We prove new upper bound theorems on the consistency strengths of SPFA(θ), SPFA(θ-linked) and SPFA(θ-cc). Our results are in terms of (θ,Γ)-subcompactness, which is a new large cardinal notion that combines the ideas behind subcompactness and Γ-indescribability. Our upper bound for SPFA(c-linked) has a corresponding lower bound, which is due to Neeman and appears in his follow-up to this paper....

Say that κ’s measurability is destructible if there exists a <κ-closed forcing adding a new subset of κ which destroys κ’s measurability. For any δ, let λδ =df The least beth fixed point above δ. Suppose that κ is indestructibly supercompact and there is a measurable cardinal λ > κ. It then follows that A1 = {δ < κ | δ is measurable, δ is not a limit of measurable cardinals, δ is not δ+ strongl...

This note gives proves two theorems. The first is that it is consistent to have ωn for every n, but not have אω . This is done by carefully collapsing a supercompact cardinal and adding square sequences to each ωn. The crux of the proof is that in the resulting model every stationary subset of אω+1∩cof(ω) reflects to an ordinal of cofinality ω1, that is to say it has stationary intersection wit...

Assume λ is a singular limit of η supercompact cardinals, where η ≤ λ is a limit ordinal. We present two forcing methods for making λ+ the successor of the limit of the first η measurable cardinals while the tree property holding at λ+. The first method is then used to get, from the same assumptions, tree property at אη2+1 with the failure of SCH at אη2 . This extends results of Neeman and Sina...

We show that the transfer property (א1,א0)→ (λ+, λ) for singular λ does not imply (even) the existence of a non-reflecting stationary subset of λ+. The result assumes the consistency of ZFC with the existence of infinitely many supercompact cardinals. We employ a technique of “resurrection of supercompactness”. Our forcing extension destroys the supercompactness of some cardinals; to show that ...