Mathematical Interpretation of Formal Systems

STUDIES IN LOGIC AND

THE FOUNDATIONS OF MA THEMA TIC S

L. E. J. B R 0 U W E R E. W. BETH A. HEYTING Editors

~c ~

~ 1971

NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM· LONDON

MATHEMATICAL INTERPRETATION OF FORMAL SYSTEMS

TH. SKOLEM G. HASENJAEGER G. KREISEL A. ROBINSON HAO WANG L. HENKIN J. l.OS

1971 NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM· LONDON

© NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM - 1955

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First edition 1955 Second edition 1971

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TH. SKOLEM

PEANO'S AXIOMS AND MODELS OF ARITHMETIC Introduction More than 30 years ago I proved by use of a theorem of Lowenheim that a theory based on axioms formulated in the lower predicate calculus could always be satisfied in a denumerable infinite domain of objects. Later one has often expressed this by saying that a denumerable model exists for such a theory. Of particular interest was of course the application of this theorem to axiomatic set theory, showing that also for this an arithmetical model can be found. As I emphasized this leads to a relativisation of set theoretic notions. On the other hand, if one desires to develop arithmetic as a part of set theory, a definition of the natural number series is needed and can be set up as for example done by Zermelo. However, this definition cannot then be conceived as having an absolute meaning, because the notion set and particularly the notion subset in the case of infinite sets can only be asserted to exist in a relative sense. It was then to be expected that if we try to characterize the number series by axioms, for example by Peano's, using the reasoning with sets given axiomatically or what amounts to the same thing given by some formal system, we would not obtain a complete characterisation. By closer study I succeeded in showing that this really is so. This fact can be expressed by saying that besides the usual number series other models exist of the number theory given by Peano's axioms or any similar axiom system. In the sequel I will first give an account as short as possible of myoId proof of this, my exposition now being a little different in some respects. Mter that I intend to show how models of a similar kind can be set up in a perfectly constructive way when we consider some very restricted arithmetical theories.

2

§ 1.

TH. SKOLEM

Preliminary Remarks

We may set up a theory of natural numbers by adding to the predicate calculus of first order some constants namely the individual constant I, the predicate = and some functions namely the successor function, denoted by an apostroph, and addition and multiplication. Further we may assume the non-logical axioms x' ¥= I

(x'=y')

-?-

(x=y)

(y ¥= I)

-?-

(Ex)(y=x')

x+ I =x' x+y'=(x+y)' x·I =x

+x

xy' = xy x =x (x = y)

-?-

(U(y)

U(I)&(x)(U(x)

-?-

-?-

U(x))

U(x'))

-?-

U(y).

Here U denotes an arbitrary propositional function. It is most natural and convenient to let the propositional functions be those which can be constructed from equations by use of the connectives &, V and - together with the quantifiers extended over individuals, i.e. numbers. The two last axioms containing U are meant as axiom-schemes so that every individual case is an axiom. This formal system of arithmetic contains Peano's axioms. Other systems could be used as well, for example the system ZI' in Hilbert-Bernays [I], p. 293. Every proposition is equivalent to one which is built by use of quantifiers on an elementary expression, namely built by use of &, V and - on equations with polynomial terms on both sides when we replace x' by x+ 1. However, we may omit the negation, because x -=j::.y may be replaced by (Ez)((x=y+z) V (y=x+z)). Further any proposition constructed by use of & and V from

PEANO'S AXIOMS AND MODELS OF ARITHMETIC

3

equations is equivalent to a single equation because of the equivalences (1)

(a=b) V (c=d)

(2)

(a=b) & (c=d)~

(ad+bc=ac+bd)

~

(a 2+b2+c2+d2=2ab+2cd)

Thus every proposition is equivalent to an expression beginning with a sequence of quantifiers followed by an equation between two polynomial terms. As to the arithmetical functions many more are definable than the polynomials. Indeed let the proposition (3)

(Xl)' .. (x",)(Ey)A(xl , • • • • ,

X""

y)

be true. Here A is a propositional function which may be arbitrarily complicated. It may for example still contain quantifiers. The word true may mean either provable in the system or that the statement is assumed as a further axiom. Now it is well known that we can prove by use of the induction axioms that if (Ey)A(y)

is true, then (Ey)[A(y) & (z)(A(z) V (y ~ z))]

follows which means that every non void set of numbers contains a least element. Therefore from (3) follows (Xl)' .. (x",)(Ey)(A(x v ' .. ,

X""

y) & (z)(A(xv . . . , X"" z) V (y ~ z))).

Then one and only one y here exists corresponding to a given m-tuple XV"" X.... This y is therefore a function !(xI , . . . , x",). Using this function we may write (3) as a formula containing no other quantifiers than those which perhaps occur in A(~, . . , X.., y) namely as A(al , ·

.. ,

am' !(a v' .. , am))'

Repeating this introduction of functions one finds (see my paper "Ober die Nichtcharakterisierbarkeit der ZaWenreihe etc." [2], that every correct formula (Xl)' . (xm)(EYI)' . (Ey,,)(~)

. . (zJl)(E~)

. . (Eu q ) •

• •

. . .A (Xv •. Xm , YI" . y", ~, ....

»

4

TH. SKOLEM

may be written with free variables only A(€Lt, . . , am' fl(€Lt,· . , am)" . , f..(a1 , ·

•,

am), bi>' . , bp , gl (€Lt, .• , am' bi> .• , bp ) , •

• ).

Since for example we have the correct formula (x)(y)«x=y) V (Ez)«x=y+z) V (y=x+z»)

a function exists, usually written

Ix - Y!, such that

(a=b) V (a=b+la-bl) V (b=a+/a-bl)

is a correct formula. Let F be the set of all arithmetical functions in this sense. It is easily seen that F is closed with regard to the operation called nesting or substitution. Let for example z=!(x, y) and y=g(u) be respectively equivalent to A(x, y, z) and B(y, u). Then it is evident that Z= f(x, g(u» is equivalent to C(x, u, z), where C(x, u, z) is the propositional function (Ey)(A(x, y, z) & B(y, u».

It is clear after these preparations that every statement can be replaced by an equivalent equation between two elements of F containing only free variables. It is evident that all true formulas may be listed as an enumerated set S. To each of them we may find an equivalent equation whose both sides are functions belonging to F. Therefore in order to prove the existence of a model N' different from N for the set S of statements it will suffice to prove the following theorem. Let S be a set of equations whose both sides are elements of a denumerable set of functions closed with regard to nesting. Assuming the equations belonging to S all valid for the natural number series N we may define a greater series N* such that by suitable extension of all notions concerning N to corresponding ones in N* all equations in S are also valid for N*. In order to establish this I need an arithmetical lemma which I shall prove first. It ought to be added that this procedure is sufficient for our purpose because it will turn out that the equivalences we used above will remain valid in N*.

PEANO'S AXIOMS AND MODELS OF ARITHMETIC

§ 2.

5

An Arithmetical Lemma

We consider an enumerated sequence of arithmetical functions (4)

Let N(1), NCZ),

NCBI

be resp. the subsets of N for which

Mt) <Mt), Mt) = Mt), fl(t) > Mt).

One at least of NU), NIZI, N(S) is infinite. Let N 1 be that with the least upper index which is infinite. Then there are for each t E N 1 at most 5 possible cases for fs(t) in relation to Mt) and Mt), namely if for example N 1 is NU) fs(t)<Mt), fs(t)=fl(t), fI(t)max (i, j). The function g(t) is steadily non-decreasing. In our applications of this lemma we will assume that all polynomials, in particular all constants, occur in (4). Then one sees that the values of g(t) cannot possess an upper bound, because the intersection of all N, is the null set.

§ 3.

The Proof of the Existence of N*

Let F be an enumerated set of arithmetical functions of one or more variables containing besides the successor, addition and multiplication all functions occurring in the left and right terms of a set S of equations with only free variables supposed true for N, F further being supposed closed with regard to nesting. Let F 1 be the denumerable subset of F consisting of the functions I,(t) of one variable. Then relations < and = can be defined between the elements of F 1 in the following way. According to the lemma a function g(t) exists such that for any two i and j one of the three relations

holds for all t>max (i, j). I put respectively

t.ct; 1.=/i' t.>t, in these three cases. It is easy to see that the relation = thus defined is an equivalence relation and that the relation < is asymmetric and transitive. The different equivalence classes of the elements of F 1 defined by = shall then constitute the diverse elements of N*. In a very simple and natural way every function I(xv' ., x,,) in F can be extended to mean a function in the domain N*. Indeed, if every X,(t) for T= 1, .. , n is E F v then y

is

E

=

I(X1(t), . . . , X,,(t))

F v because F 1 is closed with regard to nesting. Further one

PEANO'S AXIOMS AND MODELS OJ' ARITHMETIC

7

easily sees that if X,.l(t) and X".(t) for r= 1, .. , n are the same elements of N*, then and

Yl

f(Xl,l(t),

, X",l(t))

Y 2= f(X1,2(t),

, X".2(t))

=

denote the same element of N*. Thus f also defines a function in the domain N*. Clearly all elements of N also belong to N*. Indeed they are furnished by the f in F l which are constants. Further, since the relation = has been defined in N*, an tke equations constituting S have a meaning in N*. It remains to see that they are all valid in N*. Let us consider an equation in S. It has left- and right-hand terms with some free variables, say a v all' . .. Replacing these by arbitrary elements of N* we get an equation in N*. Since this is valid for every value of tin N, it is valid for every t' when t has been replaced by g(t'). A fortiori the equation takes place for all t' > the maximum of the indices which the left- and right-hand sides of the equation possess in the sequence fl(t), f2(t), . . .. Thus, remembering the definition of = in N*, we see that the equation holds good for arbitrary elements in N*. Now I will prove that the equivalence (1) remains valid in N*. The correctness of the implication (fX={3) V (y= 15) -+ (fXt5+{3y=fXy+{3t5)

for arbitrary elements fX, {3, y, 15 of N* is seen so easily that I may confine myself to the treatment of the inverse implication. Let fXt5 + {3y = fXy + (3t5. This means that for all t' > h say and t = g(t') (5)

fX(t)15(t) + (3(t)y(t) = fX(t)y(t) + (3(t)b(t).

The number h may be chosen as the maximum of the indices of the left- and right-hand sides of (5) in the sequence (4). According to (1) this implies that for every t'>h and t=g(t') (fX(t) = (3(t) V (y(t) = V 2 may be characterized by the formation rules as being of the 1 This formulation presupposes that there is no typographical distinction between free and bound variables.

16

G.

HASENJAEGER

same type. Then "suitable" is to include S(v1 ) =S(v2 ) . (Thus the domain of individuals may be cited as S(x) for an arbitrary individual variable x.) Some constant c may be characterized as being of the same type as some variable v. Then "suitable" is to include S(c) EO S(v). Some constants or variables may be characterized as designating functions (from -- to -). Then "suitable" is to be understood as requiring the designata (in the case of constants) to be such functions, and the ranges (in the case of variables) to consist of such functions. The logical constants are used here in the sense of the twovalued logic (but this could be adapted to other purposes). A model is standard (Henkin) if and only if all ranges for variables for functions are extensionally maximal. 2 Given a model S and an assignment 1> of objects to the variables and constants 3 of a system, satisfying the conditions 1>(v) EO S(v) and 1>(c)=S(c) 3, a designatum s4>(m) is attached to each wff m of the system (see for instance [5], p. 83, 84). 1> may be called an assignment over S. For wff's m of the propositional type S4>(~O is T (truth) or F (falsehood), as technical counterpart for the concept of satisfaction. m is called S-valid if S4>(m)=T for every 1> over S. 4

§ 2.

