Power-like models of set theory

E-ISSN： 1943-5886|66|4|1766-1782

ISSN： 0022-4812

Source： The Journal of Symbolic Logic, Vol.66, Iss.4, 2001-12, pp. : 1766-1782

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Abstract

A model = (M. E, …) of Zermelo-Fraenkel set theory ZF is said to be 0-like. where E interprets ∈ and θ is an uncountable cardinal, if ∣M∣ = θ but ∣{b ∈ M: bEa}∣ < 0 for each a ∈ M, An immediate corollary of the classical theorem of Keisler and Morley on elementary end extensions of models of set theory is that every consistent extension of ZF has an ℵ₁-like model. Coupled with Chang's two cardinal theorem this implies that if θ is a regular cardinal 0 such that 2^<0 = 0 then every consistent extension of ZF also has a 0⁺-like model. In particular, in the presence of the continuum hypothesis every consistent extension of ZF has an ℵ₂-like model. Here we prove: