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Some independence results for Peano arithmetic

Published online by Cambridge University Press:  12 March 2014

J. B. Paris
J. B. Paris*
Affiliation:
Manchester University Manchester, England
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Extract

In this paper we shall outline a purely model theoretic method for obtaining independence results for Peano's first order axioms (P). The method is of interest in that it provides for the first time elementary combinatorial statements about the natural numbers which are not provable in P. We give several examples of such statements.

Central to this exposition will be the notion of an indicator. Indicators were introduced by L. Kirby and the author in [3] although they had occurred implicitly in earlier papers, for example Friedman [1]. The main result on indicators which we shall need (Lemma 1) was proved by Laurie Kirby and the author in the summer of 1976 but it was not until early in the following year that the author realised that this lemma could be used to give independence results.

The first combinatorial independence results obtained were essentially statements about certain finite games and consequently were not immediately meaningful (see Example 2). This shortcoming was remedied by Leo Harrington who, upon hearing an incorrect version of our results, noticed a beautifully simply independent combinatorial statement. We outline this result in Example 3. An alternative, more detailed, proof may be found in [5].

Clearly Laurie Kirby and Leo Harrington have made a very significant contribution to this paper and we wish to express our sincere thanks to them.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 43 , Issue 4 , December 1978 , pp. 725 - 731
Copyright
Copyright © Association for Symbolic Logic 1978

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References

REFERENCES

[1]Friedman, H., Some applications of Kleene's methods for intuitionistic systems, Cambridge Summer School in Mathematical Logic, Lecture Notes in Mathematics, No. 337, Spring-er-Verlag, Berlin and New York, 1973.Google Scholar
[2]Kirby, L., Doctoral dissertation, Manchester University, 1977.Google Scholar
[3]Kirby, L. and Paris, J., Initial segments of models of Peano's axioms. Proceedings of the Bierutowice Conference, 1976, Springer-Verlag, Berlin and New York (to appear).Google Scholar
[4]McAloon, K., On iterating the “new” undecidable formulas (to appear).Google Scholar
[5]Paris, J. and Harrington, L., A mathematical incompleteness in Peano arithmetic, Handbook of mathematical logic (to appear).Google Scholar
[6]Silver, J., Harrington's version of the Paris result, mimeographed notes.Google Scholar
[7]Takeuti, G., Proof theory, North-Holland, Amsterdam.Google Scholar