The Concept of Universal Consequence

~( being a wff of the propositional type and r being a set of such wff's, m is called a universal consequence of (short: rUl-m) if and only if

Dl: for every standard S such that each element of misS-valid too.

r is S-valid,

2 The asswnption that such domains exist seems to be the only justification for considering impredicative comprehension principles, but these set-theoretical principles do not fully describe such domains. 3 This is only to simplify the notation of the definition of satisfaction in Tarski's sense. 4 "For every " is used here in the extensionally maximal sense. Certain restrictions of this may be described by modifications of S, and others are not needed here.

ON DEFINABILITY AND DERIVABILITY

17

To distinguish from a later definition the relation U~ is called standard universal consequence. The definition of universal consequence is a generalization of the following "classical" concept of consequence I~: r I~ ~l if and only if D2: for every standard S and every ep over S: if S ¢($B) = T for every $B E r, then S.p(m)=T. U~ seems to be a better description of the actual intentions of mathematicians in making deductions than I~ . The difference between U~ and I~ is this. I~ makes no distinction between free variables and constants; for under I~ each free variable is treated like a constant, in the sense that, for any interpretation a fixed (but arbitrary) denotation is given to the symbol. Since this is the treatment of the constants by U~, U~ may be considered as a generalization of I~, and UI- and l~ restricted to closed wff's coincide. On the other hand, all effective variables are implicitly generalized at the beginning of each formula by the definition of U~. Thus Ur gives the interpretation of the customary use of free variables in the universal sense, and so a kind of "legitimization" of such formulations of number theory as in HilbertBernays' Grundlagen der Mathematik I, § 6, § 7, where variables not generalizable in the language are used in the universal sense: number variables in the recursive number theory, and predicate variables for the formulation of complete induction in the wider number theory. Such variables let us call essentially tree variables. Thus U~ includes the intended concept of consequence for languages of several sorts: (i) all free symbols are regarded as potential names for the basic concepts of a theory (this case of U~ is reduced to I~), (ii) the general case (there are both extralogical constants and essential free variables, as in the above mentioned systems of number theory), (iii) there are essentially free variables but no extralogical constants.

18

G. HASENJAEGER

Our aim here is to show that case (iii) yields problems analogous to those of case (ii) arising from the incompleteness of the usual formalized number theories. In spite of the above generality of definitions from now on (until a final remark) we restrict our considerations to systems without bound function variables. These offer all problems except those arising from impredicativity. The three cases are given by the corresponding classification of the predicate symbols of a given first order language into variables and constants.

§ 3.

Incompletability Theorems for the relation of Universal Consequence

In the case (i) (where UI- reduces to 11-) a complete syntactical characterization of UI- is given by a general form of Godela completeness theorem. For case (ii) this is excluded by the fact that the class of universal consequences of the finite number-theoretical axiom system Z (Hilbert-Bernays I, p. 371) is not even arithmetically definable in the sense of Kleene-Mostowski 5, hence not axiomatizable in the usual syntactical sense. Similarly, as we shall show below, we get non-axiomatizability in case (iii). Consider, then, the special case {&} UI- 5B of universal consequence, where &,5B are wff's of the first order calculus, all predicate symbols of which (excepting at most the identity sign) are effective variables. This gives via arithmetization a relation which is not in P~2), hence not axiomatizable. Combining this with a similar result we prove

Thm. A: The restricted relation UI- is not in

P~2)

U Q~2).

Proof: It suffices to consider formulas not containing the identity sign. Let lY be a fixed formula valid exactly in the finite domains, & a fixed generally valid one, and To the class of all generally valid formulas. The well known syntactical definition of To shows that To E P~l), and by the theorems of Church and Post we have To ¢ Q~l). 6

A. Mostowski's terminology is used here; see [7].

19

ON DEFINABILITY AND DERIVABILITY

(1) Suppose U~ E P~2). Then from the equivalence (for the language without identity) ~! E

Fo if and only if

not

{2!} U~

~

we would get F o E Q~l). (2) Suppose U~ EQ~2). Then from the equivalence

2! E F o if and only if {&} U~ 2! we would get F o E Q~l). We remark that in the case (iii) the restricted UH is in the Boolean algebra generated by the elements of P~2). Thus in case (iii), as in case (ii), the question always arises whether a given concept of formal derivability yields a given universal consequence from given premises; and in both cases (when restricted to first order calculi) a proof of non-derivability may be arranged by the elementary method of re-interpreting all essentially free variables in the premises as syntactical variables, so that premises containing those variables change into schemata. But I think that the use of the following concept also in the first order case also helps us to understand the general case somewhat better. § 4.

The Concept of Universal K-Consequence

A less elementary arrangement for the intended non-derivability proofs which, however, is more instructive in the above mentioned sense can be given by adapting D1 to the given concept of formal derivability. We introduce into D1 a parameter K, denoting a class of models not necessarily standard, i.e. we define: 2! is a universal K -consequence of F (short: F U~ K 2!) if and only if D3: for every S E K such that each element of F is S-valid, 2! is S-valid too. Obviously D3 contains D1 as a special case. On the other hand D3 can be adapted by a suitable choice of K to various concepts of formal derivability in the following sense. If "F f- 2!" denotes

20

G. HASENJAEGER

that m is derivable from I', then adequacy of K for pressed by

r f- m if

and only if

r u~K m,

for all rand

f-

IS

ex-

m.

That a given K is adequate for a given f- is a completeness theorem in the sense of Henkin's completeness in the theory of types [5]. For the concept of derivability for a first order calculus (short: f-p) whose predicate symbols are characterized as variables by a substitution rule, the class K o of all models S satisfying the following condition can be taken for K. S",(A.!m(!))

E

S(Pl'I) for each

(! short for

Xl' •.• ,

ep over S. 6 xl'I)

Thus we have the

Thm. B: If rf-p

m,

then

rU~K.

m,

which justifies the underivability proofs by models E K o. For the other part of the adequacy, we get by a modification of the completeness proof of Godel (or Henkin) the

Thm. 0: If r

U~K.

m,

then

rf-p m,

7

which asserts that there are enough models for non-derivability proofs. On the other hand theorem A yields that there is no universal method to find such models. Thus instead of a detailed proof of 0 I shall discuss some special cases in which models have been found.

§ 5.

Special Cases of Non-derivability

In the case (iii) (no extra-logical constants) for standard S S-validity depends only on the cardinal number of S(x). If the • I.e.: S(P") is closed under all functions, defined by terms A!m:(!), of arguments given by 4>(v) for all variables v free in ).!m:(~). - If, as usual, ).·terms are not in the calculus, they may be adjoined in order to get simpler descriptions of this condition and of the substitution rule for predicate variables. 7 The idea of the application of that proof method is contained in § 5, (B).

21

ON DEFINABILITY AND DERIVABILITY

class of cardinals which yield models of the wff m: is called the spectrum of m: (short: Sp(m:)), we have {2t} Ub ~ if and only if Sp(m:)

c Sp(~).

Thus the questions formulated in Hilbert-Bernays I p. 124, footnote 1, and answered in [2], are examples of the problem here considered. Now I shall discuss the derivability between two formulas of the predicate calculus with identity, which is perhaps of some number-theoretical interest. Let ~(A) 8 be a single axiom for the theory of Boolean algebras and let %(B) 8 be a single axiom for the theory of fields of characteristic 2. To develop these theories the symbols A and Bare naturally to be considered as constants in the sense of § 2. It is well known that each of the formulas ~(A) and %(B) is satisfiable in precisely those finite domains whose cardinal is a power of 2, and in all infinite domains. Consider now the wff's ~(A) and %(B), where the constants A, B have been replaced by variables A, B. ~(A) and %(B) are valid for exactly the same standard models, namely those models whose domain of individuals is finite and is not a power of 2. So we have I'o.J

I'o.J

I'o.J

(1)

I'o.J

I'o.J

%(B) U~ ~ ~(A)

and (2)

~ ~(A)

Ur ~ %(B),

and the question arises if ~ ~(A) and ~ %(B) are derivable from each other. I shall treat only the case (1). For (2) see the final remarks. In order to show that ~ ~(A) is not derivable from ~ %(B) it would suffice to give a model S (E K o) for which ~ %(B) is valid but not ~ ~(A). But the lemma which would be used to construct the model 8 For simplicity I assume that each of these wff's contains exactly one predicate symbol other than identity, and that no individual constant appears among the basic concepts. In >8(A) A may be a symbol for the partial order or the Sheffer function. A general method is suggested by the following equivalence

Axyzu _ x

+y

=

z /\ x'y

=

u.

22

G. HASENJAEGER

by a general method (see below) gives an immediate proof by the methods used in [2]. Namely it suffices to show the Lemma ("), For each substitution Bjk£9l(!, a, A) there is a model 8 and an assignment cPo such that 8(a), we get S",.(Va ,...,,%(J,rm:(r, a, A)))=T.

This completes the proof. From the lemma (*) the asserted underivability can be shown in the following ways, (A) or (B). (A). The method of the "Ruokverlegung der Einsetzungen" (see Hilbert-Bernays I, p. 225) shows that each proof from %(B) can be transformed into a proof from a finite class of formulas ,..." %(A-rm:1(r, ... )), ... , ,..." %(A-rm:,,(r, ... )) without the use of further substitutions. The implication i--.

permits us to derive the above substitutions in ,..." %(B) from a single substitution ,..." %(A-rm:(r, a, A)), and a proof of ,..." ~(A) from this can be transformed into a proof of

which is excluded by the lemma (*). Note that the S in the proof of (*) is not in K o, since generally S",(A-rm:(r, a, A)) is not in S(B) for 4>(A)*4>o(A). For, 4>(A) may require fixpoints other than 4>(0). In order to get a model in K o we have to modify the preceding proof nearly in the following way. (B). A contradiction (without use of substitutions) in the following set M consisting of (a) the wff ~(A), (b) the wff's ,..." %(BfP) for all possible variables P, (c) all wff's Vr(Pr +---+ (£:(r, ... )), where P is suitable chosen to avoid circularities in the dependence of P from the parameters in (£:(r, ... ), reduces to a contradiction in some finite subset E of M. By a procedure analogous to the "Ruckverlegung" we remove all predicate variables but A in such a way that all wff's (c) become 10

For the proof see [2], p. 274. Moreover the equivalence is valid.

24

G. HASENJAEGER

logical theorems. The finitely many cases of (b) are reduced as above to a single substitution such that a contradiction in

would arise, contrary to the lemma ("). Thus M is consistent. Each known completeness proof gives a "model" (S', ep') where S'(x) is the set of the ep'(z) for all individual variables z, whereas the domains S'(R") need only to contain the ep'(R"). If we define S'(P") to be the set of the ep'(Rn) for all R",

then by the truth of the wff's (c) the S'-validity is closed under the general substitution rule (see Godel [I], Satz X, Henkin [4]; also Henkin [6]). § 6.

Final Remarks

By the construction of the model S' in (B) we learn almost nothing about S'. Thus there may be some interest in the following model So, though one essential question concerning So remains open. So is constructed from the Boolean algebra (J0 consisting of the finite sets of natural numbers and their complements in the following way. SO(x): the domain of (J0, SO(R"): the class of all n-adic relations over SO(x) invariant under the group of automorphisms of ({J0, a) for at least one finite sequence a of elements of {J0.

lt can be shown that SO-validity is closed under substitutions of formulas even containing bound predicate variables (see [3]). is not SO-valid. But the question whether Obviously ,...., ~(A) ,...., '1:(B) is SO-valid requires a better knowledge of the infinite fields of characteristic 2. 11 A positive answer in connection with a which, formalization of the intuitive prooffor {,...., '1:( B)} U~ ,...., ~(A) I suppose, needs the axiom of choice, would give another independence proof for this axiom. Knowledge of the automorphisms 11 In the symposion I asserted too much. I neglected the difference between the model 8 in (*) and the model 80.

