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Certain predicates defined by induction schemata

Published online by Cambridge University Press:  12 March 2014

Hao Wang
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It is known that we can introduce in number theory (for example, the system Z of Hilbert-Bernays) by induction schemata certain predicates of natural numbers which cannot be expressed explicitly within the framework of number theory. The question arises how we can define these predicates in some richer system, without employing induction schemata. In this paper a general notion of definability by induction (relative to number theory), which seems to apply to all the known predicates of this kind, is introduced; and it is proved that in a system L1 which forms an extension of number theory all predicates which are definable by induction (hereafter to be abbreviated d.i.) according to the definition are explicitly expressible.

In order to define such predicates and prove theorems answering to their induction schemata, we have to allow certain impredicative classes in L1. However, if we want merely to prove that for each constant number the special case of the induction schema for a predicate d.i. is provable, we do not have to assume the existence of impredicative classes. A certain weaker system L2, in which only predicative classes of natural numbers are allowed, is sufficient for the purpose. It is noted that a truth definition for number theory can be obtained in L2. Consistency proofs for number theory do not seem to be formalizable in L2, although they can, it is observed, be formalized in L1.

In general, given any ordinary formal system (say Zermelo set theory), it is possible to define by induction schemata, in the same manner as in number theory, certain predicates which are not explicitly definable in the system. Here again, by extending the system in an analogous fashion, these predicates become expressible in the resulting system. The crucial predicate instrumental to obtaining a truth definition for a given system is taken as an example.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 18 , Issue 1 , March 1953 , pp. 49 - 59
Copyright
Copyright © Association for Symbolic Logic 1953

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References

1 See p. 384 of Hilbert, D. and Bernays, P., Grundlagen der Mathematik, Vol. II (Berlin, 1939)Google Scholar. This system Z contains the quantification theory (for naturalnumbers), the axioms for identity, the Peano axioms for arithmetic, and the recursion equations for addition and multiplication. Throughout this paper, number theory will be understood to be this system Z.

2 Ibid., pp. 328–340.

3 For instance, the axioms and rules of inference on p. 88 of Quine's, W. V.Mathematical Logic (New York, 1940)Google Scholar, with ϕ, ϕ′, and ψ understood as arbitrary sentences of L1.

4 See pp. 140–145 of Quine's, Element and number, this Journal, vol. 6 (1941), pp. 135–149Google Scholar. The system is presented here in a somewhat different manner. However, it is essentially as rich as the system formulated there on p. 140, since we are only interested in finite sets (elements).

5 See the reference in the preceding footnote.

6 See the predicate H defined in the last part of Skolem's, Über Zurückführbarkeit einiger durch Rekursionen definierter Relationen auf arithmetische, Acta litterarum ac scientiarum (Szeged), vol. 8 (1936–1937), pp. 73–88Google Scholar.

7 See p. 71 of Kleene, S. C., Recursive predicates and quantifiers, Trans. Amer. Math. Soc., vol. 53 (1943), pp. 41–73CrossRefGoogle Scholar.

8 Hilbert-Bernays, op. cit., p. 368.

9 Ibid., p. 369 and pp. 338–339.

10 See Bernays, P., A system of axiomatic set theory, this Journal, vol. 2 (1937), pp. 65–77Google Scholar, and vol. 6 (1941), pp. 1–17.

11 We say that a truth definition Tr for a system L is adequate, if for each constant m which is the number of a statement (sentence containing no free variables) pm of L, the following case of the “truth schema” holds: Tr(m) = pm. If we can define Tr and prove all such equivalences in L′, then we say that L′ contains an adequate truth definition for L. It is not necessary that in L′ we can also prove all theorems of L to be true according to the definition. This weaker criterion of adequacy seems to conform to Tarski's outspoken conventions (see, for instance, § 4 of his popular article The semantic concept of truth, Philosophy and phenomenological research, vol. 4 (1944), pp. 341–376)CrossRefGoogle Scholar, although some of his assertions (especially those regarding consistency proofs) appear to be justifiable only if we require in addition that all theorems of L can be shown to be true according to the definition. In any case, when we speak of adequate truth definitions in this paper, we shall not make such additional requirements. It is thought that even if the present use of the term “adequate truth definition” disagrees with Tarski's intention, it is of enough interest to deserve separate study. We might refer to adequate truth definitions which satisfy the additional requirements on theorems as normal ones.

12 Compare the considerations on pp. 334–337 of Hilbert-Bernays, op. cit.

13 Incidentally, it follows from Theorem IV that Z′ is consistent if one of the systems L2, L4, and L5 is ω-consistent.

14 For example, as formulated by us on pp. 150–151 of the Proceedings Nat. Acad. Sci. U.S.A., vol. 35 (1949)Google Scholar.

15 An explicit formulation of the definition of the predicate S and a proof of its adequacy for defining truth would lead us too far afield for our present purpose. For outlines of the main ideas involved we refer to § 11 of Tarski's paper mentioned before under footnote 11 and his more technical paper Der Wahrheitsbegriff in den formalisierten Sprachen, Studia philosophica, vol. 1 (1935), pp. 261–405Google Scholar.

16 See Hilbert-Bernays, op. cit., pp. 254–258.

17 We are assuming that the provability of all cases of the truth schema is sufficient criterion for an adequate truth definition. That is why the strength of the theorems of the system is not relevant here. Compare footnote 11.

18 We should like to thank Professor Bernays for instructions on questions regarding the construction of truth definitions.