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Uniqueness, collection, and external collapse of cardinals in IST and models of Peano arithmetic

Published online by Cambridge University Press:  12 March 2014

V. Kanovei
V. Kanovei*
Affiliation:
Moscow Transport Engineering Institute, Moscow State University, E-mail: [email protected], [email protected]
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Abstract

We prove that in IST, Nelson's internal set theory, the Uniqueness and Collection principles, hold for all (including external) formulas. A corollary of the Collection theorem shows that in IST there are no definable mappings of a set X onto a set Y of greater (not equal) cardinality unless both sets are finite and #(Y) ≤ n #(X) for some standard n. Proofs are based on a rather general technique which may be applied to other nonstandard structures. In particular we prove that in a nonstandard model of PA, Peano arithmetic, every hyperinteger uniquely definable by a formula of the PA language extended by the predicate of standardness, can be defined also by a pure PA formula.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 60 , Issue 1 , March 1995 , pp. 318 - 324
Copyright
Copyright © Association for Symbolic Logic 1995

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References

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