ON DEFINABILITY AND DERIVABILITY

25

of a suitable infinite field of characteristic 2 is needed also for the problem whether v-c 58(A) h ,...., ~(B), for there are not enough automorphisms in the finite fields to prove a lemma analogous to (*). REFERENCES [1] K. GODEL, Die Vollstandigkeit der Axiome des Iogisohen Funktionenkalkuls, Monatshefte fur Mathematik und Physik, 34 (1930), 349-360. [2] G. IIASENJAEGER, Uber eine Art von Unvollatandigkeit des Pradikatenkalkills der ersten Stufe. Journal of Symbolic Logic, 15 (1950),273-276. [3] , Some non-standard models of impredicative comprehension axioms, in preparation. [4] L. HENKIN, The completeness of the first- order functional calculus, Journal of Symbolic Logic, 14 (1949), 159-166. [5] , Completeness in the theory of types, Journal of Symbolic Logic, 15 (1950), 81-91. [6] , Banishing the rule of substitution for functional variables, Journal of Symbolic Logic, 18 (1953), 201-208. [7] A. MOSTOWSKI, On definable sets of positive integers, Fundamenta Mathematicae, 34 (1946), 81-112. Institut fur mathematiache Logik und Grundlagenforschung der Universitab, MUnster i W., Germany.

G. KREISEL

MODELS, TRANSLATIONS AND INTERPRETATIONS

I propose to discuss these three syntactic relations between formal systems from the following points of view: their relation to the consistency problem, their relation to each other, and some of their uses in understanding informal mathematics. I shall deal not only with axiomatizable systems, i.e. those whose axioms form a (general) recursive set, but also with certain non-axiomatizable systems since they are better formalizations of such branches of mathematics as arithmetic and set theory. The systems in which these syntactic relations are established, will, in general, be mentioned explicitly. (Both the use of non-axiomatizable systems and explicit mention of metamathematical methods of proof may be regarded as a natural reaction to Godel's incompleteness theorems; for, these show (i) that axiomatizable systems are not satisfactory approximations to certain branches of mathematics, (ii) that various formalizations of these branches of mathematics are of different "strength", so that one may expect to learn essentially more from the particular proofs of a theorem than from its assertion). It turns out that most of our work is finitist lor, more precisely, is formulated in quantifier-free systems with decidable predicates and computable functions. This comes about as follows: syntactic relations can be arithmetized in elementary arithmetic, and, as is explained at length in [1], p. 113, quantifier-free proofs are particularly appropriate in arithmetic; further, proofs in the elementary 1 I do not need a definition of this word since I never try to show that some particular theorem cannot be proved by finitist means; the reader may give his own definition and see that it fits our work; if it doesn't he may wish to revise his definition.

27

MODELS, TRANSLATIONS AND INTERPRETATIONS

quantification theory of arithmetic may be replaced by quantifierfree ones, [1], pp. 122-123. It is worth noting that the treatment of non-axiomatizable systems is quantifier-free, too. The main conclusions are summarized at the end of each section. The reader is warned that current usage of the words "model", "translation", "interpretation" is not uniform. Notation

§ 1.

h h, (Z-f-), (Nw-f-), (N" -f-), (II-f-) mean in this order: can be proved in (8), (8 i ), Z (quantification theory of + and . with induction), N w (primitive recursive arithmetic), N" (ordinal recursive arithmetic of order iX), II (predicate calculus). Con 8: a formulation of "(8) is consistent" which satisfies Godel's second undecidability theorem. Prov, (m, n): m is (the number of) a proof in (8.) of the formula (with number) n. General recursive functions are denoted by Greek letters: v.(n) is the negation of n in (the numbering of) (8.); the value of a(n, m) is the term obtained by substituting the numeral Olffl) in the term n. 1\: for all functions f; /

V: there exists a function /

f.

N.R A system N" consists of the elementary calculus with free variables, defining equations for a particular primitive recursive ordering of the integers which has been established to be a wellordering with ordinal iX, and schemata for definitions and proofs by transfinite induction based on this ordering of the integers, cf. [l], p. 113. The system is determined by the particular ordering used, and not by the ordinal iX. For our purpose the non-axiomatizable (class of) systems consisting of all N", for a given iX, is unsuitable: in such a system every formula (x)(Ey)A(x, y) with recursive A can be decided since there is an effective method of constructing to every such A a primitive recursive ordering < A which is a well-ordering (of order w 2 ) if and only if (x)(Ey)A(x, y) [Markwald, Math. Annalen, 127 (1954) p. 148]. Kleene has announced (personal communication) that every arithmetic proposi-

28

G. KREISEL

tion has a similar equivalent. In other words, this class of systems is of very high degree of undecidability. § 2.

Models

The familiar consistency proofs of various geometries, the algebra of complex numbers, or-to take a modern case-of general set theory (G), [2], are obtained by means of models. This notion, which Tarski, [3], p. 20, calls "interpretation", may be defined for systems of the first order predicate calculus as follows: A system (8 1 ) has a model in (8 2) if the non-logical constants of (8 1) , i.e. its predicate symbols and function symbols, can be replaced by .expressions of (8 2) in such a way that the axioms of (81 ) go into theorems of (8 2 ) . The discussion of models is best subdivided into three groups: (8 1 ) has a finite set of axioms, (8 1 ) has an infinite, but recursive set of axioms, (81 ) is not axiomatizable. In the first case the model can be exhibited in full, in the other cases we need a syntactic proof to show that a given replacement of the non-logical constants of (8 1) constitutes a model. FROM MODEL TO CONSISTENCY

(i) If (8 1 ) is finitely axiomatizable and has a model in (8 2 ) then (N.,-H(Con8 2 --+ Con 8 1 ) , [4]. For certain systems (8 2) , if a finitely axiomatizable (8 1 ) has a model in (8 2 ) then h Con 8 1 ; e.g. if (8 2 ) is the system Z, or, more generally, if there is a normal truth definition in (8 2) for quantifierfree formulae of (8 2 ) , This is not possible for all (8 2 ) : consider a finitely axiomatizable system (82 ) which satisfies the conditions of Godel's second undecidability theorem, when h Con 8 2 is false though every system has a model in itself. (ii) If (8 1) has infinitely many axioms, which are mapped into the formulae <X2(n) of (8 2 ) and f- (Ey) Prov, [y, <X2(n)] or t- Prov, [n(n) , <x2 (n )] then t- (Con 8 2 --+ Con 8 1 ) . It should be noted that, e.g., one can prove in Z the existence of a model of (G) in Z and hence (Z-H (Con Z --+ Con G), but not (Z-H Con G.

MODELS, TRANSLATIONS AND INTERPRETATIONS

29

If (8 1 ) is not axiomatizable, we require a partially recursive function fl(n) such that if n is an axiom of (81) , fl(n) is the number of a proof in (8 2 ) of the model of n. In practice there is an important case where (8 2 ) is a subsystem of (8 1 ) and (8 1 ) and (82 ) are obtained by adding the same axioms to the axiomatizable systems (8~), (8~): in this case fl(n)=n. Below we shall consider extensions of (8') which are obtained from (8') by adding verifiable primitive recursive formulae; these non-axiomatizable systems may be handled by quite elementary means. (iii)

FROM CONSISTENCY TO MODEL

If (81 ) has a finite set of axioms and h Con 8 1 where (82 ) contains the system Z then a model of (8 1 ) may be defined in (82 ) : this is the famous Loewenheim-Skolem-Godel-Bernays completeness theorem. It is known that under the present conditions (81 ) does not generally have a recursively enumerable model, see, e.g., [6] or [7]. (The proof that the model is not necessarily recursive is given in [10].) The situation is not very different with higher order predicate calculi, [8]: if h Con 8 1 and (82 ) contains Z then a model of (81 ) may be defined in (8 2 ) where the models for the sets of (8 1 ) range over a class of arithmetic sets definable in P2 or 22 (in Mostowski's notation), [9]. MODELS OF ARITHMETIC AND SET THEORY

The construction of models has, perhaps, contributed more towards an understanding of mathematics, in particular of the impossibility of characterizing by axiomatic means the notions of "integer" and "set", than any other single work of mathematical logic. In picturesque language: every consistent system of arithmetic has models which are too large; i.e. while these models contain (representatives of) the numerals they also contain other terms like Skolem's functions; on the other hand every consistent system of set theory has models which are too small; i.e. they possess models by classes of sets, as in [9], which satisfy all the

30

G. KREISEL

closure conditions imposed by the axioms and rules of proof, yet these classes do not exhaust all "arbitrary" sets. SUMMARY

The notion of a model allows one to study relations between (axiomatizable and non-axiomatizable) systems which contain the same logical apparatus 2 (classical quantification theory), but the notion is not defined for systems with different logical character; in particular, it is not suitable for comparing essentially undecidable and decidable systems, or classical and intuitionistic systems. § 3.

Translations

Two well known consistency proofs, which do not use models, relate systems of different logical character to each other; one is the short consistency proof for the predicate calculus which shows that every theorem in this system is true in a universe consisting of a single individual, and, since the latter statement can be expressed in the propositional calculus, we have a relation between the predicate calculus and the propositional calculus; the other is G6del's proof, in primitive recursive arithmetic, which establishes that a formula is provable in Z if and only if a certain associated formula is provable in intuitionist arithmetic. Wang [11] has generalized these relations in his definition of "translation" which we reproduce with certain modifications: 3 A recursive function r(n) is an (8)-translation of the system (81 ) into (8 2 ) if

f- (Ey) Prov, (y,

n)

-'>-

(Ey) Prov, [y, r(n)]

2 Tarski [3], p. 22, footnote 17, mentions the interpretation of quantifiers but gives no details: presumably he means some such definition as

(x) +--> -, (Ex) -, .

3 (i) If A 1 of (S1) is associated with As of (Ss)' Wang requires that [-1 A 1 be equivalent to [-2 As in Def. 2 of p. 284 of [11]; he does not use this condition, which is not satisfied by the "translation" of the predicate calculus into the propositional calculus. (ii) He considers only Z-translations, which seems an artificial restriction.

MODELS, TRANSLATIONS AND INTERPRETATIONS

and

~ T[v1(n)]=v 2[T(n)]

31

(if n is a formula of (B1)).

(The definition assumes that the systems (B 1 ) and (B 2 ) contain symbols for negation). If (B) is externally consistent ([12]) with respect to recursive functions, the existence symbol may be eliminated: ~ Prov. (p, n) ~ Prov, [n(p), T(n)]. TRANSLATION TO CONSISTENCY

If (B 1 ) has an (B)-translation into (B2 ) then

~

(Con B 2 ~ Con B 1 ) .

