| On a question of Hamkins and Löwe (PRELIMINARY) (2008) | |||||||||||||||
Abstract | |||||||||||||||
| 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 κ +-repeat 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 +. | |||||||||||||||
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