25
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 particu- larly 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.

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G. KREISEL

MODELS, TRANSLATIONS AND INTERPRETATIONS

I propose to discuss these three syntactic relations betweenformal systems from the following points of view: their relation tothe consistency problem, their relation to each other, and some oftheir uses in understanding informal mathematics. I shall deal notonly with axiomatizable systems, i.e. those whose axioms form a(general) recursive set, but also with certain non-axiomatizablesystems since they are better formalizations of such branches ofmathematics as arithmetic and set theory. The systems in whichthese syntactic relations are established, will, in general, bementioned explicitly. (Both the use of non-axiomatizable systemsand explicit mention of metamathematical methods of proof maybe regarded as a natural reaction to Godel's incompletenesstheorems; for, these show (i) that axiomatizable systems are notsatisfactory approximations to certain branches of mathematics,(ii) that various formalizations of these branches of mathematicsare of different "strength", so that one may expect to learnessentially more from the particular proofs of a theorem thanfrom its assertion).

It turns out that most of our work is finitist lor, more precisely,is formulated in quantifier-free systems with decidable predicatesand computable functions. This comes about as follows: syntacticrelations can be arithmetized in elementary arithmetic, and, as isexplained at length in [1], p. 113, quantifier-free proofs are particu-larly appropriate in arithmetic; further, proofs in the elementary

1 I do not need a definition of this word since I never try to show thatsome particular theorem cannot be proved by finitist means; the readermay give his own definition and see that it fits our work; if it doesn't hemay wish to revise his definition.

MODELS, TRANSLATIONS AND INTERPRETATIONS 27

quantification theory of arithmetic may be replaced by quantifier-free ones, [1], pp. 122-123. It is worth noting that the treatmentof 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.

§ 1. Notation

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 . withinduction), N w (primitive recursive arithmetic), N" (ordinalrecursive arithmetic of order iX), II (predicate calculus).

Con 8: a formulation of "(8) is consistent" which satisfiesGodel'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 withfree variables, defining equations for a particular primitive recursiveordering of the integers which has been established to be a well-ordering with ordinal iX, and schemata for definitions and proofsby transfinite induction based on this ordering of the integers,cf. [l], p. 113. The system is determined by the particular orderingused, and not by the ordinal iX. For our purpose the non-axio-matizable (class of) systems consisting of all N", for a given iX, isunsuitable: in such a system every formula (x)(Ey)A(x, y) withrecursive A can be decided since there is an effective method ofconstructing 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 hasannounced (personal communication) that every arithmetic proposi-

28 G. KREISEL

tion has a similar equivalent. In other words, this class of systemsis of very high degree of undecidability.

§ 2. Models

The familiar consistency proofs of various geometries, thealgebra of complex numbers, or-to take a modern case-ofgeneral set theory (G), [2], are obtained by means of models. Thisnotion, which Tarski, [3], p. 20, calls "interpretation", may bedefined for systems of the first order predicate calculus as follows:

A system (81) has a model in (82) if the non-logical constants of(81) , i.e. its predicate symbols and function symbols, can be replacedby .expressions of (82) in such a way that the axioms of (81) gointo theorems of (82) .

The discussion of models is best subdivided into three groups:(81) has a finite set of axioms, (81) has an infinite, but recursiveset of axioms, (81) is not axiomatizable. In the first case the modelcan be exhibited in full, in the other cases we need a syntacticproof to show that a given replacement of the non-logical constantsof (81) constitutes a model.

FROM MODEL TO CONSISTENCY

(i) If (81) is finitely axiomatizable and has a model in (82) then

(N.,-H(Con82 --+ Con 8 1) , [4].

For certain systems (82) , if a finitely axiomatizable (81) has amodel in (82) then h Con 81 ; e.g. if (82) is the system Z, or, moregenerally, if there is a normal truth definition in (82) for quantifier-free formulae of (82) , This is not possible for all (82) : consider afinitely axiomatizable system (82) which satisfies the conditions ofGodel's second undecidability theorem, when h Con 8 2 is falsethough every system has a model in itself.(ii) If (81) has infinitely many axioms, which are mapped intothe formulae <X2(n) of (82) 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., onecan 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

(iii) If (81) is not axiomatizable, we require a partially recursivefunction fl(n) such that if n is an axiom of (81) , fl(n) is the numberof a proof in (82) of the model of n. In practice there is an importantcase where (82) is a subsystem of (81) and (81) and (82) are obtainedby 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 primitiverecursive formulae; these non-axiomatizable systems may behandled by quite elementary means.

FROM CONSISTENCY TO MODEL

If (81) has a finite set of axioms and h Con 8 1 where (82) containsthe system Z then a model of (81) may be defined in (82) : this isthe famous Loewenheim-Skolem-Godel-Bernays completeness the-orem. It is known that under the present conditions (81) does notgenerally have a recursively enumerable model, see, e.g., [6] or[7]. (The proof that the model is not necessarily recursive is givenin [10].)

The situation is not very different with higher order predicatecalculi, [8]: if h Con 81 and (82) contains Z then a model of (81)

may be defined in (82) where the models for the sets of (81) rangeover a class of arithmetic sets definable in P2 or 22 (in Mostowski'snotation), [9].

MODELS OF ARITHMETIC AND SET THEORY

The construction of models has, perhaps, contributed moretowards an understanding of mathematics, in particular of theimpossibility of characterizing by axiomatic means the notions of"integer" and "set", than any other single work of mathematicallogic. In picturesque language: every consistent system of arith-metic has models which are too large; i.e. while these modelscontain (representatives of) the numerals they also contain otherterms like Skolem's functions; on the other hand every consistentsystem of set theory has models which are too small; i.e. theypossess 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, yetthese 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 thesame logical apparatus 2 (classical quantification theory), but thenotion is not defined for systems with different logical character;in particular, it is not suitable for comparing essentially un-decidable and decidable systems, or classical and intuitionisticsystems.

