The Isomorphism Property for Nonstandard Universes

The κ -isomorphism property (IP κ ) 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 , IP κ is equivalent to what we will refer to as the κ -resplendence property. Earlier, in [6], Jin asked if IP κ is equivalent to IP ℵ0 plus κ -saturation. He answered this question positively for κ = ℵ 1 . 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 V 0 be the set of hereditarily finite sets and for n ω , then ( V ω , ∈) is the standard model of ZC, where V ω = ⋃ n . V n .