CONSISTENCY TO TRANSLATION (LIMITATION)

There are systems (B), (B 1 ) , (B 2 ) such that ~ (Con B 2 ~ Con B 1 ) , but (B 1 ) has no (B)-translation into (B 2 ) . Let (B) be the system Z, (B1 ) the system (Q) of [3], p. 51, (B2 ) the propositional calculus IIo. Then, by [5], (Z-H Con Q and hence (Z-H (Con IIo ~ Con Q), but (Q) cannot be Z-translated into any decidable system, such as IIo : if T(n) is the proposed translation, let T1(n)=O if (the formula with number) T(n) is provable in IIo' T1(n) = 1 if T(n) is not so provable; then there is a term q of (Q) such that T1(q) = 1 has the number q, and by the diagonal argument we get a contradiction. (See the remark below.) MODEL TO TRANSLATION

If (B 1 ) has a model in (B 2 ) and if this model is established in (B) then (B 1 ) has an (B)-translation into (B 2 ) . TRANSLATION TO MODELS (LIMITATION)

There are finitely axiomatizable systems (B 1 ) and (B 2 ) which can be N w-translated into each other, but have no models in each other. Take for (B1 ) the elementary quantification theory of addition, for (B 2 ) that of multiplication (or of some suitable monadic predicate). Then, since these systems are decidable (in N w) we can map Ai of (Bi) into 0 = 0 if h Ai and into 0 i= 0 if h Ai is false. But it is known that the systems mentioned above have no models in each other. ; Xl' . X", Yl' .Y,,) for all m. The reason for this somewhat lengthy definition of "w-consistency" is given on p. 41 of [7]. Incidentally, the proof below gives a solution to problem 2 of JSL 17 (1952), p. 160. An analogous but simpler argument establishes the w-consistency of the elementary quantification theory of addition and multiplication (without induction). We show that this argument cannot be formalized in Z while it is well known that the consistency of this quantification theory can be proved in Z. Since the proof uses the notion of a computable functional it seems desirable to begin with a section on this notion. § 5.

Computable Functionals

Informal Idea. (i) By a "computable function" with integral arguments and values is meant a method which provides for any integer 0("> an integer O(m,,> (its value). It should be noted that this definition applies to representations of integers by numerals only: e.g. the function lX which satisfies the relations lX(O) = 1, lX(n+ 1) = 0 is computable; yet, if in some particular system (Ex)A(x) is undecidable, these defining conditions do not decide the value of lX[,u.,A(X)] in the system concerned; in other words, representations of integers by ,u-symbol expressions are (properly) ignored in the definition of computability. (ii) By a "computable functional" (whose arguments range over integral valued functions of the integers) with integral values is meant a method which provides for any such function f an integer o(m,>; the method should be such that once a sufficiently large, but finite number of values of f has been computed the value of m, should be determined; it is not assumed that f is computable,

40

G. KREISEL

i.e, that a method of computing t has been given in advance. The notion of a computable functional is not as definite as that of a computable function since there is no analogue to the representation of integers by numerals. However, we shall see below that in practice this difference is less serious than appears. Standardization. Kleene has given standard forms for computable functions and functionals: (i) primitive recursive functions ~(l; m, n), i(n) such that, if tP is a computable function there is an integer l~ with the properties

(a) for all n there is an m such

~(l.p;

o(n), o(m))=o

(b) for all n, tP(o(n))=i{,u.,[f)(l.p; o(n), x)=O]}; (ii) a primitive recursive predicate T(l; m), such that if (/J is a computable functional there is an integer l.p with the properties

(a') For all t there is an m such that T[l.p, t(m)) where t(m) denotes

.II p:(i)+\ Pi being the i th prime (Po=2).

,<m

(b

/)

For all [, (/J(f) = j{,ullT [l .p, t(y)]}.

Observe that, if these conditions are satisfied, for any given [, one can verify this fact a posteriori by means of a finite procedure: if there is an n satisfying (a') one need only calculate the values of t for arguments ~ n, and then one can choose the smallest n satisfying condition (a /). The values of t for arguments exceeding n do not enter into the calculation of the functional. The indefiniteness of the notion derives from the fact that the (crucial) condition (a') may be satisfied for all t of some large class (e.g. all arithmetically definable functions), but not for certain impredicative functions. In practice the analogy with computable functions is to some extent restored by the indefiniteness of the notion of a "correct proof" for condition (a): e.g. most of the proofs of condition (a) proceed by ordinal induction with respect to some ordering - be the yth pair of integers in some ordering of pairs of integers, and, for each numeral O(1Il, let p(n) be the number of the formula (Ey)(z) {, 0(0(11), y, z) V Prov {Yl' VC1[0(1I), s(a, a), Y2]}}'

with the free variable a. Let q(O(1Il) denote the expression s[p(O(1Il), p(O(1Il)].

Then (the value of) q(O(1I)) is the number of (Ey)(z){,O(O(1Il, y, z) V Prov {Yl' Ya[O, b2)] V Prov {"Pl(o(nl, b2), ya[o(nl, q(o(nl), "P2(0(111, b2)]}

(iii).

If n is the number of the functional N(f), we have, rewriting (iii), ,own), N('fJb,),

n; N('fJb,)]

V Prov {NI('fJ",) , va[o(n), q(o(nl), N 2 ('fJb,m

(iv).

46

G. KREISEL

But, as in the discussion of interpretations above, we get from (iv) (Ey)(z){, O(O W(c)

is deducible from K. Then i.e. VI/'- ... r-: Vm:>Z

also in deducible from K. We write VI/'- ..• r-; Vm= V(a I, ... , ak)

where we have distinguished all the constants a v ... , a k which belong to M' but which do not belong to M. Accordingly, these constants are contained neither in X nor in any of the elements of K. It then follows from the rules of the calculus of predicates that the statement [(3YI)'" (3Yk)V(YI' ... , Yk)]:>Z

also is deducible from K. We propose to show that the statement X

=

(3YI) ... (3Yk) V(YI' ... , Yk)

satisfies the conditions of the theorem.

ORDERED STRUCTURES AND RELATED CONCEPTS

55

Indeed, V(Yv ... , Yk) has the required form since V(~, ... , a k) is a conjunction of elements of N'. Also, X holds in M' since V(a 1 , •.• , a k ) holds in that structure. On the other hand it is not possible that X hold also in M. For in that case there would exist constants bv ... , bk in M such that V(b v ... , bk ) holds in M. Since V does not contain any quantifiers it would then follow further that V(b v ... , bk ) and hence X = (3Yl) ... (3Yk) V(Yl' ... , Yk) holds also in all extensions of M. X would accordingly be deducible from N. But X ~ Z is deducible from X, and so the statements X, X ~ Z and hence Z would all be deducible from K u N. This is contrary to the assumption that the pair {M; Z} is undecidable. Thus X does not hold in M; the theorem is proved. As stated above, model-completeness does not in general entail completeness in the ordinary sense. However, it can be shown that if the model-complete set of statements X possesses a primemodel then K is also complete in the ordinary sense. The model M of the set of statements K is said to be a prime model of K if every model of K is isomorphic to an extension of M. § 3.

Applications Let K be a set of axioms for the concept of an algebraically closed field formulated in terms of the relations of equality, addition, and multiplication. We wish to show that X is model-complete. For this case, we may interpret the main theorem as follows. Consider any finite system of equations and inequalities which belong to the following types: 0 DELz, .•. asserting existence of number sets such that DELi is provable in L, if and only if i :S j. In particular, we know that L o may be a system such that all predicative number sets expressible in Q are provable in L o. When such is the case, the formulas DELo, DELI' DELz, '" define a sequence of impredicative number sets which are in a sense of monotone increasing complexity. Let Q be the same and L be an arbitrary subsystem with finitely many axioms. Mostowski proves in [15] the further theorem: Theorem IV. If Con (L) is the arithmetic formula expressing the consistency of L, then Con (L) is a theorem of Q. A more detailed proof of the same theorem which depends more closely on the constructions of Wang [26] is given in McNaughton [14]. In his proof, McNaughton uses the enumeration of sets of L, as described above, to provide in Q a "transformed" normal truth definition of L (terminology of p. 270, Wang [28]). This is accomplished by saying, instead of formulas being satisfied by finite sequences of sets of L, that formulas are satisfied by finite sequences of their corresponding integers in the enumeration.

§ 5.

Certain Special Methods of Enumeration

In a system of set theory, there are axioms which assert unconditionally that such and such sets exist, and also axioms which assert that if such and such sets exist, then certain other sets related to them in certain definite manners also exist. For example, the axiom of infinity in the Zermelo set theory belongs to the first kind, while the axiom of power set, the axiom of sum set, etc. are

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HAO WANG

of the second kind (given a set, its power set exists; given a set, its sum set exists). There are also axioms which do not seem to generate new sets, e.g., the axiom of extensionality. The general method of enumeration discussed in § 3 amounts to treating all the axioms formalistically and in a uniform manner. There is a strong temptation to approach the matter on a more intuitive basis. For example, we are inclined to disregard the axiom of extensionality and just enumerate the sets required to exist by the axioms which assert set existence conditionally or unconditionally. When we enumerate such a model, a special proof is usually necessary to establish that the axiom of extensionality is also satisfied. In certain special cases, on account of peculiarities of the system under consideration, we can even disregard many of the axioms which assert set existence and construct a model by enumerating only those sets which are required to exist by a few of the axioms of the system. Whenever we face such a situation, we have to make special efforts, on the one hand, to establish that the other axioms of set existence are also satisfied, but, on the other hand, since fewer axioms are directly involved in constructing the model, the model thus obtained is more transparent and often more informative. An interesting example is the system G presented in Godel [4]. Godel enumerates a model for G by taking into account merely the axioms A4, BI-B8, and using an additional clause which takes care of the union of all sets enumerated up to a given stage. On account of the last clause, Godel's enumeration uses transfinite ordinals and proves deep results about the system G. If we omit his last clause, we get a model which is denumerable (by using just positive integers) and rather simple. This simpler model is described in Wang [24] and Myhill [16]. It obviously satisfies A4 and BI-B8. This fact is employed in Rosser-Wang [18] (p. 128) to prove the consistency of G relative to the Zermelo-Fraenkel set theory. That the model should satisfy also all other axioms of Gis, though plausible, not at all obvious. Indeed, this is true only

ON DENUMERABLE BASES OF FORMAL SYSTEMS

65

because of many peculiar properties of the system G. This truth is established in Suszko [23J and Myhill [16J, independently of each other. Suszko and Myhill have proved that the consistency of G is not impaired if we add as an additional axiom the assertion that all sets belong to the simple model which satisfies A4 and BI-B8. This result is similar to the general Theorem I proved above. But it is, on account of special properties of G, stronger in that a simpler enumeration function is used and there is no need to add Hilbert's s-operator. Incidentally, Suszko's proof uses auxiliary metalinguistic concepts which can be dispensed with by using the arithmetization of syntax. Myhill's presentation of his proof is rather condensed. There are places where the speaker is not able to follow his steps of argument. It does not appear justified to consider the major significance of this result about G as proving that there are no nameless sets. In the first place, there are other known methods of giving every set in G a name, e.g., by using Hilbert's s-operator and then taking the denumerable totality of all the s-terms of Ge- In the second place, those who believe in the existence of nameless sets would assert, truly I think, that there are in any case sets which are not available in the system G, and that, therefore, although there are no nameless sets in G, there might still be nameless sets. We note incidentally that the more direct method of enumeration considered in this section is not satisfactory when impredicative definitions are involved in the introduction of new sets. For example, suppose we have an axiom (3x)(y)[y

EX

=

(3z) --, Y

E

zJ.

Superficially this says merely that there exists a set of members y defined by the condition (3z) --, y E z. Yet on account of the fact that the set x thus introduced also falls under the range of values of the bound variable z, no model consisting of a single set could satisfy the axiom. - When the defining condition contains additional free variables, the situation gets more complex of course. Therefore, if we apply the method of the present section to cases 5

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HAO WANG

where the axioms contain impredicative defining conditions, it is not necessarily true that the sets enumerated are enough to form a model. The method of § 3 is free from this difficulty, because it begins by bringing all quantifiers in the defining conditions to the beginning of the axioms and then proceed in a uniform manner.

§ 6.

