On a question of Hamkins and Löwe (PRELIMINARY)

Hamkins and Löwe asked whether there can be a model N of set theory with the property that N ≡ N [H] whenever H is a generic collapse of a cardinal of N onto ω. We obtain a lower bound, a cardinal κ with a κrepeat point, for the consistency of such a model. We do not know how to construct such a model, under any assumption. We do construct, from a cardinal κ with o(κ) = κ, a model N which satisfies the desired condition when H is the collapse of any successor cardinal. We also give a much weaker lower bound for this property. Joel Hamkins and Benedikt Löwe have asked, in connection with results reported in [1], whether there can be a model N of ZFC set theory such that N [H] ≡ N whenever H is the generic collapse of any cardinal onto ω. This note gives some partial results related to this question. In the positive direction we have the following partial result: Theorem 1. Suppose there is a cardinal κ with o(κ) = κ. Then there is, in a generic extension, a model N of ZFC with the property that N [H] ≡ N whenever H is a generic collapse of some successor cardinal λ of N onto ω. The following result gives a lower bound, much weaker than the hypothesis of Theorem 1, for consistency strength of the conclusion of that theorem: Theorem 2. Suppose that V ≡ V [H] for any cardinal λ and any generic H ⊂ Coll(ω, λ). Then there is an inner model in which {λ : o(λ) > α } is stationary for all ordinals α. The conclusion of Theorem 2 does not imply the existence of a model with a cardinals κ such that o(κ) = κ, and so is much weaker than the hypothesis of Theorem 1. In addition if λ is a regular cardinal, regular, or if λ = Ω, the class of all ordinals, then the proof of Theorem 2 does not imply that o(λ) > 1. To state our lower bound for the full property which Hamkins and Löweasked for, we need a definition: Definition 3. We define the notion of an α-repeat point by recursion on α: A measure U on a cardinal κ is an α-repeat point if for every α′ < α and every set x ∈ U there is a α′-repeat point U ′ / U with x ∈ U ′.