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On axiomatizability within a system

  • William Craig (a1)

Let C be the closure of a recursively enumerable set B under some relation R. Suppose there is a primitive recursive relation Q, such that Q is a symmetric subrelation of R (i.e. if Q(m, n), then Q(n, m) and R(m, n)), and such that, for each m ϵ B, Q(m, n) for infinitely many n. Then there exists a primitive recursive set A, such that C is the closure under R of A. For proof, note that , where f is a primitive recursive function which enumerates B, has the required properties. For each m ϵ B, there is an n ϵ A, such that Q(m, n) and hence Q(n, m); therefore the closure of A under Q, and hence that under R, includes B. Conversely, since Q is a subrelation of R, A is included in C. Finally, that A is primitive recursive follows from [2] p. 180.

This observation can be applied to many formal systems S, by letting R correspond to the relation of deducibility in S, so that R(m, n) if and only if m is the Gödel number of a formula of S, or of a sequence of formulas, from which, together with axioms of S, a formula with the Gödel number n can be obtained by applications of rules of inference of S.

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[1]Craig, William, A theorem about first order functional calculus with identity, and two applications, Ph.D. thesis, Harvard University, 1951.
[2]Gödel, Kurt, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I., Monatshefte für Mathematik und Physik, vol. 38 (1931), pp. 173198.
[3]Kleene, S. C., Finite axiomatizability of theories in the predicate calculus using additional predicate symbols.Memoirs of the American Mathematical Society, no. 10, pp. 2768.
[4]Mostowski, Andrzej, On definable sets of positive integers, Fundamenta Mathematicae, vol. 34 (1946), pp. 81112.
[5]Mostowski, Andrzej, On a set of integers not definable by means of one-quantifier predicates, Annates de la Société Polonaise de Mathématique, vol. 21 (1948), pp. 114119.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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