Principlesfof Choicerand Hilbert's e-Operator

Clearly there is some connection between the axiom of choice and rules governing Hilbert's s-operator. Occasionally, the s-rule is referred to as a generalized principle of choice. This is misleading. For example, if the axiom of choice were a special case of the s-rule, why does the consistency of the axiom of choice not follow from the s-theorems according to which application of the s-rules can be dispensed with if the s-operator occurs in neither the axioms nor the conclusions? Indeed, if the axiom of choice were derivable from the s-rule, we would, by the s-theorems, be able to derive the axiom of choice from the other axioms of set theory. But, as we know, the independence of the axiom of choice is an unsolved problem. Of course, there is at least one difference between the s-rule and the axiom of choice. The former makes a single selection, while the latter requires that a simultaneous choice from each member of a given set be made and that all these selected items be put together to generate a new set. Hence, there is no reason to suppose that, in general, the axiom of choice follows from the s-rule. If the following (x)(3y)(z){z

E

y

=

(3w)[w

EX

& z=e..(u

E

w)]}

happens to be a theorem in a certain system of set theory, then the axiom of choice does follow from the s-rule in that system. But then it would be highly unlikely that the s-operator did not appear in the axioms of the system. There are also cases where, although the s-rule would yield the desired result, the axiom of choice would not. For example, in the Zermelo theory we can infer "(x)R(x, e,;Rxy)" from "(x}(3y)Rxy" by the s-rule, but we cannot infer "there exists I, (x)R(x, Ix)" from

ON DENUMERABLE BASES OF FOBMAL SYSTEMS

67

"(x)(3y)Rxy" by the axiom of choice, on account of the absence of a universal set in Zermelo's theory In recent years it has become rather customary to use in quantification theory systems of natural deduction, more or less along the line of Gentzen and JaSkowski. To get definiteness, let us consider the system of Quine [17]. In this system, similarly as in several other systems, an attemp is made to formalize the following common mode of mathematical reasoning: "There exist y such that Fy. Let x be one such and consider whether this x has such and such properties." In Quine [17]. the step is translated simply into "From (3y)Fy, infer Ez", Since the x chosen is not an arbitrary object but rather an arbitrary object which happens to have the property F, the variable "x" in the above formalization cannot be expected to behave like ordinary variables in mathematical logic, As a result, Quine is led to the introduction and use of several concepts and restrictions: flagging a variable, no variable is to be flagged twice, alphabetical order of variables, finished deductions. It would seem more natural to introduce the s-operator and dispense with the various restrictions. Thus, let us modify the system of Quine [17] in the following manner: strengthen UI and EG by permitting s-terms as substitution instances of the bound variables, and replace EI by: "From (3x)Fx, infer F(e.cFx)". In the resulting system, there is no longer need to flag variables, every line of a proof is valid, and there is no longer need to consider the alphabetical order of variables. Completeness of the system can easily be verified similarly as in Quine [17], and it becomes an easy affair to prove soundness. The system thus obtained is quite similar to systems in. HilbertBernays [9]. The second s-theorem can be proved for this system in a manner which is more direct than the method used in HilbertBernays [9] and is along the line of Gentzen's argument. For this purpose, we have to assume as an additional primitive rule the derivable rule "From Fx::> p, infer (3x)Fx::> p if x is not free in p". This rule is no longer derivable if we omit EI. Thus, suppose we have a derivation in the above system of q from

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HAO WANG

PI' ... , Pm' and none of q, PI"" Pm contains Hilbert's s-symbol.

The problem is to show that we can find in the above system another derivation of q from PI' ... Pm such that Hilbert's e-symbol does not occur in the proof. The rules to be applied are A, TF, Cd, the strengthened UI and EG, the revised EI, and VG (cf, Quine [17]). Suppose that e"Flx, ... , e"F"x

are all the s-terms introduced by EI, arranged in the order in which they appear in the proof. It follows that if instead of using EI, we just write down each of FI(e"Flx), , F"(e"F,,x), we get again a proof of q, but this time from PI' , Pm' together with FI(e"Flx), ... , F,.(e"F"x). The problem now is to find another proof in which the additional premises are dispensed with. Consider the last of these: F"(e"F,,x). By the rule Cd, we can prove "Ff&(e"Ff&x):> q" from PI> ... , Pm' FI(e"Flx) , ... , F"_I(e"F"_IX) without using EI. Throughout the proof, we can replace e"F"x everywhere by a new variable 'v' (say) and obtain a proof of "F"v:> q". By the additional rule assumed especially for this purpose, we can then derive "(3v)F"v:> q" without using EI. But "F"(e,,Ff&x)" was originally introduced by EI. Its premise "(3x)Ff&x" is, therefore, derivable from PI>'" Pm' FI(e"Flx), ... , F"_I(e,,Ff&_lx) already. Hence, since "(3v)F"v:> q" is also derivable from the same, q is already derivable from PI' ... , Pm' FI(e"Flx), ... , Ff&_I(e"Ff&_lx) without using EI. Hence, we have succeeded in getting rid of the premise Ff&(e"Ff&x), Repeating the same process, we can get rid of the other premises Ff&-I(e",Ff&_lx), ... , FI(e"Flx) one by one and obtain a proof of q from PI' ... Pm in which EI is not applied. In the resulting proof, there may still appear s-terms introduced by A, T F, and V I. Suppose that e"Hlx, ... , e~,;.c

are all the s-terms in the resulting proof, arranged in the order in throughout which their last occurrences appear, Replace e~,;.c the proof by a new free variable, and we get again a proof of q

ON DENUMERABLE BASES OF FORMAL SYSTEMS

69

from Pt' ... , Pm' in which EI is not used. Repeat the same with eJ]k-tX, and so on. We finally get a proof of q from P» ... , Pm in which no s-terms appear. This completes the proof of the second s-theorem. § 7.

Unintended Interpretations of Formal Systems

The attempt to find a comprehensive formal logic in which we can develop all ordinary mathematics has led to several interesting axiom systems for set theory. Usually a system is said to be categorical if any two models for the system must be isomorphic. It is well-known that none of these systems of set theory is categorical, nor could any of them be made categorical by adding new axioms. In the first place, since each of these systems contains indenumerable sets, there is by the Lowenheim-Skolem theorem, always some model for each system which is denumerable and therefore different from the intended model. Secondly, by Godel's incompletability theorem, each such system must necessarily contain undecidable sentences; therefore, there exist at least two non-isomorphic models for each system such that the undecidable propositions are true in one and false in the other. There remain the questions of relative categoricity. Vaguely the concept is merely to consider whether two interpretations which agree in certain special aspects are isomorphic. A more exact definition can be given in two parts. Definition. (a) A system 8 is categorical relative to a subsystem 8 t of 8 if and only if any two models of 8 which contain isomorphic submodels of 8 t are isomorphic. (b) If P v ... , P k are obtainable in a system 8, 8 is categorical relative to the predicates P v ... , P k , if and only if, any two models of 8 in which the interpretations of Pt, ... , P k are respectively isomorphic are isomorphic. For example, 8 may be a set theory containing only the membership primitive predicate and PtX, P'J1j, Paxy may be respectively the property of being a natural number, the property of being zero, the successor relation; the system 8 is said to be categorical relative to its natural numbers, if and only if, in any two models of 8, isomorphic interpretations of P v P 2 , P a yield isomorphic interpretations of

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the membership predicate. Similarly, a set theory may also be categorical relative to its ordinal numbers. Some simple questions of relative categoricity are studied in Wang [29]. These questions are simple because they are concerned with rather weak systems of set theory. It is not obvious that stronger systems can be treated entirely similarly. There are, however, questions which do not appear too difficult. Is the system G described in Godel [4], when extended by adding "V =L" as a new axiom, categorical relative to the ordinal numbers 1 Is the system G, when extended by adding an axiom of limitation along the line of § 3 above (resp. along the line of § 5 above), categorical relative to the natural numbers1 Probably the most frequently discussed unintended models are the models of set theory associated with the Lowenheim theorem or the Skolem paradox. A few comments on this topic may not be completely out of place here. Of course, the paradox of Skolem is pretty closely related to Cantor's notion of the indenumerable. The classical diagonal argument was given and accepted in the absolute sense, i.e., without reference to any formal system as logicians now understand it. There is absolutely no law which would correlate all sets of positive integers one-by-one with all positive integers. For, as you will recall, every enumeration of sets of positive integers always leaves out a set K which differs from the n-th set in that n belongs to the n-th set if and only if it does not belong to K. Let S be, for instance, Zermelo's set theory. On the one hand, we seem to be able to define in S the set of all sets of positive integers and prove in Sits indenumerability. On the other hand, the Lowenheim-Skolem theorem tells us that every set definable in S must be denumerable, if S is consistent. Hence, the paradox. The paradox can be dissolved either by admitting the inadequacy of formalism to intuition or by refusing to countenance any (absolutely) indenumerable sets. The Lowenheim-Skolem model is an unexpected interpretation which disagrees with our intention. The system S is not categorical because it admits both the intended and the Lowenheim-Skolem

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interpretations. What is more, the system S is not faithful to Cantor's notion which does not admit the Lowenheim-Skolem interpretation. The remedy is to keep the intuitive interpretation firmly before our mind as we work with the formalism S. It is, of course, not necessary that we always restrict the means of communication to formal systems. The fact that we do in a sense understand the informal diagonal argument proves the possibility of communication by other means. According to this line of reasoning, the Skolem paradox is one among the many arguments to show that intuition cannot be completely formalized. A more drastic solution is to reject absolute indenumerability. The diagonal argument merely shows that given any enumeration of sets of positive integer, there is a set of positive integers not in it. Unless we assume that there is a universal set which contains as members all sets of positive integers, we cannot get an absolutely indenumerable set out of the argument. True, once we give a. denumerable set and claim it to be including the totality of all sets of positive integers, we can be soundly refuted by the diagonal argument. But if we admit this and still contend that all sets are denumerable, the argument is quite powerless to refute us. When such a position is adopted, the Skolem paradox automatically disappears. The resulting theory of mathematics would become something like the ramified theory of types, except that it is not necessary to restrict the type or order indices to finite numbers. Vaguely we feel that each formal system is constructed with a unique intended model, which may be called the standard model, in mind. The speaker shares with many the discomfort over the unqualified notion of a standard model. The notion of standard model relative to certain preassigned interpretations of certain specific notions is easier. There remains, nevertheless, the question of specifying the preassigned interpretations. For example, one might feel that there is obviously only one standard model of set theory because the membership predicate is preassigned on interpretation. If, however, we specify the preassigned interpretation by stating explicitly certain conditions to be fulfilled, it is quite

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possible that if we get one model which satisfies these preassigned conditions, we can get more. Since we cannot formalize entirely the membership relation, the explicit specification of the preassigned interpretation may also involve insurmountable difficulties. Even if we leave the interpretation to our intuition, there is no assurance that our intuition is sharp enough to be capable of singling out a unique model that is standard. In this connection, the situation with number theory is much better than the situation with set theory. The standard interpretation of positive integers can be specified, for example, by emphasizing that every positive integer is either 1 or obtainable from 1 by applying the operation of adding 1 a finite number of times. Indeed, the Hilbert-Bernays truth definition already provides a standard model of number theory. For set theories such as Zermelo's, we know of neither standard models nor any models which could supply a comparable amount of information. § 8.

Completeness of Quantification Theory

This section contains a brief review of the history of the Lowenheim theorem and a proof of the completeness of quantification theory which appears to be more straightforward than the standard proofs. The following result is proved in Lowenheim [12]: (i) Every quantificational formula, if satisfiable in any (nonempty) domain at all, is satisfiable in a denumerable domain. In Skolem (19], Lowenheim's proof is simplified and an extension is obtained: (ii) H a denumerable number of quantificational formulas are

simultaneously satisfiable in any domain at all, they are simultaneously satisfiable in a denumerable domain. Skolem's proofs of (i) and (ii), as well as L6wenheim's original proof of (i), all make use of the axiom of choice. A proof of (i) is given in Skolem (20] in which the axiom of choice is not applied. Finally, in Skolem (21], improved proofs of (i) and (ii) are given. These proofs are very interesting.

