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. As a corollary, SPFA(c-linked) and PFA(c-linked) are each equiconsistent with the existence of a $\Sigma _{1}^{2}$ -indescribable cardinal. Our (...) upper bound for SPFA(c-c.c.) is a $\Sigma _{2}^{2}$ -indescribable cardinal, which is consistent with V = L. Our upper bound for SPFA(c⁺-linked) is a cardinal κ that is $(\kappa ^{+},\Sigma _{1}^{2})$ -subcompact, which is strictly weaker than κ⁺-supercompact. The axiom MM(c) is a consequence of SPFA(c⁺-linked) by a slight refinement of a theorem of Shelah. Our upper bound for SPFA(c⁺⁺-c.c.) is a cardinal κ that is $(\kappa ^{+},\Sigma _{2}^{2})$ -subcompact, which is also strictly weaker than κ⁺-supercompact. (shrink)

Journal of Mathematical Logic, Volume 22, Issue 02, August 2022. We introduce bounded category forcing axioms for well-behaved classes [math]. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe [math] modulo forcing in [math], for some cardinal [math] naturally associated to [math]. These axioms naturally extend projective absoluteness for arbitrary set-forcing — in this situation [math] — to classes [math] with [math]. Unlike projective absoluteness, these higher bounded category forcing (...) axioms do not follow from large cardinal axioms but can be forced under mild large cardinal assumptions on [math]. We also show the existence of many classes [math] with [math] giving rise to pairwise incompatible theories for [math]. (shrink)

We show that large fragments of MM, e. g. the tree property and stationary reflection, are preserved by strongly -game-closed forcings. PFA can be destroyed by a strongly -game-closed forcing but not by an ω2-closed.

We examine the differences between three standard classes of forcing notions relative to the way they collapse the continuum. It turns out that proper and semi-proper posets behave differently in that respect from the class of posets that preserve stationary subsets of \.

We introduce the resurrection axioms, a new class of forcing axioms, and the uplifting cardinals, a new large cardinal notion, and prove that various instances of the resurrection axioms are equiconsistent over ZFC with the existence of an uplifting cardinal.

It is possible to control to a large extent, via semiproper forcing, the parameters measuring the guessing density of the members of any given antichain of stationary subsets of ω1 . Here, given a pair of ordinals, we will say that a stationary set Sω1 has guessing density if β0=γ and , where γ is, for every stationary S*ω1, the infimum of the set of ordinals τ≤ω1+1 for which there is a function with ot)<τ for all νS* and with {νS*:gF} (...) stationary for every α<ω2 and every canonical function g for α. This work involves an analysis of iterations of models of set theory relative to sequences of measures on possibly distinct measurable cardinals. As an application of these techniques I show how to force, from the existence of a supercompact cardinal, a model of in which there is a well-order of H definable, over H,, by a formula without parameters. (shrink)