§ 3. Translations

Two well known consistency proofs, which do not use models,relate systems of different logical character to each other; one isthe short consistency proof for the predicate calculus which showsthat every theorem in this system is true in a universe consistingof a single individual, and, since the latter statement can beexpressed in the propositional calculus, we have a relation betweenthe predicate calculus and the propositional calculus; the other isG6del's proof, in primitive recursive arithmetic, which establishesthat a formula is provable in Z if and only if a certain associatedformula 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 (82) if

f- (Ey) Prov, (y, n) -'>- (Ey) Prov, [y, r(n)]

2 Tarski [3], p. 22, footnote 17, mentions the interpretation of quanti-fiers 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 [-1A 1be equivalent to [-2As in Def. 2 of p. 284 of [11]; he does not use this con-dition, which is not satisfied by the "translation" of the predicate calculusinto the propositional calculus. (ii) He considers only Z-translations, whichseems an artificial restriction.

MODELS, TRANSLATIONS AND INTERPRETATIONS 31

and ~ T[v1(n)]=v2[T(n)] (if n is a formula of (B1)).

(The definition assumes that the systems (B1) and (B2) containsymbols for negation). If (B) is externally consistent ([12]) withrespect to recursive functions, the existence symbol may beeliminated:

~ Prov. (p, n) ~ Prov, [n(p), T(n)].

TRANSLATION TO CONSISTENCY

If (B1) has an (B)-translation into (B2) then ~ (Con B2 ~ Con B1) .

CONSISTENCY TO TRANSLATION (LIMITATION)

There are systems (B), (B1 ) , (B2) such that ~ (Con B2 ~ Con B1 ) ,

but (B1 ) has no (B)-translation into (B2 ) . 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, suchas IIo: if T(n) is the proposed translation, let T1(n)=O if (theformula with number) T(n) is provable in IIo' T1(n) = 1 if T(n) isnot so provable; then there is a term q of (Q) such that T1(q)= 1has the number q, and by the diagonal argument we get a con-tradiction. (See the remark below.)

MODEL TO TRANSLATION

If (B1 ) has a model in (B2) and if this model is established in (B)then (B1) has an (B)-translation into (B2 ) .

TRANSLATION TO MODELS (LIMITATION)

There are finitely axiomatizable systems (B1) and (B2) whichcan be N w-translated into each other, but have no models in eachother. Take for (B1) the elementary quantification theory ofaddition, for (B2 ) that of multiplication (or of some suitable monadicpredicate). Then, since these systems are decidable (in N w) wecan 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 modelsin each other. <I

4 Naturally, the systems (81 ) and (82) are so chosen that the notion of amodel for one system in the other is defined.

32 G. KREISEL

Remark 1. The examples above, which establish limitations,are artificial because the comparison system (8) is stronger thaneither (8 1) or (82)' It would be interesting to decide whetherBernays' set theory with class variables but without the axiomof infinity (cf. appendix of [7]) can be Z-translated into Z: for,if it can we should have a finitely axiomatizable system containingZ which can be Z-translated into Z without having a model in Z(see [5]); if it cannot be Z-translated into Z we should be able toprove in Z the consistency of a system from the consistency of Zwithout there being a Z-translation into Z.

Remark 2. If (82) is a quantifier-free system, the notion of"negation" presents a difficulty: either, if Al (of (81) ) is related toA 2(a ) we relate ,AI to ,A2(O(<l<») with some particular numeral IX,

hence "AI to A2(O<<l<I), and then we should have a translationof (81) into numerical arithmetic; this is impossible for any essenti-ally undecidable system (81) ; or we relate, Al to ,A2(a ) withthe free variable a. Though I do not know whether such a trans-lation is always possible for axiomatizable systems (81) , it isobvious that it is impossible if (81) is an arithmetic to which allverifiable primitive recursive formulae may be added as axioms.

COMPARISON OF FORMAL SYSTEMS

Apart from the consistency of (81) relative to (82)' a translationof (81) into (82) seems to provide, in general, little information ofany interest about (81) ; this is particularly true of the translationof the predicate calculus into the propositional calculus. To makefurther progress it is desirable to analyze this translation moreclosely:

Speaking informally, the translation weakens the formula whichis translated: "A is identically true" is replaced by "A is true in auniverse with a single individual" (This notion of a weakening canbe defined for any systems (81) , (82) with respect to a commonextension (8): f- (AI -+ A 2) but not f- (A 2 -+ AI)') In general, atranslation replaces a proposition Al of (81) by a weaker one, A 2 ,

of (82): the formula A 2 does not "express" the "full content"of AI'

MODELS, TRANSLATIONS AND INTERPRETATIONS 33

These informal criticisms will be made clearer in the next sectionand will there be formulated syntactically.

