...,h,,). Since this expression has no variables its value for any a, is

.,

constant and equal t o f (hl,. . . ,h,)=f(@(kJ,. . @(k,))=r(@). If I' is the expression (1 1) and we abbreviate Si:; r, as r;, then and therefore

$,--., .... kh,n r = F ; r ;

. . . . a

::::

r:,

kl,'."knr(~l)=f(r;(~,,), . .,h, ...,r;pj).

%I..

Since I';.(@l)=I'j(@)by the inductive assumption, we obtain

Lemma 3.58 'is thus proved.

Lemma 3.59. I f E is a sentence with exactly the free variables X k l , . . ., x b n and if
(16)

{@ does not satisfy E ]= - r

On the other hand E'=R,

(rl',...,r,')

(17) {E' is false}= { f o r every

(I',

(a), ...,L(@)). and

a, I-r (rI'pl),. ..,rn'pl))]}.

176

[29], 259

FOUNDATIONAL STUDIES

By lemma 3.58 rj'((a1)=rj((a) for any a1.Hence the right side of (16) is equivalent to the right side of (17) and we obtain the contention of the lemma for sentences of rank 0. Assume that the lemma holds for sentences of ranks less than p + 1 and let E be a sentence of rank p 1. We must consider separately the cases (13) and (14). If E is the sentence (13) then the inductive assumption entails the followin'g elquivalences

+

{(a

does not satisfy E} = {(a satrsfies El and E 2 } = {El' and E i are not false}.

Since Ell and E i have no free variables, the fact that a se. quence satisfiesE,' o r Ei does not depend at all on the terms of the 'sequence Ccf. lemma 3.52). This means that if a se'quence satisfies EL or E2), then every sequence does the same. We obtain therefore {(a

does not satisfy E } = {E'l is true } and {E'z is true}.

Since 'evid,ently { E i is true} and {E,' is true} = {El' I E i is false } we obtain fimnally {@ does not satisfy E]={E' is false}.

It remains ,to consider the case (14). The assumption that does not satisfy E is in this case equivalent to the following statement: i f a1 is any sequence such that

@

(18)

(al( j ) = @ ( j ) for j f k,

@,(k)G r(@),

then (a does not satisfy E S . Using the 'inductive assumption we transfdrm this statement into the following one: if Q1 fulfills

the conditions (18), 'then the sentence E;I'= SL Ej, where h

= @,(k)

is ialse. Since h may be any integer 4I'(@we ) finally obtain (19){@ does not satisfy E } = { E ; is false for h = l , 2 , ...,r((a)}.

[29], 260

A CLASSIFICATION OF LOGICAL SYSTEMS

177

On the other hand E'=(ZxJpEj an,d this sentence is false if and only if no a2and Q3 exist such that

By leimma 3.58 I"(@2)has the constant value r(@)for any QDz. If we put Q 3 ( k ) = h we infer from lemma 3.55 that (20) is equivalent to the following statement: if Q4 is asequence such that a4( j ) = a, ( j ) for j =i= k, then
that Q4(j)=Q3(j) for j # k and

@a

satisfies E i } .

we see that the right Since we can take in particular Q 4 = Q 3 side of this e1quivalencle says the satme as the right side of (19). Lemma 3.59 is thus ppoved. 3.6. Evaluation of the class of S,. We begin with the following

Theorem 3.61. Each sentence without free variables is either true or false. Moreover the set o f (numbers corresponding to) the true sentences without free variables as well as the set o f (numbers corresponding to) the false sentences without free variables are both recursively enumerable. Proof. If E has no free variables and there exists a selquence

iP which satisfies E , then according to lemma 3.52 any other

sequence does the same and E is true. The first (part of the theorem is thus proved. In order to prove tlie second part we shall ,indicate a finitary method which will allow us to decide whether any effectively given sentence 'without free variables is true or false. As we have remarked i n section 1.3 the existence of such a method entails the recursive enumerability of the considered sets. Suppose that E is a sentence of rank 0 without free variables, i. e. that E is identical with (12) where the num,kr-theoretic expressions Yl, ..., Y,j have no free variables. The values of

178

FOUNDATIONAL STUDIES

[29], 261

ri (a) are

then constants indepemdent of @ which we know to calculate effectively. Denoting t h a n by pi we infer from the definition of truth tlhat

( E is true} = ( r (pi,. .., . p n ) } ¶

.

