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A note on a question of Woodin

Apter Arthur W.

Bulletin of the Polish Academy of Sciences. Mathematics

2023

Vol. 71, no 2

115--122

A question of Woodin from the 1980s asks, assuming there is no inner model of ZFC with a strong cardinal, if it is possible for there to be a model M of ZFC such that M ⊨ “2אω > אω+2 and 2אn = אn+1 for every n < ω”, together with the existence of an inner model N∗ ⊆ M of ZFC such that for the γ, δ satisfying γ = (אω) and δ = (אω+3)M,N∗ ⊨ “γ is measurable and 2γ ≥ δ”. We show that this is the case for a choiceless version of Woodin’s question, where we assume AC fails in M but holds in N∗. We also prove analogous results for אω1 and אω2 . The methods used allow for equiconsistencies in certain cases.