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Flipping properties in arithmetic

Published online by Cambridge University Press:  12 March 2014

L. A. S. Kirby
L. A. S. Kirby*
Affiliation:
Princeton University, Princeton, New Jersey 08540
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Extract

Flipping properties were introduced in set theory by Abramson, Harrington, Kleinberg and Zwicker [1]. Here we consider them in the context of arithmetic and link them with combinatorial properties of initial segments of nonstandard models studied in [3]. As a corollary we obtain independence resutls involving flipping properties.

We follow the notation of the author and Paris in [3] and [2], and assume some knowledge of [3]. M will denote a countable nonstandard model of P (Peano arithmetic) and I will be a proper initial segment of M. We denote by N the standard model or the standard part of M. X ↑ I will mean that X is unbounded in I. If X ⊆ M is coded in M and M ≺ K, let X(K) be the subset of K coded in K by the element which codes X in M. So X(K) ⋂ M = X.

Recall that M ≺IK (K is an I-extension of M) if M ≺ K and for some c∈K,

In [3] regular and strong initial segments are defined, and among other things it is shown that I is regular if and only if there exists an I-extension of M.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 47 , Issue 2 , June 1982 , pp. 416 - 422
Copyright
Copyright © Association for Symbolic Logic 1982

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References

REFERENCES

[1]Abramson, F. G., Harrington, L. A., Kleinberg, E. M. and Zwicker, W. S., Flipping properties: A unifying thread in the theory of large cardinals, Annals of Mathematical Logic, vol. 12 (1977), pp. 25–58.CrossRefGoogle Scholar
[2]Kirby, L. A. S., Ph. D. Thesis, Manchester, 1977.Google Scholar
[3]Kirby, L. A. S. and Paris, J. B., Initial segments of models of Peano's axioms, Set theory and hierarchy theory. V, Lecture Notes in Mathematics, vol. 619, Springer-Verlag, Berlin and New York, 1977, pp. 211–226.CrossRefGoogle Scholar
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[6]Paris, J. B., A hierarchy of cuts in models of arithmetic, Proceedings of the 1979 Karpacz Conference, North-Holland, Amsterdam, pp. 312–332.Google Scholar
[7]Paris, J. B. and Kirby, L. A. S., Σn-collection Schemas in arithmetic, Logic Colloquium '77, North-Holland, Amsterdam, 1978, pp. 199–210.CrossRefGoogle Scholar