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Skolem merely assumes that the quantificational formulas be given in prenex normal form. But, for brevity, let us consider a single formula in the Skolem normal form:

Skolem takes H(I, ... , 1; 2, ... , n+ 1) as HI' and then considers all the m-tuples of the positive integers no greater than n+ 1, ordering them in an arbitrary manner with the m-tuple (1, ... , 1) as the first. If (k il , ... kim) is the i-th m-tuple, then Hi is Hi-I> & H(k il

•••

kim; n(i-l)+2, ... , ni+ 1).

He then considers all the m-tuples of the positive integers used so far, and again couples the i-th m-tuple with the n-tuple consisting of the n consecutive integers starting with n(i - 1) + 2. In this way he defines a sequence of quantifier-free conjunctions HI> H"" ... all gotten from the formula A. (Cf. Skolem [21], p. 23 if and Skolem [22], p. 28 if). His proof of (i) contains two parts: (iii) If a quantificational formula A is satisfiable at all, then none of the formulas -, HI> -, Hz, ... is a tautology. (iv) If none of the formulas -, HI> -, H 2 , •• , is a tautology, then the formula A itself is satisfiable in the domain of positive integers. A little while later, Herbrand proved the following theorem (Herbrand [7], and Herbrand [8], p. 112): (v) If -, A is not a theorem of quantification theory (in other words, if the system determined by A is consistent), then none of -'-l HI> -, Hz, ... is a tautology. From (iv) and (v), the completeness of quantification theory follows as a corollary: (vi) If A is consistent, then A is satisfiable in the domain of positive integers; therefore, if -, B is not satisfiable in any domain (i.e., B is valid), then -, -, B (and therewith B) is a theorem of quantification theory. In his considerations, Herbrand also uses prenex rather than

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Skolem normal forms. Actually (v) is only a rather unimportant part of Herbrand's important theorem which he considers to be a finitary interpretation of the Lowerheim theorem: (vii) A formula A is consistent (i.e., .. A is not a theorem of quantification theorem) if and only if, none of -. HI> .. H s, ... is a tautology; moreover every proof yields effectively a proof in the "Herbrand normal form". Soon after, Gadel independently proved (iv)-(vi) in a peculiarly clear and exact manner (Godel [2]). His proof of (iv) which is widely known through Hilbert and Ackermann is rather different from Skolem's (for Skolem's argument, cf. Hilbert-Bernays [9], pp. 186188 and Wang [25]). From (vi), it follows that given any axiom system S with a finite number of special axioms and formulated in quantification theory, S contains no contradictions if and only if it has a model in the domain of positive integers. Gadel also extends this to apply to cases where there are a denumerable infinity of special axioms: (viii) If a system formulated in quantification theory is consistent, then it has a model in the domain of positive integers. Another extension of (vi) given in Godel [3] is the completeness of the extended quantification theory obtained by including identity and axioms for identity. In Henkin [5], an interesting alternative proof of (viii) is given in which Henkin introduces the method of constructing maximum consistent extensions of sets of formulas, presumably inspired by similar constructions in modern algebra. The method may be described as follows. All the quantificational formulas are enumerated in a definite manner so that each is correlated with a unique positive integer in the standard ordering: qI> qs, etc. If a consistent set So of quantificational formulas is given, its maximum consistent extension is constructed in the following manner. If So plus q1 is consistent, then S1 is So plus q1; otherwise, S1 is the same as So' In general, if S" plus q,,+1 is consistent, then S,,+1 is S" plus q,,+1; Otherwise, S,,+1 is the same as S". The maximum consistent extension is the union of the sets S•.

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Henkin succeeds in proving (viii) by application of this method and a device of introducing denumerably many suitable new symbols. If we include Hilbert's s-symbol and its associated apparatus, we do not need these new symbols because certain s-terms would serve their purpose. Thus, we get a rather simple proof of (viii) and therewith of (vi) in the following manner. Let So be a consistent set of closed quantificational formulas. Add the s-expressions and the s-rule. Then So remains consistent by the second s-theorem, Construct the maximum consistent extension (call it S) of So within the enlarged framework. Call a closed formula true or false according as whether it belongs to S or not. We assert that hereby we get a model of So in the denumerable domain D of all constant s-terms (i.e., those containing no free variables). In the first place, if P is an arbitrary predicate with n arguments and all , an are n constant s-terms, then either P(~, ... , an) or -, P(~, , an), but not both, is true, because S is maximal consistent. In the second place, "neither p nor q" is true if and only if neither p is true nor q is true, again because S is maximal consistent. In the third place, (3x)F(x) is true if and only if there is some constant e-term a such that F(a) is true, because "(3x)F(x) = F(e",Fx)" belongs to S. Moreover, since So is a subset of S, all formulas in So are true. Hence, we get (viii). In particular, So may contain a single formula, and (vi) follows immediately. It may be noted in passing that we could, if we wish, also think of the s-expressions as expressions for Skolem functions. Thus, for example, given a formula (3y)F(x, y), the s-expression e"F(x, y) of course corresponds to the Skolem function f(x) such that F(x, f(x». Corresponding to constant s-terms, we introduce Skolem functions with zero arguments. If we look at the matter in this manner, we may describe the above proof as an extension of Skolem's proof for (ii). A further sharpening of (viii) and the Lowenheim theorem is given in Hilbert-Bernays [9} and extended in Wang [25]. Thus,

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given Zermelo's set theory or some other system 8, we can as usual use an arithmetic proposition Con (8) to express the consistency of 8. For example, we can assume that Con (8) says in effect that the proposition 0= 1 is not a theorem of 8. Let N. be obtained from elementary number theory (Peano axioms and recursive definitions of addition and multiplication) by adding Con (8) as a new axiom. The result is then as follows. We can exhibit an arithmetic predicate P in N. such that if we replace the membership predicate of 8 throughout by P, then all theorems of 8 turn into theorems of N." This is sharper than the usual forms because it answers clearly the questions of expressibility and provability of the models. Moreover, if 8 is consistent, then the proposition Con (8) is not only true but obtainable from numerically true (verifiable) propositions by using merely one quantifier. Another extension of the various theorems given above is to the many-sorted quantification theories. As is indicated in Wang [27], such extension is entirely straightforward. The contrast between one-sorted and many-sorted quantification theories is more basic than that between first-order and higherorder predicate calculi. While higher-order predicate calculi and theories formulated therein all are many-sorted systems, there are many-sorted theories such as geometry dealing with points, lines, and planes which are not formulated in higher-order predicate calculi. While many-sorted quantification theories treat all predicates on an equal basis, higher-order predicate calculi reserve a special pedestal for the membership relation. Indeed, an n-th order predicate calculus is nothing but a n-sorted theory which contains a single axiom schema (the axiom of comprehension) asserting that every meaningful formula in the system defines a set. An immediate corollary of (vi) and (viii), useful for the study of completeness, is: (ix) Every sentence of a system 8 formulated in quantification theory, if true in every model of 8, is a theorem of 8. Thus, for simplicity, let us assume that 8 has a single special axiom A. If p is true in every model of 8, then "A :J p" is valid

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or true in every model (of quantification theory). For, if a model is not one of S, A is false and therefore "A :J p" is true; if a model is one of S, then p is also true in it by hypothesis. Hence, by (vi), "A :J p" is a theorem of quantification theory and, therefore, p is a theorem of S. A similar proof is available if S has infinitely many axioms. A similar result holds for many-sorted theories. Such results are of interest if we wish to make a given system complete by introducing a suitable definition of completeness. Thus, if we wish to make a system S complete, all we have to do is to define completeness to mean provability of every formula which is true in all models of S. By (ix), it follows that S is complete by this definition. This somewhat trivial trick seems to provide a clue for methods of manufacturing nontrivial concepts of completeness. Thus, given a system S, (ix) leads us to look for significant properties common and peculiar to all models of S. If, for a given system S, it happens that there are such properties, then the system S is complete in the interesting sense that every formula which is true in all models possessing these properties is provable in S. For each particular system S, special considerations are required to find suitable properties and prove the completeness of S relative to them. The concept of completeness of predicate calculi, introduced in Henkin [6], seems to be a good example. § 9.

Arbitrary Functions

What is an arbitrary set? What is an arbitrary function? There appear to be two extreme answers. On the one end, there are those who would like to deal only with computable or recursive functions. On the other end, there are those who do not hesitate to accept all the higher infinities in Cantor's theory. In the speaker's opinion, not enough attention is paid by logicians to the middle position presented and defended, perhaps not very clearly, by the semiintuitionists such as Borel, Lebesque, and Lusin. For most purposes, computable functions are sufficient in the theory of positive integers. Indeed, Ackermann and Kreisel have succeeded in interpreting quantifiers in the arithmetic of positive

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integers by computable functions. Yet it seems misguided to attempt to build up analysis or set theory solely on the basis of computable functions. The desire to see exactly how much function theory we can get by using solely computable functions is apparently innocent. It could, nonetheless, be harmful if, through prolonged exclusive occupation with computable functions, one were led to refuse to countenance other mathematical functions. In this connection, it is of interest to note that, although Brouwer does not seem willing to admit any function which is not computable, he does permit the use of free choice sequences and spreads which are not generated by computable functions (see e.g., Kleene [11]). On the other hand, there are those who would rather keep the whole Cantorian theory of the transfinite, either for its beauty or for its usefulness. It seems, however, indisputable, that the human mind is at present quite incapable of forming a clear and distinct idea of the indenumerable or the impredicative. A beauty which everybody fails to comprehend is probably a luxury which we can go without, at least insofar as mathematics is concerned. As for the application in "useful" mathematics, the speaker is of the opinion that there is no need to go beyond the denumerable. When we seem to need the indenumerable in function theory and measure theory, indenumerability relative to certain restricted means is usually sufficient. At one time or another, Weyl, Russell, Chwistek all attempted to develop mathematics, as far as possible, on a basis that does not appeal to the indenumerable or the impredicative. In recent years Lorenzen has successfully carried the program further, using nothing which resembles a formal system. In Wang [30], the speaker discusses a theory which is an extension of the ramified theory of types to the transfinite so that for any acceptable ordinal number iX of Cantor's second number class, there are variables and sets of order or type iX. In the speaker's opinion, the theory is essentially equivalent to the body of methods which Lorenzen admits and applies. The theory has on the lowest order (the O-th order) a denumerable

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totality consisting of (say) all the positive integers or all the finite sets built up out of the empty set. On the first order are these same sets plus sets of them which can be defined by properties referring at most only to the totality of all sets of the O-th order (or, in other words, by formulas which contain no bound variables of the first or a higher order). Similarly, for every positive integer n, the sets of order n + 1 includes all sets of order n together with sets of them defined by properties referring at most only to the totality of all sets of the n-th order. The sets of order t» includes all and only sets of the finite orders. For any ordinal number (X + 1, the sets of order (X + 1 are related to those of order (X in the same way as the sets of order n+ 1 are to those of order n (n a nonnegative integer). For any ordinal number (X which is the limit number of a monotome increasing sequence PI> P2' ... of ordinals, the sets of order (X are related to the sets of order PI> P2' ... in the same way as the sets of order ware related to those of finite orders. An important question is the nature of the ordinal numbers employed in describing the theory. For example, as Godel remarks in his discussion of Russell's mathematical logic, if we permit sufficiently many ordinals introduced by impredicative definitions, then we have more or less the ordinary impredicative axiomatic set theory. Since we wish to use exclusively predicative sets, we should use all and only ordinals definable by predicative definitions. It seems difficult to get a more exact characterization of these ordinals. For instance, the following method, though reasonable, is inadequate because we can again get new constructive ordinals by the diagonal procedure. Thus, let R be that part of the theory which contains only the sets and variables of finite orders. This system R, as we know, can be described exactly. Assume that an arithmetization of the syntax of R is given. Consider all the functions of positive integers which can be proved to exist in R. Each function has a defining formula which has a Godel number. The Godel number of the defining formula of a function will be called simply the Godel number of the function. We can then define ordinals in Rand represent them by positive integers in the following manner.