{ E is false)3 ( - r (pl,. . , PA}. Since r is a primitrive recursive relation, we possess a-metlhhod enabling us to decide whether r holds between any given integers. Hence there exists a method enlabling to deulde the truth or the falsity of E. Let us assume thk existence of a method of this kind for sentemces of ranks 4 p and consider an expression E of rank p 1. If E has the form (13), then we have

+

( E is frue}"{E, is false)or(E,is false], ( E is false}= (Elis true} and ( E a is true) and it follows at once from these equivalences and fram the inductive assumption that there exists a method enabling us to decide *the truth o r the falsity of E. Suppose now that E has the form (14). Since E is supposed to have no free vari~bles,it follows that E, has at most one free variable x1: and that I' has no free variables at all. Hence I'(@) does not depend on and has a constant value, say R. By lemma 3.57 E is bme if and only if at least m e of the sentences Ei = Sl; E3 ( i = I,2,. .,n) is true, and E is false if and only if all the sentences E , are false. Since n can be calculated in a finite number of steps and the s e a t a c e s E i are of ranks lower than p + 1, it follows thad there exists a method enabling us to decide !he truth o r the fdsity of E . T h e o m 3.61 is thus proved.

.

Theorem 3.62. The complement of the set o f true sentences is recursively enumerable. The proof is based on the following lemma which we shall not prove in detail: the (number correspondi'ng to the) e x p ~ s -

[29], 262

A CLASSIFICATION OF LOGICAL SYSTEMS

179

sion S kbn1 , . . .,, vkn E is a primitive recursive function f of (the bn number corresponding to) E an'd of the numbers .+

Demote by A the set of (numbers of) false sentences without free variables. Acc'ording to lemmas 3.52 and 3.59 the sentence E is not true if and only if there exist integers h and k such that f(h,k,E)€ A. The set of all triples (h,k,E) such that f(h,,k,E) € A is recursively enumerable accolrdinig to theorems 3.61 and 1.62 and it is known that an existential quantifier applied to a recursively enumerable predicate yields again a recursively enumerable predicate". Hence the set of E such that E is not true is recursively enumerable Theorelm 3.62 gives an evaluation of the Gass of S, from above. The next theorem gives an evaluation from below:

Theorem 3.63. The set of (numbers corresponding to) true sentences of the system S, is not recursively enumerable. Proof.29 Let f be any primitive recursive fluunction such that the values of f are exclusivetly (the numbers corresponding to) true sentences of S e . We have to show that there exists a true sentence E such that f ( n ) $= E for any n. To this end we consider bhe fjunctions s and n such that for each (number of an) exipressim E the values s(E) and n ( E ) be numbers of the expressions S k E and SAE I SAE. It is not difficult to prove that the functions s and n are primitive recursive 30. Let I be an abbreviation for R = . We consider the following sentence E, :

(Ps(xA F h J )

I

I

I t 3 ~ 3 ) a (Fn(xJi ( F n( X J r

)I 1

Ff(x,))I

I [(~xA

(x3)

M o s t o w s k i [9],P. 90. It can be shown in the same way that the complement of the set oP all false sentences is also recursively enumerable. 2* This proof has been obtained by an easy adaptation to the syden, s, of a method due to Rosser [la]. so Compare the functions Sb and Neg defined in Ci o d e 1 [6], PP. 1& and 182. 27

28

180

[29], 263

FOUNDATIONAL STUDIES

and denote by the same symlbol E, the number corresponding to this sentence. Substituting in E, the E,-th numeral for the variable x we

obtain a sent,ence

E = Sk. E, with the num,ber s(E,). Evidently

We shall show that E # f(n) for each there is an integer n such that

R.

Suppose in fact that

Since the values of f are (hhe numbers correspondilng to) true sentences it would f,ollow that E is true and in particular that ajny sequence Q, with CP (2) = n satisfies E. Using the definition of satisfaction we infer that either

or

(23) is equivalent to @

safisfies (ax3),, I (Fn(E,), Ff(x3))

and hence to the ,existence of a sequence CPl such that (h)= @ (h) for h # 3, (3)< CP (2) = n,

~91,264

A CLASSIFICATION OF LOGICAL SYSTEMS

181

Put aI(3) = p. We see that our assumption implies the existence of an integer p < n such that if a sequence Q, with Q, (2) = n satisfies the sentence I (F, (E,), Ff (x3)),then a sequence alwith al(3)=p satisfies the sentenceI (Fn(Eo),Ff(x.J). Now we have

(a satisfies Z (Fa@,,),Ff(x2)))= (s(E,) = f(n)}, {@I

satisfies I (Fn(Eo), Ff(x3))} = (n(E,)= f(p)]

and therefore our assumption implies the following: i f s(E,) = f ( n ) , then there is a p < n such that n(E,) = f @ ) . Since E = s(Eo) we infer from (21) that the hypothesis of this implication is true and hence that there exists a p such that n(Eo)= f ( p ) . Hence n(E,) is th4enbumikir of a true statemlent. But n(E,) is the number of S ~ p E o l S ~ oi.Ee. o ,o f E IE. We arrive thus at the conclusion that E and E I G, are simultaneously true which is of course impossible. The equation (21) is thus false for every value of n:

(24)

E*ffn) for n = 1, 2,

. . ...

We further show that E is true. To this end we consider an arbitrary sequence and put Q(2) =.n ErDm (24) we obtain s(Eo)=I=f (no and hence (22) and (23)imiplies, as we knlow, that