Uniqueness, collection, and external collapse of cardinals in ist and models of peano arithmetic

Journal of Symbolic Logic 60 (1):318-324 (1995)

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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
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DOI 10.2307/2275523
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The Syntax of Nonstandard Analysis.Edward Nelson - 1988 - Annals of Pure and Applied Logic 38 (2):123-134.

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