Incompatible bounded category forcing axioms

We introduce bounded category forcing axioms for well-behaved classes $\Gamma$. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe $H_{\lambda_\Gamma^+}$ modulo forcing in $\Gamma$, for some cardinal $\lambda_\Gamma$ naturally associated to $\Gamma$. These axioms naturally extend projective absoluteness for arbitrary set-forcing--in this situation $\lambda_\Gamma=\omega$--to classes $\Gamma$ with $\lambda_\Gamma>\omega$. 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 $V$. We also show the existence of many classes $\Gamma$ with $\lambda_\Gamma=\omega_1$, and giving rise to pairwise incompatible theories for $H_{\omega_2}$.Comment: arXiv admin note: substantial text overlap with arXiv:1805.0873

Incompatible bounded category forcing axioms

We introduce bounded category forcing axioms for well-behaved classes Γ. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe Hλ+Γ modulo forcing in Γ, for some cardinal λΓ naturally associated to Γ. These axioms naturally extend projective absoluteness for arbitrary set-forcing — in this situation λΓ=ω — to classes Γ with λΓ>ω. 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 V. We also show the existence of many classes Γ with λΓ=ω1 giving rise to pairwise incompatible theories for Hω2

Alternative Cichoń Diagrams and Forcing Axioms Compatible with CH

This dissertation surveys several topics in the general areas of iterated forcing, infinite combinatorics and set theory of the reals. There are two parts. In the first half I consider alternative versions of the Cichoń diagram. First I show that for a wide variety of reduction concepts there is a Cichoń diagram for effective cardinal characteristics relativized to that reduction. As an application I investigate in detail the Cichoń diagram for degrees of constructibility relative to a fixed inner model of ZFC. Then I study generalizations of cardinal characteristics to the space of functions from Baire space to Baire space. I prove that these cardinals can be organized into two diagrams analogous to the standard Cichoń diagram show several independence results and investigate their relation to cardinal invariants on omega. In the second half of the thesis I look at forcing axioms compatible with CH. First I consider Jensen\u27s subcomplete and subproper forcing. I generalize these notions to larger classes which are (apparently) much more nicely behaved structurally. I prove iteration and preservation theorems for both classes and use these to produce many new models of the subcomplete forcing axiom. Finally I deal with dee-complete forcing and its associated axiom DCFA. Extending a well-known result of Shelah, I show that if a tree of height omega one with no branch can be embedded into an omega one tree, possibly with uncountable branches, then it can be specialized without adding reals. As a consequence I show that DCFA implies there are no Kurepa trees, even if CH fails