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

E-ISSN： 1943-5886|60|1|318-324

ISSN： 0022-4812

Source： The Journal of Symbolic Logic, Vol.60, Iss.1, 1995-03, pp. : 318-324

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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.