Power-like models of set theory

Journal of Symbolic Logic 66 (4):1766-1782 (2001)
Abstract
A model M = (M, E,...) of Zermelo-Fraenkel set theory ZF is said to be θ-like, where E interprets ∈ and θ is an uncountable cardinal, if |M| = θ but $|\{b \in M: bEa\}| 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 ℵ 1 -like model. Coupled with Chang's two cardinal theorem this implies that if θ is a regular cardinal θ such that $2^{ then every consistent extension of ZF also has a θ + -like model. In particular, in the presence of the continuum hypothesis every consistent extension of ZF has an ℵ 2 -like model. Here we prove: THEOREM A. If θ has the tree property then the following are equivalent for any completion T of ZFC: (i) T has a θ-like model. (ii) $\Phi \subseteq T$ , where Φ is the recursive set of axioms {∃ κ(κ is n-Mahlo and "V κ is a Σ n -elementary submodel of the universe"): n ∈ ω}. (iii) T has a λ-like model for every uncountable cardinal λ. THEOREM B. The following are equiconsistent over ZFC: (i) "There exists an ω-Mahlo cardinal". (ii) "For every finite language L, all ℵ 2 -like models of ZFC(L) satisfy the scheme Φ(L)
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DOI 10.2307/2694973
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References found in this work BETA
The Tree Property at Successors of Singular Cardinals.Menachem Magidor & Saharon Shelah - 1996 - Archive for Mathematical Logic 35 (5-6):385-404.
Aronszajn Trees on ℵ2 and ℵ3.Uri Abraham - 1983 - Annals of Pure and Applied Logic 24 (3):213-230.

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