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On the completeness of some transfinite recursive progressions of axiomatic theories

Published online by Cambridge University Press:  12 March 2014

Jens Erik Fenstad
Jens Erik Fenstad*
Affiliation:
University of Oslo
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Extract

The well-known incompleteness results of Gödel assert that there is no recursively enumerable set of sentences of formalized first order arithmetic which entails all true statements of that theory. It is equally well known that by introducing sufficiently nonconstructive rules, such as the ω-rule of induction, completeness can be re-established.

Starting from the work of Turing [4] Feferman in [1] developed another method, viz. the study of transfinite recursive progressions of theories, for closing the gap between Gödel (recursively enumerable sets of axioms yield incompleteness) and Tarski (number-theoretic truth is not arithmetically definable).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 33 , Issue 1 , 26 April 1968 , pp. 69 - 76
Copyright
Copyright © Association for Symbolic Logic 1968

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References

[1]Feferman, S., Transfinite recursive progressions of axiomatic theories, this Journal, vol. 27 (1962), pp. 259–316.Google Scholar
[2]Kreisel, G., Shoenfield, J. and Wang, H., Number-theoretic concepts and recursive wellorderings, Archie für Mathematische Logik und Grundlagenforschung, vol. 5 (1960), pp. 42–64.CrossRefGoogle Scholar
[3]Shoenfield, J., On a restricted ω-rule, Bulletin de l' Academie Polonaise des Sciences, vol. 7 (1959), pp. 405–407.Google Scholar
[4]Turing, A. M., Systems of logic based on ordinals, Proceedings of the London Mathematical Society, sec. 2, vol. 45 (1939), pp. 161–228.CrossRefGoogle Scholar