Oonclusion. The notion of "translation" applies to a wider classof pairs of systems than that of a "model". However, two importanttopics in the foundations of arithmetic cannot be treated bymeans of this notion: there are no translations of elementaryarithmetic into quantifier-free systems although these systemshave a special claim to attention. A translation into (82) does notgenerally extract from a formula Al of (81) all that-naivelyspeaking - one can learn from a proof of Al in (81) about thesystem (82) ,

§ 4. Interpretations

GENTZEN'S Hauptsatz or HERBRAND'S theorem establish an in-teresting connection between the predicate calculus II and thepropositional calculus Ilo; the contrast between this work and thetranslation discussed in the last section will make our criticismsmuch clearer. I consider a formula (x)(Ey)(z)A(x, y, z) (or m:) whereA(x, y, z) is quantifier-free. (i) m: can be proved in III precisely ifthere are quantifier-free terms yo(a), ... y..(a, a l ... a,,) of III suchthat

A[a, yo(a), Ut] VA[a, Yl (a; Ut), a2] V .. VA[a, y,,(a; Ut· .a,,), a..+1] (i)

is an identity (or: a theorem of IIo)'Read: either the function yo(a) satisfies A [a, yo(a), Ut] for all Ut,

or it does not when a l =~; then Yl(a;~) satisfies A [a, Yl(a; al ) , a2 ]

for all a2 etc.From each formula of the Herbrand disjunction (i) one can

derive m: by means of classical logic: the disjunction is strongerthan m: (in classical logic) in contrast to the translation of IIIinto IIo. (ii) Since the analysis above of III by Ilocannot be extendedto other systems (see [13], p. 114) it is desirable to give anotherformulation of Herbrand's theorem which is intuitively obvious:

Suppose m: were false in the sense that if x=xo and Z= j(y) then-, A [xo, y, j(y)] for all y. To prove m: in classical logic, it is sufficientto show that this is impossible, i.e. to show A (Ey)A[x, y, j(y)];

I3

34 G. KREISEL

one does this by constructing a functional Y(f, x), i.e. a term con-taining the symbols f and x, such that for all x and f we have

A{x, Y(f, x), f[Y(f, x)]}. 5

These considerations can be sharpened in two directions: (a) thefunctionals Y which are needed for our purpose, can be enumeratedin advance and depend, of course, on the particular system, (b) thefunctionals are such that in order to prove ~ we need not assumeA{x, Y(f, x), f[Y(f, x)]} for all functions, i.e. one does not need thenotion of an arbitrary function, but (e.g. in arithmetic)

(x) (n) A {x, Y(f", x), f"[Y(f,,, x)]} -+~,

where I" ranges over a suitable sequence of functions (a base, [1],p. 120). 6 We can thus list the functionals Yv Y2 , •• , and insteadof the sequence of disjunctions in (i) we associate with ~ a sequence

.. A{a, Y.(f, a), f[Y.U, am, ••

In general, each such formula is stronger than ~ (classically),since ~ can be derived from A{a, Y.U, a), I[Y.U, am by classicallogic, while one cannot read off from ~ which particular functionalis needed.

This discussion of Herbrand's theorem suggests three generaltypes of relations between systems (81) and (82) :

Definition: A recursive function 1(a,n) is a (i) disjunctive, (ii)complete disjunctive, (iii) strong disjunctive interpretation if thevalue of 1(a, n) is the number of a formula A~") of (82) when a is aformula Al of (81) ,

I In classical logic, 2( f-> ~ (x)(Ey) A[x, y. f(y)] f-> ¥(x)(z) A [x, f(x), z],

the two equivalences being dual to each other. As explained below, theequivalence ~(x)(Ey)A[x,y,f(Y)] f-> ¥(x)(z)A[x,f(x),z] does not hold con-

structively, if "function" is interpreted differently in existential and universalpropositions. In particular, if 2( holds classically, t\(x)(Ey) A [x, y, f(x)]

1holds constructively (intuitionistically), but not always ¥(x)(z) A [x, f(x), z].

• The values of these functionals are determined when the values of fare given for a suitably large, but finite set of arguments of t. e.g, if Y(f, x)is f{fU(x)]}. For details, see the appendix.

MODELS, TRANSLATIONS AND INTERPRETATIONS 35

and (i) if h Al then, for some n, h A~~)if L I A then for each n A (,,) is "false'",1 l' , 2 '

(ii): (i) and: if, for some n, f-2 A~") then hAl;(iii): (i) and: if (B2) is a subsystem of (B1) then Al can be provedfrom A~") in (B1) (for each n) (A~") is "stronger" than AI)'

If (B2) is quantifier-free "A~") is false" means that there is a sub-stitution for the free variables of A~") which reduces this formulato a false numerical formula.

In symbols (p, a, n are free variables):

f- Prov, (p, a) -+ Prov, {n(p), 1[a, n'(p)]}

f- Prov. [P, 'Jf1(a)] -+ Prov, {nl(p), V2{0'2[1(a, n), n~(p, n)]}},

where 0'2 is the Godel substitution function for (B2) .

TRANSLATION TO INTERPRETATION

An (B)-translation is an (B)-disjunctive interpretation (of course,not necessarily complete).

INTERPRETATION TO TRANSLATION (LIMITATION)

There is a (complete) interpretation of the extension of Z byverifiable primitive recursive formula in the same extension ofN., but there is no translation in this direction.

INTERPRETATION TO CONSISTENCY

From an (B)-interpretation of (B1) in (B2) we get f- (ConB2-+ ConB1).

CONSISTENCY TO INTERPRETATION

Every axiomatizable consistent system has a complete, 7 strongdisjunctive interpretation in numerical arithmetic: associate withAl the sequence Prov, (n, a), i.e, "AI can be proved in (B1) " : there-

7 Note that the trivial interpretation of JSL 16, p. 248, § 16, is notcomplete: with Al is associated the formula I PrOVI [p,1I1(a)] with thefree variable p; if this can be proved in (8t) (for one a) I-t Con 8 1, Now if(x) I PrOVI (x, q) (or 0) has the number q, (Z - f-)(Con 8 1 -+- 0) and,hence, if (8,) contains Z, also f-t 0: 0. of (8t) is associated. in the trivialinterpretation with -, 0 of (81) ; this formula cannot be proved in (81) , but,0 can be proved in (8a).

