Journal of Symbolic Logic

Uniqueness, Collection, and External Collapse of Cardinals in Ist and Models of Peano Arithmetic

V. Kanovei

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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 ${\tt\#}(Y) \leq n {\tt\#}(X)$ 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.

Article information

Source
J. Symbolic Logic, Volume 60, Issue 1 (1995), 318-324.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183744692

Mathematical Reviews number (MathSciNet)
MR1324515

Zentralblatt MATH identifier
0820.03034

JSTOR
links.jstor.org

Citation

Kanovei, V. Uniqueness, Collection, and External Collapse of Cardinals in Ist and Models of Peano Arithmetic. J. Symbolic Logic 60 (1995), no. 1, 318--324. https://projecteuclid.org/euclid.jsl/1183744692


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