Alternative Cicho\'n Diagrams and Forcing Axioms Compatible with CH

TL;DR

This dissertation explores alternative Cichoń diagrams, their relation to effective cardinal characteristics, and investigates forcing axioms compatible with CH, including new models and preservation theorems for subcomplete and dee-complete forcing.

Contribution

It introduces generalized Cichoń diagrams for various reduction concepts and degrees of constructibility, and develops new classes of forcing notions with iteration and preservation results.

Findings

Existence of Cichoń diagrams for various reduction concepts

New models of subcomplete forcing axiom constructed

DCFA implies no Kurepa trees even if CH fails

Abstract

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 Cicho\'n diagram. First I show that for a wide variety of reduction concepts there is a Cicho\'n diagram for effective cardinal characteristics relativized to that reduction. As an application I investigate in detail the Cicho\'n 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 $f:\omega^\omega \to \omega^\omega$. I prove that these cardinals can be organized into two diagrams analogous to the standard Cicho\'n diagram, prove several independence results and investigate the relation between these cardinals and the standard cardinal invariants on omega. In the second half of the thesis I look at forcing axioms compatible with CH. First I consider Jensen's 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 $\omega_1$ with no branch can be embedded into an $\omega_1$-tree, possibly with uncountable branches, then it can be specialized without adding reals. As a consequence I give a fleshed out proof that DCFA implies there are no Kurepa trees, even if CH fails.