36 G. KREISEL

fore the interpretation of non-axiomatizable systems as aboveoffers the most interesting problems.

USES OF INTERPRETATIONS

It seems best to confine the discussion of the uses of inter-pretations to those of the particular interpretation describedabove, the so called no-counterexample-interpretation. The notionitself is so general that its main use seems to consist in the generalorientation which it affords and in negative results, e.g, [13]remark to theorem 1 and theorem 3; further it allows one toformulate Mostowski's question whether there exists an analogueto Herbrand's theorem for intuitionist logic, as follows: does thereexist a disjunctive interpretation of the intuitionist predicatecalculus in the intuitionist propositional calculus which remainsinvariant for every extension of these two systems by the sameconsistent set of quantifier-free axioms?

The no-counter example-interpretation has been useful bothin logic and in mathematics.

!J-consistency. A proof of the co-consistency of Z in a sharp formcan be given by means of our interpretation. Since the proofrequires a certain amount of formal detail it is relegated to theappendix.

Computable Functionals are brought into prominence by the no-counter example-interpretation, and are handled by constructivemeans. The notion of a computable functional seems to deserveattention because it presents difficulties which do not touchcomputable functions.

Intuitionism. For me at least, the no-counterexample-inter-pretation seems to provide a better approach to Brouwer's ideasthan the formal systems given by Heyting; in particular, thepassage from arithmetic to intuitionistic analysis with its freechoices becomes very natural. - Brouwer means by "function" acomputable function; therefore when he wishes to use (somethinglike) the concept of an arbitrary function, he needs a new word,and chooses "freie Wahlfolge". He does not speak of the existenceof a "Wahlfolge", nor of propositions about all "functions". - An

MODELS, TRANSLATIONS AND INTERPRETATIONS 37

intuitionistic proof of (x)(Ey)(z)A(x, y, z) or V(x)(z) A [x, !(x), z)t

requires the existence of a computable function !; a refutation onthe other hand, i.e. a proof of 1\(Ex) (Ez) , A [x, !(x), z) requires

tfor every Wahl!olge ! the existence of numbers x t and Zt such that,A[xt , !(x t ) , Zt]. One cannot expect these two possibilities to beexhaustive (yet another formulation of the failure of the law ofexcluded middle). 8

Remark. I believe that this asymmetry in the use of "function"in existential and universal propositions is characteristic ofBrouwer's writings. But there is also an independent justificationfrom the point of view of computing techniques. "Existence ofa function" in connection with a computation can refer only tosome method of computation having been made available; on theother hand a universal proposition about functions may enterinstructions as follows: we may have a random element in themachine, and the instruction may state that at some particularstage the machine should print 0 if the random sequence has aproperty and I if it the sequence does not have the property: toshow that this instruction is not circular (for any random sequence)requires a proof about all freie Wahlfolgen. (Examples of suchproofs are given in the appendix).

Arithmetic and Analysis. Especially convergence theory andcompactness theory, where non-constructive proofs occur, isgreatly unified if the results are expressed in quantifier-freeformulae and proved by quantifier-free methods. The eliminationof non-constructive methods is explained in various publications,but it is worth while giving a simple example. It is known that(even) for (computable) reals iX, f3 one cannot generally decidewhether iX = f3 or iX< f3 or f3 < iX; but suppose that the ordering of

8 The no-counter example-interpretation retains the law of excludedmiddle in the following form: it replaces (x)(Ey)(z)A(x, y, z) by 1'(x)(Ey)

A [x, y, I(y)] and its negation by 1\(Ex')(Ez') ,A[x', g(x'), z"]. Now, if1/

for x = Xo and I = 10 we have, A [xo, y, lo(y)] for all y then, for any g,take x' =xo'z' =fo[g(xo)]: so, if ~(x)(Ey)A[x,y,/(Y)] can be refuted,

1\(Ex')(Ez') , A[x', g(x'), z'] can be proved.1/

38 G. KREISEL

the continuum has been used in the proof of a quantifier free formulaB[n, A.(n)] V B[n, ~(n)] V B[n, ~(n)] (whose variable n ranges overthe integers) as follows:

(X={3 --+ B[n, A.(n)], (X<{3 --+ B[n, ~(n)], (3<(X --+ B[n, ~(n)].

Consider any n; B[n, A.(n)] is decidable; if it is false we have aproof of (X =1= {3: by calculating (X and {3 to a sufficient degree ofaccuracy (depending on the proof of (X={3 --+ B[n, A.(n)]) we candecide (X < {3 or {3 < (x, and then proceed. (The argument assumesthat the methods of proof used are externally consistent.)

I learnt recently that some 30 years ago Skolem recommendedthe use of quantifier-free methods in arithmetic: though he didnot mention function variables, I believe that the no-counter-example-interpretation is in the spirit of Skolem's recommendation.

CONCLUSION

One may take one of two views of the no-counter example-interpretation: since any classical theorem ~ can be proved byfinitist methods in a version which, classically speaking, is at leastas strong as ~, a preference for finitist methods is confirmed bythe fact that classical methods are no stronger than finitist ones;and a preference for classical methods is confirmed by the factthat they do not lead to (quantifier-free) formulae to which afinitist could object. My own view on such matters is: one man'smeat is another man's poison.

