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Incompleteness along paths in progressions of theories1

Published online by Cambridge University Press:  12 March 2014

S. Feferman  and
C. Spector
S. Feferman
Affiliation:
Institute for Advanced Study and Stanford University, Ohio State University
C. Spector
Affiliation:
Institute for Advanced Study and Stanford University, Ohio State University
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Extract

We deal in the following with certain theories S, by which we mean sets of sentences closed under logical deduction. The basic logic is understood to be the classical one, but we place no restriction on the orders of the variables to be used. However, we do assume that we can at least express certain notions from classical first-order number theory within these theories. In particular, there should correspond to each primitive recursive function ξ a formula φ(χ), where ‘x’ is a variable ranging over natural numbers, such that for each numeral ñ, φ(ñ) expresses in the language of S that ξ(η) = 0. Such formulas, when obtained say by the Gödel method of eliminating primitive recursive definitions in favor of arithmetical definitions in +. ·. are called PR-formulas (cf. [1] §2 (C)).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 27 , Issue 4 , December 1962 , pp. 383 - 390
Copyright
Copyright © Association for Symbolic Logic 1962

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Footnotes

2

National Science Foundation Fellow at Institute for Advanced Study, 1959–60. The research of this author was sponsored in part by the Office of Ordnance Research under Contract DA-04-200-ORD-997 at Stanford University.

3

The research of the second author (who died July 29, 1961) was supported by a grant from the National Science Foundation of the U.S.

1

The main results of this paper were communicated to the International Congress for Logic, Methodology and Philosophy of Science at Stanford University, August 24-September 2, 1960.

We wish to thank Professors K. Gödel and G. Kreisel for helpful comments on a draft of this paper.

References

[1] Feferman, S., Transfinite recursive progressions of axiomatic theories,this Journal , Vol. 27 (1962), pp. 259–316.Google Scholar
[2] Kleene, S. C., On the forms of predicates in the theory of constructive ordinals, American journal of mathematics, Vol. 66 (1944), pp. 41–58.CrossRefGoogle Scholar
[3] Kleene, S. C., On the forms of predicates in the theory of constructive ordinals (second paper). American journal of mathematics, Vol. 77 (1955), pp. 405–428.CrossRefGoogle Scholar
[4] Kleene, S. C., Quantification of number-theoretic functions, Compositio mathematica, Vol. 14 (1959), pp. 23–40.Google Scholar
[5] Spector, C., Hyperarithmetical quantifiers, Fundamenta mathematica, to appear.Google Scholar
[6] Spector, C., Recursive well-orderings, this Journal , Vol. 20 (1955), pp. 151–163.Google Scholar