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The isomorphism property for nonstandard universes

Published online by Cambridge University Press:  12 March 2014

James H. Schmerl
James H. Schmerl*
Affiliation:
Department of Mathematics, University of Connecticut, Storrs, Connecticut 06268, E-mail: [email protected]
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Extract

The κ-isomorphism property (IPκ) for nonstandard universes was introduced by Henson in [4]. There has been some recent effort aimed at more fully understanding this property. Jin and Shelah in [7] have shown that for κ < ⊐ω, IPκ is equivalent to what we will refer to as the κ-resplendence property. Earlier, in [6], Jin asked if IPκ is equivalent to IPℵ0 plus κ-saturation. He answered this question positively for κ = ℵ1. In this note we extend this answer to all κ. We also extend the result of Jin and Shelah to all κ. (Jin also observed this could be done.)

In order to strike a balance between the generalities of model theory and the specifics of nonstandard analysis, we will consider models of Zermelo set theory with the Axiom of Choice; we denote this theory by ZC. The axioms of ZC are just those of ZFC but without the replacement scheme. Thus, among the axioms of ZC are the power set axiom, the infinity axiom, the separation axioms and the axiom of choice.

Let(V, E) ⊨ ZC. If a ∈ V, we let *a = {x ∈ V: (V,E) ⊨ x ∈ a}. In particular, i ∈ *ω iff i ∈ V and (V,E) ⊨ (i is a natural number). A subset A ⊆ V is internal if A = *a for some a ∈ V.

The standard model of ZC consists of those sets of rank at most ω + ω. In other words, if we let V0 be the set of hereditarily finite sets and for n < ω, then (Vω, ∈) is the standard model of ZC, where Vω = ⋃n<ω. Vn.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 60 , Issue 2 , June 1995 , pp. 512 - 516
Copyright
Copyright © Association for Symbolic Logic 1995

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References

REFERENCES

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