APPENDIX

We shall prove the w-consistency of Z 9 in the following form:if a formula

(Ey)(x1)(EYl)· .. (x,.)(Ey,.)A(y; XI • • X,., Yl .. y,.)

can be proved in Z there is an ordinal recursive functional tP(p) of

9 R. Gandy has informed me that hr. possesses a proof of the w-consistencyof Z which is based on Gentzen's work.

MODELS, TRANSLATIONS AND INTERPRETATIONS 39

order <e2 such that p[l.P(p)] is not (the number of) a proof in Z ofthe formula

--, (xl)(EYl)' . (x,,)(EY,,)A [l.P(p); Xl' . X", Yl' .y,,].

In other words, given a function p whose arguments and values areintegers, it is not true that the value of p(m) is the number of aproof of --, (Xt)(EYl)' . (x,,)(Ey,,)A(o(m>; Xl' . X", Yl' .Y,,) for all m. Thereason for this somewhat lengthy definition of "w-consistency" isgiven on p. 41 of [7].

Incidentally, the proof below gives a solution to problem 2 ofJSL 17 (1952), p. 160.

An analogous but simpler argument establishes the w-consistencyof the elementary quantification theory of addition and multi-plication (without induction). We show that this argument cannotbe formalized in Z while it is well known that the consistency ofthis quantification theory can be proved in Z.

Since the proof uses the notion of a computable functional itseems desirable to begin with a section on this notion.

§ 5. Computable Functionals

Informal Idea. (i) By a "computable function" with integralarguments and values is meant a method which provides for anyinteger 0("> an integer O(m,,> (its value). It should be noted that thisdefinition applies to representations of integers by numerals only:e.g. the function lX which satisfies the relations lX(O) = 1, lX(n+ 1)= 0is computable; yet, if in some particular system (Ex)A(x) is unde-cidable, these defining conditions do not decide the value oflX[,u.,A(X)] in the system concerned; in other words, representationsof integers by ,u-symbol expressions are (properly) ignored in thedefinition of computability.

(ii) By a "computable functional" (whose arguments rangeover integral valued functions of the integers) with integral valuesis meant a method which provides for any such function f aninteger o(m,>; the method should be such that once a sufficientlylarge, but finite number of values of f has been computed the valueof 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. Thenotion of a computable functional is not as definite as that of acomputable function since there is no analogue to the representationof integers by numerals. However, we shall see below that inpractice this difference is less serious than appears.

Standardization. Kleene has given standard forms for compu-table functions and functionals:

(i) primitive recursive functions ~(l; m, n), i(n) such that, if tP isa 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 acomputable 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 [, onecan verify this fact a posteriori by means of a finite procedure: ifthere is an n satisfying (a') one need only calculate the values of tfor arguments ~ n, and then one can choose the smallest n satisfyingcondition (a/). The values of t for arguments exceeding n do notenter 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 certainimpredicative functions.

In practice the analogy with computable functions is to someextent restored by the indefiniteness of the notion of a "correctproof" for condition (a): e.g. most of the proofs of condition (a)proceed by ordinal induction with respect to some ordering -<: toestablish the well-ordering character of <, one has to show thatfor all functions t there is an n such that.., [f(n+ 1) -< t(n)), i.e.

MODELS, TRANSLATIONS AND INTERPRETATIONS 41

a proposition of the same form as condition (a'). 10 Below we shallbe concerned with two kinds of classes of functionals: the first areso simple that their computability is apparent, the second are sodefined that they are computable for all functions if they arecomputable for arithmetically definable functions (provided theproofs of computability employ the standard substitution rules):thus the vagueness in some contexts of the concept of "arbitraryfunction" does not spoil the work below.

(If the reader feels that there can be no "constructive" treatmentof propositions about arbitrary functions since the class of arbitraryfunctions includes highly non-constructive entities, he shouldeither consider simple examples from the elementary calculus withfree variables or should compare the situation with the propositionalcalculus; an identity of the latter is a schema: provided the truthvalues of the constituent propositions have been decided the truthvalues of the compound proposition is decided, and it is truewhatever truth values have been assigned to the constituentpropositions; the propositions about arbitrary functions which weconsider have a similar property: they are decided provided thevalues of the functions concerned have been assigned at a suitablefinite number of arguments: only now the (number of these)arguments which are used in deciding a given proposition maydepend on the particular function used; the proposition is anidentity if it is true for an arbitrary choice of values of the function.The fact that we often use "definitions" of functions which do notprovide an immediate method of calculation or, on inspection, turnout not to be unambiguous definitions at all, has a perfect analoguewith "definitions" of properties (propositional calculus).)

10 More precisely thus: analysis of such an argwnent often shows thatthe argwnent does not need well-ordering with respect to arbitrary (des.cending) sequences, but only with respect to sequences of a restricted type;e.g, in the consistency proof of [15] only primitive recursive sequencesappear. But the proof that the particular orderings < used in [15] are well

1>orderings, makes no use of this restriction; and the fact that the orderingsare well orderings with respect to arbitrary sequences is needed, e.g., in thefull statement of the no-counterexample-interpretation of arithmetic.

42 G. KREISEL

Ordinal Becursioe Functionals. The following simple class offunotionals suffices for our w-consistency proof.

Let I, denote an ordinal recursive function of order < <X • 4>(/; m, n)is the maximum of the numbers m and I(x) for x ~ n.

w(/; m, n) is a kind ofiteration functional which may be computedby means of the following relations:

w(f; m, 0)=(/>(/; m, 0), w(f; m, n+ 1)=4>[/; m, w(f; m, n)].

The notion of an ordinal recursive functional of order <X whosearguments are the function variable I and the individual variablem is defined inductively as follows:

The variable m and a numeral 0("1 are such functionals; if t:P, lJIaresuch functionals so are rp(f; t:P, lJI), cot]; t:P, lJI}, 1(t:P), li(t:P) , and,u",[x ~ t:P & A(f, x)] where A is a formula of the predicate calculuswhose non-logical constants are symbols for ordinal recursivefunctionals of order <x, and whose quantifiers are bounded.

