Equiconsistencies at subcompact cardinals

Itay Neeman, John Steel
2015 Archive for Mathematical Logic  
We present equiconsistency results at the level of subcompact cardinals. Assuming SBH δ , a special case of the Strategic Branches Hypothesis, we prove that if δ is a Woodin cardinal and both 2(δ) and 2 δ fail, then δ is subcompact in a class inner model. If in addition 2(δ + ) fails, we prove that δ is Π 2 1 subcompact in a class inner model. These results are optimal, and lead to equiconsistencies. As a corollary we also see that assuming the existence of a Woodin cardinal δ so that SBH δ
more » ... s, the Proper Forcing Axiom implies the existence of a class inner model with a Π 2 1 subcompact cardinal. Our methods generalize to higher levels of the large cardinal hierarchy, that involve long extenders, and large cardinal axioms up to δ is δ +(n) supercompact for all n < ω. We state some results at this level, and indicate how they are proved. MSC 2010: 03E45, 03E55. We dedicate this paper to Rich Laver, a brilliant mathematician and a kind and generous colleague. §1. Introduction. We present equiconsistency results at the level of subcompact cardinals. The methods we use extend further, to levels which are interlaced with the axioms κ is κ +(n) supercompact, for n < ω. The extensions will be carried out in a sequel to this paper, Neeman-Steel [7], but we indicate in this paper some of the main ideas involved. Our reversals assume iterability for countable substructures of V . By a strictly short extender we mean an extender F which maps its critical point strictly above its strength. Let SBH δ be the statement that for every countable H ≺ V δ , the good player has a winning strategy in the full iteration game of length ω 1 + 1 on the transitive collapse of H, with only strictly short extenders allowed, and with the iteration trees restricted to linear compositions of normal, non-overlapping, plus 2 trees. This is a special case of Strategic Branches Hypothesis of . Recall that a sequence ⟨C α | α ∈ Z ⊆ δ⟩ is a coherent sequence on Z if C α is club in α and α ∈ Lim(C β ) ∩ Z → C α = C β ∩ α, where Lim(C β ) is the set of limit points of C β . A thread through a coherent sequence is a club C ⊆ δ, so that for every α ∈ Lim(C) ∩ Z, C α = C ∩ α. The statement that there is a coherent sequence on δ that cannot be threaded is denoted P(δ). We will be
doi:10.1007/s00153-015-0465-4 fatcat:nlk4dcs2qjfuhg6hmk4yhuxjya