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(1) 1 represents the ordinal number O. (2) If m represents IX, 2m represents IX + 1. (3) If {IX..} is a monotone increasing sequence of

ordinal numbers, for each n, m.. represents IX.. , and k is the Godel number of the function of R which enumerates the sequence {ml' m 2, ... } then 3.5 k represents the limit ordinal of {IX..}. In this way, we get a totality of ordinal numbers N l . Let us expand the system R by permitting all numbers in N l to serve as indices of sets and variables. Call the resulting system R l and arithmetize its syntax. We can then define ordinals in R l and represent them by positive integers, as with R. In this way, we get a new totality N 2 of ordinal numbers from which we can construct a system R2 • And so on. Since the initial system R is well-defined, and since there is a definite method of extending a given system to get a new one, we have defined both a definite totality of ordinal numbers and also a comprehensive theory which is the union of all the partial theories. It is not easy to say much that is positive about these ordinals. They all belong to Cantor's second number class, and they are all constructive in the sense that no impredicative definitions are employed in their definitions. Moreover, there are more such ordinals than the constructive ordinals of Church and Kleene (see Kleene [l0]). Indeed, there are more ordinals in N l (the set of ordinals defined from R) than their constructive ordinals. Thus, the representation of ordinals of R is the same as theirs except that under condition (3) they use only recursive functions, while we use functions of R. But it is well-known that all recursive functions are available in R while many functions which can be proved to exist in R are not recursive. Hence, there are more constructive ordinals by the definition given above than the Church-Kleene ordinals. Instead of or in addition to attempting to build up mathematics on a predicative basis, one may try to prove the consistency of systems of impredicative set theory. It is well-known that the problem is very difficult. Let us examine a little closely the nature of the difficulties. For example, if we take the system described in Gadel [4] and

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replace the axiom of infinity by an axiom giving the empty set 0, the resulting system (call it S) has a simple denumerable model, as is well-known. Suppose now we add a simple axiom of the form (1)

(3Y)(x)[x

E

Y ;:; (3X)FXx],

where FXx contains no other free variables besides X and x, no other class variables besides X. Let us try to get a model for the extended system by enlarging the known model for the original system. It can happen that (1) is true already in the given model. This is of course not true for most of the interesting cases. So let us assume that (1) is not true in the given model. The natural thing to do would appear to be the addition of a new class X to the model of the original system S. For a given set x, if there is a class X in the original model, such that FXx is true, x E X is of course true. But it can happen that F Xx is false for every X in the original model, then the truth of x E X would depend on the truth of F K». Since F K» can contain x E X as a part, it can happen that F K» is true if and only if x E X is not true. As a result, in order to get a model for the system S plus (1), we must introduce other classes in addition to the class X. In certain simple cases, it is not difficult to introduce a new class X and get a model of S plus (1). This is in general true for cases where the new class X does not affect the truth of (3X)FXx: in other words, either F K» is false, or else there is already a class X in the original model for S such that F Xx is true. In such a case the truth of x E X is determined univocally by the original model for S and by introducing X we get easily a model for the extension. Similarly, such simple extensions are also possible when several bound class variables occur in the defining condition but the truth of the defining condition is not affected by the new class introduced by the condition. For example, consistency proofs of the number theory Z of HilbertBernay cannot be formalized in S but can be formalized in an extension of S with two new axioms, one of the form (1) and one with two general class quantifies in the defining condition. These 6

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two axioms do satisfy the requirement of the preceding paragraph. As a result, we can extend the model of S by adding two new classes and obtain a model for the extension of S which contains two apparently impredicative axioms of class existence. This is not surprising since, as we know, the same consistency proof can also be formalized in other systems which contain no such axioms but include suitable induction axioms or predicative classes of one level higher. But the example may be of interest in so far as it illustrates that many impredicative definitions can be replaced by predicative conditions. ADDITIONAL NOTE

Mr. Richard Montague has read the paper and offered useful criticisms which have induced several changes in the text. In addition, he has called my attention to an error with regard to the proofs of Theorem IV, Theorem V, and the second half of Theorem II. In these theorems I speak of "finding the functions [, g, h" in the system S or in the system Q. I failed to distinguish two senses in which the functions [, g, h can be found in a system. The proofs are valid only if the functions t, g, b are part of the original system and thereby characterized by the original axioms of the system. If, as is often the case, the functions are introduced as descriptions by contextual definitions, then the proofs are no longer valid because these contextual definitions amount to new axioms while the proofs only take care of the original axioms. The reader is asked to bear in mind this distinction and to note that the reference to Wang [26] in the paragraph immediately following the statement of Theorem IV is incorrect, because in the treatment of Wang [26], the functions i, g, h are gotten by using contextual definitions and Mr. Montague's criticism applies. (All these proofs can be rectified by a supplementary argument because in the cases concerned the relative consistency of to Q can be proved.) I should like to stress two points here. In the first place, the criticism does not affect the proofs of Theorem I, Theorem III, and the first half of Theorem II, because the use of e-expressions

Q:

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does make the functions i, g, h. part of the original system. Hence, the main conclusions which are affected by Mr. Montague's criticism are correctly established if we use e-expressions. In the second place, even if we do not use the e-expressions, the questionable conclusions of Wang [26] and the present paper can all be justified by the extension of Wang [26] in McNaughton [14], independently of how t, g, h are obtained. Thus, as is stated in the text, a proof of Theorem VI is given in McNaughton [14]. Theorem IV is an immediate corollary. Theorem V also follows from Theorem VI. Thus, Con (L o) is a theorem of Q and only a finite number of axioms of Q are used in the proof of Con (Lo). Let L 1 be L o plus these axioms. Similarly, Con (L 1 ) is by Theorem VI a theorem of Q, and let L 2 be L 1 plus the finitely many axioms employed in the proof of Con (L 1 ) . And so on. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

ALONZO CHURCH, Introduction to Mathematical Logic, Princeton (1944). KURT GODEL, Monatsh. Math. Phys., 37 (1930), 349. , Proc. Nat. Acad. Sci. U.S.A., 25 (1939), 220. , Consistency of Continuum Hypothesis, Princeton (1940). LEON HENKIN, J. Symbolic Logic, 14 (1949), 159. , J. Symbolic Logic, IS (1950), 81. JACQUES HERBRAND, Comptes rendus Acad. Sci. (Paris), 188 (1929), 1076. , Recherches aur la theorie de la demonstration. (1930). DAVID HILBERT and PAUL BERNAYS, Grundlager der Mathematik, vol. 2, (1939). S. C. KEENE, J. Symbolic Logic, 3 (1938), 150. , Proc. Int. Congo Math., 1950, (1952). LEOPOLD LOWENHEIM, Math. Annalen, 76 (1915), 447. ROBERT McNAUGHTON, J. Symbolic Logic, 18 (1953), 265 (review of Wang [26]). , "A Non-Standard Truth Definition", Proc. Am. Math. Soc., 5 (1954), 505. A. MOSTOWSKI, Fund. Math., 39 (1952), 133. JOHN MYHILL, Proc. Nat. Acad. Sci. U.S.A., 38 (1952), 979. W. V. QUINE, J. Symbolic Logic, IS (1950), 93. BARKLEY ROSSER and HAO WANG, J. Symbolic Logic, IS (1950), ll3. TH. SKOLEM, Vidensk. Skrifter I, Mat. naturw. Klasse, Oslo, No.4, (1920).

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[20] TH. SKOLEM, Proc. 5th Scand. Congo Math., 1922, (1923), 217. [21] , Vidensk. Skrifter I, Mat. naturw. Klasse, Oslo, No.4, (1929). [22] , Lea entretiens de Zurich, 1938, (1941), 25. [23] ROMAN SUSZKO, Studia philosophica, 4 (1951), 301. [24] llAo WANG, Proc. Nat. Aead, Sci. U.S.A., 36 (1950), 479. [25] , Methodos, 3 (1951), 217. [26] , Math. Annalen, 125 (1952), 56. [27] , J. Symbolic Logic, 17 (1952), 105. [28] , Trans. Am. Math. Soe., 73 (1952), 243. [29] , Math. Annalen, 126 (1953), 385. [30] , "The formalization of mathematics", J. Symbolic Logic, (1954), 241. Department of Philosophy, Emerson Hall, Harvard University, Cambridge, Mass. U.S.A.

L. HENKIN

THE REPRESENTATION THEOREM FOR CYLINDRICAL ALGEBRAS In this paper we shall deal with certain algebraic structures introduced and studied by Alfred Tarski and his student F. B. Thompson. Unfortunately these concepts are not widely known, being presently available in the literature only in abstract form [1]. We shall, therefore, precede a description of our own work by an account of the basic ideas of Tarski and Thompson, mentioning only those concepts and results necessary to render intelligible the later sections of this paper.

§ 1. Basic Concepts Consider first a first-order formalism g:. Such a formalism is comprised of certain primitive symbols together with rules for forming these symbols into terms and well-formed formulas. The symbols include quantifiers, truth-functional connectives such as a negation sign and an implication sign, an equality sign, parentheses, a list of individual variables Xv x 2' X a, ... , and finally a list of constants divided into three sorts: individual constants, predicate constants, and operation symbols. We presume that the reader is familiar with the way in which formulas are constructed from these symbols [2]. In connection with such a formalism g: we often have occasion to consider a model 9)(. Such a model consists of a domain of individuals 1, X, together with a function which assigns to each constant g of the formalism a value g. If g is an individual constant then g is an element of X; if g is an n-ary predicate constant then 1

This may be an arbitrary non-empty set of elements.

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L. HENKIN

{j is a set of ordered n-tuples of elements of X; if g is an n-ary operation symbol then {j is an n-ary operation which associates

an element of X with each ordered n-tuple of elements of X. Given a formalism g: and a model 9R there is defined the important relation of satisfaction which holds between certain wellformed formulas G and elements IX of X'" (i.e, infinite sequences IX each term of which is an element of X); and we presume that the reader is familiar with this definition 2. There is thus associated with each wff G a certain subset G* of XO> defined by the condition that IX E G* if and only if IX satisfies G. Let A be the class of all sets G* (for all wffs G of the formalism The class A is closed under the operation U of union, since it results from the definition of satisfaction that (0 V H)* =G* u H* for any wffs G and H. Similarly A is closed under the operation 1 of complementation (with respect to X"'), since (,....., G)* (G*). From these observations it follows that the system 2{= (A, u, n, I) is a Boolean algebra of sets in which the zero element is the empty set (i.e., (,....., (:11. =xI ))*), and the unit element is XO> (i.e., (Xl =xl )*). Among the elements of A are the diagonal elements do;, defined by the law di j = (x.=x i)*, i, j = 1,2,3, .... Furthermore, in addition to the Boolean operations on A we distinguish certain unary operations, the (outer) cylindrifications C i , defined by the law 3 C, (G*) = ((3x.)G)*, t.: 1,2,3, ... , for any wff G. The system ~=(A, u, n, I , dij, C.) is an example of a highly proper (wdimensional) cylindrical algebra (with diagonal elements). More generally, let 'YJ be any ordinal number and X any set. By X'I we mean the set of all functions 4 whose domain is 'YJ (we

m.