It should be noted that the definition is easily modified so thatinstead of restricting / i to a class of ordinal recursive functionalswe can take them to be general recursive or even arithmeticallydefinable. The modifications necessary for "representing" thesefunctionals in arithmetic (see below) are obvious.

N.B. To distinguish those functionals which are obtained byletting t, range over the class of general recursive functions, fromKleene's uniformly general recursive functionals, we shall callthem general recursive I-functionals ("I" for: iteration). TheI-functionals are a proper subclass of Kleene's class.

Arithmetic Functionals (cf. [14], p. 3). The informal notion isthis: if (x1)(EYl)' . (x),,(Ey,,) A(~ . .x"' Yl' .Y,,), say m, is "true" then

A (EYl)" (Ey,,)A[xl YZ(Yl)' ·Y,,(Yl· ·Y,,-l), Yl' .y,,],q... u"

and

is a computable functional, where Y is the number of the n-tuple<lh, ·Y,,> in some linear ordering. Plainly, the functional iscomputable for "all" functions if it is computable for arithmetically

MODELS, TRANSLATIONS AND INTERPRETATIONS 43

definable functions; in particular if the expression above is com-putable for

m=p,., -, (EYI)(x2)(EY2)' . (xn)(EY,,)A(xl •• X"YI' .Y,,), =m1

g2(a) = p,., -, (EY2)(xa)(EYa)' . (x,,)(EY,,)A(mx2.. x" a Y2' .y,,), etc.,I

then it is computable for all functions.The informal notion is defective because it depends on the

vague notion of a "true" arithmetic formula. When this conditionis sharpened in the form that 2l is provable in a system (8) ofarithmetic, the problem arises what conditions (8) must satisfy (inaddition to being a consistent extention of recursive arithmetic).

For instance (8) may be externally consistent with respect tothe class of general recursive functions (see [14], Theorem IV),without being w-consistent. In this case, if I- 2l, then our functionalis computable for all general recursive functions, but not necessarilycomputable in the naive sense set out at the beginning of thissection. This explains the apparent paradox of [14]: there it isshown (para. 23) that any externally consistent system (8) has ano-counterexample-interpretation by means of functionals of thesystem (8) itself, while (para. 17) there are externally consistentextensions of Z which do not have such an interpretation bymeans of computable functionals; in particular, these extensionsdo not have such an interpretation by means of general recursive1-functionals.

The notion of an arithmetic functional allows one to constructa computable functional which is not a general recursive 1-functional.

Outline of the Proof. Observe that the class of general recursivefunctions can be enumerated in elementary arithmetic by meansof a 2-quantifier formula, and similarly the class of general recursive1-functionals can be so enumerated. Observe next that there is aprimitive recursive enumeration 17,,(a) of functions which areultimately zero. By means of a Godel substitution function weconstruct a formula (x)(Ey)(z)A(x, Y, z) which expresses thefollowing proposition: for each integer x, if x is the number of ageneral recursive I-functional XU) then there is a function 17" and a

44 G. KREISEL

proof III Peano's arithmetic of the formula ",A{O"',X(17011)'17011[X(17011)]}". The formula (x)(Ey)(z)A(x, y, z) is true; otherwisethere would be a general recursive I-functional X(f) with number xsuch that, for all y, there is no proof in Peano's arithmetic of"-, A {O"', X(t]01I), 17011[X (t]01I) ]}": since general recursive I -functionalsare computable, we should have A{O"', X(t]I/)' 171/[X(17I/)]}' and hence(x)(Ey)(z)A(x, y, z). Evidently the argument just given cannot beformalized in any extension of Z by defining relations for recursivefunctions, since to any such system the method of [15] can beapplied, which yields general recursive I-functionals. The reasonis that though (it can be proved in Z that) the formula

(Ey)(z)A{O"', y, z}

can be proved in Z for each numeral 0"', one cannot infer in theextensions of Z mentioned above, that (x)(Ey)(z)A(x, y, z). As acorollary it follows that in these systems one cannot prove (allinstances of) the principle

(x)(Ey) Provo [y, £x(x)] --* (x)~(x)

where £x(x) is the number of the formula "~(O"')"; not even if ~(n)

is restricted to 2-quantifier formulae.Details. Consider the class of I-functionals where t, ranges over

all the functions j,ul/[l)(l, n, y) = 0] l = 0, 1, 2, .. i.e. not only overthose l which satisfy (x)(Ey)[l)(l, x, y)= 0]. For any givenintegernonecan determine systematically (i) whether n is the number of sucha functional, (ii) the parameters ll' .. lk<nl of the functions whichappear in the defining equations for the functional. It is a generalrecursive I -functional if and only if each of these parameterssatisfies the condition (x)(Ey) [l)(li' x, y) = 0], i :s:; k(n). Since i isbounded, one can express the condition "n is the number of a generalrecursive I -functional" by a formula (x)(Ey)O(n, x, y) with primitiverecursive C.

By the definition of general recursive I -functionals there is apartially recursive function expressed by a term tp(n, m) of Z withthe following property: if n is the number of such a functional, sayI(f), then tp(n,m) can be evaluated inZ foreachm, and tp(n,m) =I(17m)'

MODELS, TRANSLATIONS AND INTERPRETATIONS 45

Here 'fJm is the mth function in some linear ordering of functions

of the integers which are ultimately zero.There is a primitive recursive function a(n, a, b) with the following

property: if a is the number of a formula (Ey)(z)9l(y, z) thena(n, a, b) is the number of the formula 9l{rp(O(1I>, O(b)), 'fJb["P(O(1Il, O(bl)]},where 0(11) denotes the nth numeral (0', 0", Oil', .. ). sia, a) is Godel'ssubstitution function.

