The Proper Forcing Axiom, Prikry forcing, and the Singular Cardinals Hypothesis

The purpose of this paper is to present some results which suggest that the Singular Cardinals Hypothesis follows from the Proper Forcing Axiom. What will be proved is that a form of simultaneous reflection follows from the Set Mapping Reflection Principle, a consequence of PFA. While the results fall short of showing that MRP implies SCH, it will be shown that MRP implies that if SCH fails first at kappa then every stationary subset of S_{kappa^+}^omega = {a < kappa^+ : cf(a) = omega} reflects. It will also be demonstrated that MRP always fails in a generic extension by Prikry forcing.Comment: 7 page

The bounded proper forcing axiom and well orderings of the reals

We show that the bounded proper forcing axiom BPFA implies that there is a well-ordering of P(ω_1) which is Δ_1 definable with parameter a subset of ω_1. Our proof shows that if BPFA holds then any inner model of the universe of sets that correctly computes N_2 and also satisfies BPFA must contain all subsets of ω_1. We show as applications how to build minimal models of BPFA and that BPFA implies that the decision problem for the Härtig quantifier is not lightface projective

Fat subsets of P kappa (lambda)

For a subset of a cardinal greater than ω1, fatness is strictly stronger than stationarity and strictly weaker than being closed unbounded. For many regular cardinals, being fat is a sufficient condition for having a closed unbounded subset in some generic extension. In this work we characterize fatness for subsets of Pκ(λ). We prove that for many regular cardinals κ and λ, a fat subset of Pκ(λ) obtains a closed unbounded subset in a cardinal-preserving generic extension. Additionally, we work out the conflict produced by two different definitions of fat subset of a cardinal, and introduce a novel (not model-theoretic) proof technique for adding a closed unbounded subset to a fat subset of a cardinal

Category forcings, MM+++MM^{+++}, and generic absoluteness for the theory of strong forcing axioms

We introduce a category whose objects are stationary set preserving complete boolean algebras and whose arrows are complete homomorphisms with a stationary set preserving quotient. We show that the cut of this category at a rank initial segment of the universe of height a super compact which is a limit of super compact cardinals is a stationary set preserving partial order which forces $MM^{++}$ and collapses its size to become the second uncountable cardinal. Next we argue that any of the known methods to produce a model of $MM^{++}$ collapsing a superhuge cardinal to become the second uncountable cardinal produces a model in which the cutoff of the category of stationary set preserving forcings at any rank initial segment of the universe of large enough height is forcing equivalent to a presaturated tower of normal filters. We let $MM^{+++}$ denote this statement and we prove that the theory of $L(Ord^{\omega_1})$ with parameters in $P(\omega_1)$ is generically invariant for stationary set preserving forcings that preserve $MM^{+++}$. Finally we argue that the work of Larson and Asper\'o shows that this is a next to optimal generalization to the Chang model $L(Ord^{\omega_1})$ of Woodin's generic absoluteness results for the Chang model $L(Ord^{\omega})$. It remains open whether $MM^{+++}$ and $MM^{++}$ are equivalent axioms modulo large cardinals and whether $MM^{++}$ suffices to prove the same generic absoluteness results for the Chang model $L(Ord^{\omega_1})$.Comment: - to appear on the Journal of the American Mathemtical Societ

Generalizing random real forcing for inaccessible cardinals

The two parallel concepts of "small" sets of the real line are meagre sets and null sets. Those are equivalent to Cohen forcing and Random real forcing for aleph_0^aleph_0; in spite of this similarity, the Cohen forcing and Random Real Forcing have very different shapes. One of these differences is in the fact that the Cohen forcing has an easy natural generalization for lambda 2 while lambda greater than aleph 0, corresponding to an extension for the meagre sets, while the Random real forcing didn't seem to have a natural generalization, as Lebesgue measure doesn't have a generalization for space 2 lambda while lambda greater than aleph 0. In work [1], Shelah found a forcing resembling the properties of Random Real Forcing for 2 lambda while lambda is a weakly compact cardinal. Here we describe, with additional assumptions, such a forcing for 2 lambda while lambda is an Inaccessible Cardinal; this forcing is less than lambda-complete and satisfies the lambda^+-c.c hence preserves cardinals and cofinalities, however unlike Cohen forcing, does not add an undominated real

Combinatorial Properties and Dependent choice in symmetric extensions based on L\'{e}vy Collapse

We work with symmetric extensions based on L\'{e}vy Collapse and extend a few results of Arthur Apter. We prove a conjecture of Ioanna Dimitriou from her P.h.d. thesis. We also observe that if $V$ is a model of ZFC, then $DC_{<\kappa}$ can be preserved in the symmetric extension of $V$ in terms of symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$, if $\mathbb{P}$ is $\kappa$-distributive and $\mathcal{F}$ is $\kappa$-complete. Further we observe that if $V$ is a model of ZF + $DC_{\kappa}$, then $DC_{<\kappa}$ can be preserved in the symmetric extension of $V$ in terms of symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$, if $\mathbb{P}$ is $\kappa$-strategically closed and $\mathcal{F}$ is $\kappa$-complete.Comment: Revised versio