=,

2 A precise definition of satisfaction was first given in Tarski [3]. A short account of this definition appears in Mostowski [4]. In terms of the relation of satisfaction other important semantical notions such as truth and consequence may be defined. 3 This definition is justified by the fact that if G and H are two different wffs such that G* = H*, then ((3xilGl* = ((3xilHl*, as easily follows from the definition of satisfaction. 4 These functions may be identified with sequences of type TJ whose terms are in X.

THE REPRESENTATION THEOREM FOR CYLINDRICAL ALGEBRAS

87

identify 'YJ with the set of ordinals less than 1]), and whose values are in X. By the diagonal element d il of X'7, i, j = 1,2, ... 8 shows that it is essentially the direct product of the algebras >81 and >8 2 , Indeed, any direct product of cylindrical algebras is again a e.a., since the axioms for c.a.'s are universal equations. On the other hand an algebraic study shows that every finite-dimensional h.p.c.a. is simple and directly indecomposable. These observations explain why we cannot expect every c.a. to be isomorphic to an h.p.o.a.. And at the same time they suggest a natural way to broaden the concept of an h.p.o.a. so as to obtain a new family of structures which will be closed under the formation of direct products. These new structures are called proper cylindrical algebras (p.c.a.'s). The proper c.a.'s are defined in the same way as the h.p.c.a.'s, except that the unit element is not restricted to be a set of the form X''', but may be a union of disjoint sets of this form (all with the same value for TJ). The diagonal element d i ; is the set of all sequences (of type TJ) of this unit element whose ith and jth terms are identical. The cylindrification operator Ci' applied to any subset Y of this unit element, yields as value the set of all sequences of the unit element which differ from a sequence of Y at most in the ith term. An TJ-dimensional p.o.a. is thus a system >8 = ... , On de chaque Par cette interpretation iX devient vraie ou algebre donnee fausse. Nous disons que iX est vraie ou fausse dans r. Ce fait nous permet d'introduire deux operations, l'une sur les algebres l'autre sur les sous-ensembles X C E. La premiere operation assigne a chaque algebre rEm: l'ensemble E(r) compose de toutes les formules iX E E qui sont vraies dans r, la seconde assigne a chaque sous-ensemble X C E la classe m(X) de toutes les algebres rEm: dans lesquelles chaque formule de X est vraie. L'operation E(·) peut etre prolongee encore de telle faeon qu'elle soit applicable non seulement aux algebres mais aussi aux sousclasses m: o C m:. On pose dans ce but

m

°

rEm.

rEm,

E(m: o )

=

IT E( r).

rE'lr.

L'ensemble E(mo ) est alors compose de toutes les formules iX E E qui sont vraies dans chaque algebre E Pour XC E, m(X) est une sous-classe de m. Ce n'est certes pas chaque sous-claese o C m qui peut etre presentee sous la forme mo = m(X), XC E. Les sous-classes qui se laissent presenter sous cette forme sont dites classes E-definissables.

r mo'

m

100

JERZY Z.OB

Des fois il est plus commode d'exprimer cette definition de Ia faeon suivante: (1.1) La classe ~o C ~ est E-definissable (E-def.) si et seulement si ~o = ~(E(~o»' Une classe ~o C ~ qui peut etre presentee sous la forme ~o = ~(X), X etant un ensemble fini, est dite axiomatisable, ou (selon Tarski [23]) classe arithmetique. Les operations ~( . ) et E( . ) sont respectivement du type 211 --+ 2w et 2~ --+ 2E • Neanmoins depuis longtemps est connue une operation dans E, c.-a.-d. une operation du type 211 --+ 211 (Tarski [19]), nommee operation de consequence. Cette operation assigne it ohaque sousensemble XC E, l'ensemble Cn(X), de toutes les formules <X E E qui se laissent deduire elementairement (o.sa-d. uniquement sur Ia base du calcul fonctionnel d'ordre premier) des formules de X. La theorie generale de cette operation a ete etudiee par Tarski d'une faeon detaillee, c'est pourquoi je trouve superflu de m'en occuper. Je rappelle seulement qu'un ensemble XC E est dit non-contradictoire, si Cn(X) # E; un systeme, si Cn(X) = X; un systeme axiomatisable, si X = Cn(X) = Cn( Y), Y etant un ensemble fini. Enfin pour un systeme X, Ie systeme

,X

IT

=

C,,(X+Y)-E

Cn(Y)

est dit pseudo-complementaire de X. Je rappelle aussi les theoremes suivants: (1.2)

Cn(X)

2

XCY

Cn(Y)

[19].

Y 1InI;

(1.3)

Un systeme X est axiomatisable si et seulement si I

X = Cn(ep)

~(X)

= ~(Cn(X».

X· (1.4)

[21].

En posant ~l (~o) = ~(E(~o»' on introduit dans la olasse ~ une operation du type 2~ -+ 2~ qui ressemble a. celle de consequence, definie dans E. Une analogie entre les classes E-def. et les systemes est alors frappante. Ces deux types d'ensembles sont definis par

101

LES CLASSES DEFINISSABLES D'ALGEBRES

une condition de la forme F(A)=A. On obtient les classes E-def. en prenant pour F I'operation mI, tandis que pour obtenir lea systemes on prend pour F l' operation On. U est connu d'ailleurs que, aussi bien les systemes que les classes E-def. forment une structure (lattice) Brouwerienne. En tenant compte de (1.4) et de la definition de I'operation m(·) on peut facilement prouver que cette operation est un homomorphisme dualiste de ces structures. Le fait que cet homomorphisme eat un Isomorphisme, resulte du theoreme de GOdel, qui est habituellement exprime: (1.5)

Pour un ensemble X non-contradictoire,

m(X)~,p,

Car une consequence de ce theoreme est que

E(m(X»=x, pour les systemes X,

(1.6)

ce qui montre l'existence de I'operation inverse. Le theoreme sur l'isomorphisme de la structure des systemes et la structure des classes E-def. n'est, a vrai dire, qu'une formulation differente du theoreme de Godel. Avec I'aide de (1.2) et (1.3) il s'en suit que:

me,

Ilm, , = ,p, mI., ..., ml,.=,p.

(1. 7) Si t E Test une famille de classes E-def. et alors il exist un ensemble fini ~, ... , tn ET, tel que

m

(1.8) Pour qu'une classe E-def. o soit axiomatisable il faut et il suffit, que la difference m- o soit une classe E-def.

m

Le theorems (1.7) est nomme (Tarski [24]) theoreme de compacite des classes E-def. (compactness theorem). U est simplement une traduction dans la langue des classes E-def. du theorems (1.2). Le theoreme (1.8) caracterise les classes axiomatisables et il est une traduction analogue de (1.3). II est a remarquer que (1.7) (Rasiowa [14]) ainsi que (1.8) (Henkin [3], p. 419) furent demontres independamment de (1.2) et (1.3). L'algebre des systemes et la theorie des structures Brouweriennes ont ete si profondement examinees par Tarski et ses collaborateurs ([21], [22]), [11], [12]), que en vue de ce que nous avons deja mentionne sur ce sujet, on ne peut s'attendre a trouver des pro-

102

JERZY l.OS

blemes nouveaux dans l'algebre des classes E def.. c.-a.-d. dans la theorie qui s'occupe uniquement des classes E-def., sans faire usage d'autres notions. En realite, les problemes nouveaux lies aux classes E-def. se developpent principalement dans d'autres directions. J'ai !'intention de vous rendre compte en lignes generales, de trois de ces directions. La direction premiere s'occupe it caracteriser des classes E-def. par moyen de certaines operations sur les algebres, telles quelles p. ex. le produit direct. La seconde traite de conditions sous lesquelles certaines classes definies par voie non-elementaire sont des classes E-def. La troisieme, enfin, contient les problemes d'extension des algebres. Cette classification n'a pas de pretention d'etre complete, en plus, les directions mentionnees ne sont pas disjointes. Nous verrons dans la suite de proches connections qui les lient. Je voudrais encore remarquer que, tant dans les considerations preeedentes, que dans celles qui suivent, la supposition que nous nous occupons des algebres, dans la plupart des cas n'est pas essentielle. On pourrait aussi bien parler plus generalement des modeles de theories elementairea arbitraires. Les difficultes d'une telle generalisation apparaissent la. seulement, ou on parle de produits directs, d'homomorphismes et de congruences. Le sens de ees notions, quant aux algebres, est tres clair; leur generalisation aux modeles presente des diffioultes. 2. Designons par R les equations de E, c.va-d. les formules de la forme {}=T, ou {}, T sont des "polynomes" formes de signes 0l> ... ,0" et de variables xl> x 2 ' ••• (nous appelons de tels "polynomes" des termes), et par 0 designons ees formules de E qui sont formees sans quantificateurs (les formules ouvertes). Evidemment RCOCE. Posons R(SJio)=R.E(SJio) et O(SJio)=O.E(SJio)' La olasse SJio est dite equationellement definissable (R-def.), si SJio= SJi(R(SJio))' et definissable par formules ouvertes (O-def.), si SJio= SJi(O(SJio))' Les classes R-def. ont ete examinees par Birkhoff [2], qui a trouve les conditions neoessaires et suffisantes pour qu'une classe donnee SJio soit R-def. Comme c'est oonnu, il faut et il suffit pour

103

LES CLASSES DEFINISSABLES D'ALGll:BRES

ce but, que ~o soit close par rapport au produit direct, aux sousalgebres et a. la division par les congruences, c.-a.-d. que (2.1)

L'algebre produit

(2.2) a. ~o.

Chaque sous-algebre

~rt

des algebres T', E ~o,appartient

ro'

d'une algebre

r E ~o'

(2.3) Chaque algebre quotient Tl--: d'une algebre tient a. ~o' 1

r

a. ~o.l appartient E ~o'

appar-

Le theorems de Birkhoff donne l'exemple typique d'une solution complete d'un probleme qui appartient a. la premiere direction mentionnee. Les autres problemes, pour la plupart, ne se laissent point resoudre par un theoreme de forme si simple que celui de Birkhoff. P. ex. pour les classes O-def. nous avons le theoreme suivant (Los [6, 7]): (2.4) Pour qu'une classe E-def. ~o soit O-def. il faut et il suffit, que ~o soit close par rapport aux sous-algebres (c.-a.-d. que la condition (2.2) soit remplie). Neanmoins ce n'est pas une oaracteristique des classes O-def. analogue a celIe de Birkhoff, car elle permet seulement de distinguer les classes O-def. parmi les classes E-def. Pour une caraeteristique complete des classes O-def. il est neoessaire d'introduire la notion du champ logique. On appelle ainsi les triples P = (B, (J, T'», ou B est un corps de Boole 2, r= (A, Ov ... ,0..) une algebre de ~ et enfin e(x, y) est une fonction definie pour x, YEA, a. valeurs qui appartiennent a. B et qui remplit les conditions (x, Y, z, xv,." x k,' Yv ... , Yk, E A): e(x, y)=ep si et seulement si X=Y; e(x, Y)=e(Y, x); e(x, y)+e(y, z):J e(x, z), k;

e(0,(x1 ,

••• ,

x kj), O,(Yv ... , Yk;» C

!

i-I

e(x i, Yi)' i

=

1, 2, ... , n.

1 ~Tt denote le produit direct des algebres T t et Tj,..." I'algebre quotient (voir p. ex. G. Birkhoff, Lattice theory, New York, 1948, Foreword on algebra, vii-ix). 2 Selon l'usage nous denotons une algebre de Boole par une seule lettre - B. Pareillement un groupe par G.

104

JERZY LOB

Si B est un Corps a deux elements, Pest dit champ simple sur r. Pour un algebre donnee T, il n'y a evidemment qu'un seul champ simple sur r. Le champ Po=