Let < YI' Y2> be the yth pair of integers in some ordering of pairsof integers, and, for each numeral O(1Il, let p(n) be the number ofthe 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<1Il, q(O<1Il), Y2]}} (i)

For each numeral 0(11) (i) can be proved (in Z) as follows:If (i) were false, n would be the number of a computable func-

tional since (y)(Ez)O(O(1Il, Y, z). Hence "P(O'"l, m) is computable. Also

, Prov {bI> vo [0(11), q(O'"l), b2]}

with free variable b. But, for each b2, Ya[O(1II, q(O(1II), b2] is thenumber of the formula

,{, 0[0(11), "P(O(1Il, O(",,), 'fJb, "P(0(1Il, O(b,))] V Prov {"Pl(O(1Il, O(b,»),

ya[O(1Il, q(O(1II), "P2(0(1I), O(b,»)]}} (ii)

But, by a straightforward extension of [12], pp. 312-324, since "Pis computable, we can prove in Z: for each b2 if (ii) is not provablein Z then

,0[0(11), "P(O(1I), b2), n: "P(o(n>, 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<fl), y, z) V Prov {Yv va[O<fll, q(O(fl», Y2]}' i.e. (i).

It follows that the functional

flll{' C(n, Y, f(y)] V Prov {Yl' va[n, q(n), Y2]}

is computable, but not a general recursive I-functional.Representation of Functionals in Arithmetic. Let Zk be the

extension of Z by defining relations for the general recursivefunctions i: I ~ i ~ k, I consider the representation of a generalrecursive functional in which the functions t, occur. By therepresentation of such a functional 8 we mean a term XU) withthe following properties: (i) XU) is a term of the predicate calculusmade up of the function variable [, the symbols t. and the non-logical constants of Z, (ii) for each term t(a) of Z,. the definingrelations for 8 (as given in the section on ordinal recursive funo-tionals) can be proved in Zk when t(a) is substituted for f(a) andX(t) for 8(1).

Note that this requirement is metamathematical, and is to beverified by a syntactic argument.

The representation has the following two properties which arethe formal equivalent in arithmetic of the informal requirementof computability.(i) Given any term t1(a) of Zk there is a term t of Zk such thatfor any term !:a(a) of Zk

(x)[x ~ t ~ t1(x)= !:a(x)] ~ X(t1)= X(!:a)

can be proved in Zk (by means of a proof whose rank depends onthe rank of t1) . Informally speaking this means that if the valuesof t1 , !:a agree over the range x ~ t the values 8(t1) , 8(!:a) are equal:only a finite number of values of t1 are needed to calculate thevalue of 8(t1) .

(ii) Given terms t1(a) and t of Zk there is a term t' of Zk such that

x ~ t ~ t1(x)= 1'Jt'(x)

can be proved in Zk' Informally speaking this means that to any

MODELS, TRANSLATIONS AND INTERPRETATIONS 47

function t1 of Zk there is a function 11t' which is ultimately zero,and has the property that E(11t,)=E(t1) .

The representation of funotionals is established in detail in [14],&-7. It should be observed that this representation applies, inparticular, to ordinal recursive functionals of order 8 (or higherorders too), since the substitution method of [15] applies to anyextension of Z by verifiable quantifier-free formulae. The repre-sentation solves Problem 2 of JSL 17 (1952), p. 160. (When Istated the problem I did not realise that the functionals consideredcould be represented in a quantifier-free extension of Z, but thoughtthat one would need a principle of ordinal induction of order 8 forquantified formulae of Z: no substitution method is known for thisextension of Z).

§ 6. (a)-Consistency

Without loss of generality we may confine ourselves to formulaewith three quantifiers: (Ex)(y)(Ez)A(x, y, z).

Lemma 1. There is an ordinal recursive functional EU, m) oforder 8 such that if -, (y)(Ez)A(n, y, z) is proved in Z by a proofwith number m then

-, A {o<nl, EU, Olm), t[EU, o<m))]}

is verifiable for all [.The functional is obtained from the substitution method in [15],

see, e.g., JSL 17, p. 49. Note that we do not assume that theformula is provable in some specific system.

Lemma 2. Ife denotes ,u,,(y)(Ez)A(x, y, z), 5(a) denotes ,u.A(e,a,z)and XU, m) is a representation of EU, m) in the extension Z. of Zby the defining relations for the ordinal recursive functions occur-ring in E, then

A{e, X[5, p(e)], 5{X [5, p(e)]}}

can be proved in Z. to which the function symbol p has been added.The substitution method applies to this extension of the system

Z., and thus we get a functional c[)(p) (actually of order <82), anda functional lJI(p, a) such that if c[)(p) is substituted for n, p[c[)(p)]for m and lJI(p, a) for t(a) the formula of lemma 1 is false. This

48 G. KREISEL

means that p[(l)(p)] is not the number of a proof of the formula

-, (y)(Ez)A[(l)(p), y, z].

This proves the w-consistency of Z in the sense explained atthe beginning of the appendix.

Remark 1. Observe that the functional (l) is of order ~ e while,if (x)(Ey)(z)A(x, y, z) has been proved in Z there is a functionalBU, a) of order <e such that A{a, BU, a), f[BU, a)]}. The reasonis clear: "w-consistency" refers to proofs with numbers p(m) whoserank (number of quantifiers) is unbounded; therefore if the sub-stitution method were applied to each of these proofs p(m), theresulting functionals would be of order exceeding any ordinal < e.

