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 give a simplified treatment of revised countable support (RCS) forcing iterations, previously considered by Shelah (see [Sh, Chap. X]). In particular we prove the fundamental theorem of semi-proper forcing, which is due to Shelah: any RCS iteration of semi-proper posets is semi-proper.

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.

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 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)

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)