Remark 2. It is interesting to consider the analogous proofof thew-consistency of the elementary quantification theory of additionand multiplication without induction. Since all functionals of thepredicate calculus may be enumerated by means of a functionalof order w 2 one might expect to be able to prove the w-consistencyof arithmetic without induction in Z itself. This is not true. What istrue is that for each k one may prove in Z the w-consistency offormulae with ::( k quantifiers of the elementary quantification ofaddition and multiplication, e.g. by means of a truth definition asin [5]: this depends on the fact that a proposed example p(m) ofan w-inconsistency, i.e. a sequence of proofs p(m) of -, S2!(o(m))together with a proof of (Ex)S2!(x), may be restricted in advance;if 2{(Olm)) is of rank k and if it can be proved in the predicatecalculus at all, it can be proved by a sequence of formulae none ofwhich contains more than k quantifiers. When one applies theanalogue to lemma 2 one gets functionals of order Wk (where WI = w

and wn+! =w;n) because in the representation of the functional E in Zone applies induction to formulae with k quantifiers (i.e, s-formulaeof the second kind of rank k are used, if the proof is formalized withthe help of the s-symbol). Thus one cannot expect to prove thew-consistency of arithmetic without induction in Z itself.

This result may be established by means of a ve... y elegantobservation due to Gentzen:

Gentzen's Lemma. A proof in Z of a formula A may b ; replaced

MODELS, TRANSLATIONS AND INTERPRETATIONS 49

by a proof of A in which there is only a single application of therule of induction.

If induction has been applied to the formulae A1(x), .. , Ak(x)in this order, take a variable a not occurring in the proof, andapply the principle of induction to the formula 2£(x), namely[a=1 ~Al(X)] & ... & [a=k~Ak(x)], i.e. use:

2£(0) & (x)[2£(x) ~ 2£(x + 1)] . ~ (x)2£(x). (i)

Form this formula, A1(x), . .Ak(x) can be derived in elementaryarithmetic without induction: A1(0), A1(x) ~ A1(x+ 1) can be soderived; we substitute 1 for a in (i) and hence we get (x)A1(x); A 2(O)

can be derived in elementary arithmetic from (x)A1(x) and so canA 2(x) ~ A 2(x+ 1), hence (x)A2(x) can be derived from (i), and soforth.

This argument can evidently be formalized in Z itself. And fromit we prove, again in Z, that Con Z follows from the w-consistencyof arithmetic without induction. For, if the formula 0 = 1 could beproved in Z it could be proved by means of a single application ofinduction (i). Thus, since 0 -:f' 1 can be proved in arithmetic withoutinduction we should have a proof in this system of a formula

2£(0) & (x)[2£(x) -+ 2£(x+ 1)] & (Ey) --, 2£(y),

or: a proof of (Ey) --, 2£(y) together with proofs of 2£(0), 2£(1), 2£(2), ..Thus arithmetic without induction would be eo-inconsistent.Since Con Z cannot be proved in Z, the w-consistency of arith-

metic without induction cannot be proved in Z, as was suggestedby the analysis of our w-consistency proof above.

Remark 3. As has been pointed out, the externally consistent,but w-inconsistent system of para. 24 in [14] has a no-counter-example-interpretation by means of functionals of the system itself.The functionals are not, e.g., general recursive, and the presentw-consistency proof cannot be applied to that system.

REFERENCES

[1] G. KREISEL, A variant to Hilbert's theory of the foundations ofarithmetic, The British Journal for the Philosophy of Science, 4, 14(1953), 107-129.

50 G. KREISEL

[2] W. ACKERMANN, Die Widerspruchsfreiheit der allgemeinen Mengen-lehre, Mathematische Annalen, 114 (1937).

[3] A. TARsKI, A. MOSTOWSKI, and R. M. ROBINSON, Undecidable Theories,Studies in Logic and the Foundations of Mathematics, Amsterdam(1953).

[4] A. MOSTOWSKI, On models of axiomatic systems, Fundamenta Mathe-maticae, 39 (1952), 133-158.

[5] G. KREISEL and H. WANG, Some applications of formalized consistencyproofs, Fundamenta Mathematicae.

[6] A. MOSTOWSKI, On a system of axioms which have no recursivelyenumerable arithmetic model, Fundamenta Mathematicae, 40 (1953),56-61.

[7] G. KREISEL, Note on arithmetic models for consistent formulae of thepredicate calculus, II, Proceedings of the XIth international congressof Philosophy, XIV, 39-49. Amsterdam (1952).

[8] L. HENKIN, Completeness in the theory of types, Journal of SymbolicLogic, 15 (1950), 81-91.

[9] G. KREISEL, Remark on complete interpretations by models, Archivfur mathematische Logik und Grundlagenforschung.

[10] , Note on arithmetic models for consistent formulae of thepredicate calculus, Fundamenta Mathematicae, 37 (1950), 265-285.

[11] H. WANG, Arithmetic translations of axiom systems, Transactions ofthe American Mathematical Society, 71 (1951), 283-293.

[12] D. HILBERT and P. BERNAYS, Grundlagen der Mathematik, Vol. II(1939), Berlin.

[13] G. KREISEL, On the concepts of completeness and interpretation offormal systems, Fundamenta Mathematicae, 39 (1952), 103-127.

[14] , Some concepts concerning formal systems of number theoryMathematische Zeitschrift, 57 (1952), 1-12.

[15] W. ACKERMANN, Zur Widerspruchsfreiheit der reinen Zahlentheorie,Mathematische Annalen, 117 (1940), 162-194.

Department of Mathematics, University of Reading, England.