I investigate the relationships between three hierarchies of reflection principles for a forcing class $Γ$: the hierarchy of bounded forcing axioms, of $Σ^1_1$-absoluteness and of Aronszajn tree preservation principles. The latter principle at level $κ$ says that whenever $T$ is a tree of height $ω_1$ and width $κ$ that does not have a branch of order type $ω_1$, and whenever $P$ is a forcing notion in $Γ$, then it is not the case that $P$ forces that $T$ has such a branch. $Σ^1_1$-absoluteness serves as an intermediary between these principles and the bounded forcing axioms. A special case of the main result is that for forcing classes that don't add reals, the three principles at level $2^ω$ are equivalent. Special attention is paid to certain subclasses of subcomplete forcing, since these are natural forcing classes that don't add reals.
Suppose that $T^*$ is an $ω_1$-Aronszajn tree with no stationary antichain. We introduce a forcing axiom PFA($T^*$) for proper forcings which preserve these properties of $T^*$. We prove that PFA($T^*$) implies many of the strong consequences of PFA, such as the failure of very weak club guessing, that all of the cardinal characteristics of the continuum are greater than $ω_1$, and the $P$-ideal dichotomy. On the other hand, PFA($T^*$) implies some of the consequences of diamond principles, such as the existence of Knaster forcings which are not stationarily Knaster.
We show that under certain circumstances wide Aronszajn trees can be specialized iteratively without adding reals. We then use this fact to study forcing axioms compatible with CH and list some open problems.
We prove that it is consistent that Club Stationary Reflection and the Special Aronszajn Tree Property simultaneously hold on $ω_2$, thereby contributing to the study of the tension between compactness and incompactness in set theory. The poset which produces the final model follows the collapse of an ineffable cardinal first with an iteration of club adding (with anticipation) and second with an iteration specializing Aronszajn trees. In the first part of the paper, we prove a general theorem about specializing Aronszajn trees on $ω_2$ after forcing with what we call $\mathcal{F}$-Strongly Pr…
A Kaufmann model is an $ω_1$-like, recursively saturated, rather classless model of $\mathrm{PA}$ or $\mathrm{ZF}$. Such models were constructed by Kaufmann under the combinatorial principle $\diamondsuit_{ω_1}$ and Shelah showed they exist in $\mathrm{ZFC}$ by an absoluteness argument. Kaufmann models are an important witness to the incompactness of $ω_1$ similar to Aronszajn trees. In this paper we look at some set theoretic issues related to this motivated by the seemingly naïve question of whether such a model can be "killed" by forcing without collapsing $ω_1$. We show that the answer to …
We investigate properties of trees of height $ω_1$ and their preservation under subcomplete forcing. We show that subcomplete forcing cannot add a new branch to an $ω_1$-tree. We introduce fragments of subcompleteness which are preserved by subcomplete forcing, and use these in order to show that certain strong forms of rigidity of Suslin trees are preserved by subcomplete forcings. Finally, we explore under what circumstances subcomplete forcing preserves Aronszajn trees of height and width $ω_1$. We show that this is the case if CH fails, and if CH holds, then this is the case iff the bounde…
We show that $\mathsf{PFA}$ (Proper Forcing Axiom) implies that adding any number of Cohen subsets of $ω$ will not add an $ω_2$-Aronszajn tree or a weak $ω_1$-Kurepa tree, and moreover no $σ$-centered forcing can add a weak $ω_1$-Kurepa tree (a tree of height and size $ω_1$ with at least $ω_2$ cofinal branches). This partially answers an open problem whether ccc forcings can add $ω_2$-Aronszajn or $ω_1$-Kurepa trees. We actually prove more: We show that a consequence of $\mathsf{PFA}$, namely the guessing model principle, $\mathsf{GMP}$, which is equivalent to the ineffable slender tree proper…
We show how to force, with finite conditions, the forcing axiom PFA(T), a relativization of PFA to proper forcing notions preserving a given Souslin tree T. The proof uses a Neeman style iteration with generalized side conditions consisting of models of two types, and a preservation theorem for such iterations. The consistency of this axiom was previously known by the standard countable support iteration, using a preservation theorem due to Miyamoto.
Given a tree $T$ of height $ω_1$, we say that a ladder system colouring $(f_α)_{α\in \limω_1}$ has a $T$-uniformization if there is a function $φ$ defined on a subtree $S$ of $T$ so that for any $s\in S_α$ of limit height and almost all $ξ\in {dom} (f_α)$, $φ(s\upharpoonright ξ)=f_α(ξ)$. In sharp contrast to the classical theory of uniformizations on $ω_1$, J. Moore proved that CH is consistent with the statement that any ladder system colouring has a $T$-uniformization (for any Aronszajn tree $T$). Our goal is to present a fine analysis of ladder system uniformization on trees pointing out th…
It is well known that pretameness implies the forcing theorem, and that pretameness is characterized by the preservation of the axioms of $\mathsf{ZF}^-$, that is $\mathsf{ZF}$ without the power set axiom, or equivalently, by the preservation of the axiom scheme of replacement, for class forcing over models of $\mathsf{ZF}$. We show that pretameness in fact has various other characterizations, for instance in terms of the forcing theorem, the preservation of the axiom scheme of separation, the forcing equivalence of partial orders and their dense suborders, and the existence of nice names for …
Various theorems for the preservation of set-theoretic axioms under forcing are proved, regarding both forcing axioms and axioms true in the Levy-Collapse. These show in particular that certain applications of forcing axioms require to add generic countable sequences high up in the set-theoretic hierarchy even before collapsing everything down to $\aleph\_1$. Later we give applications, among them the consistency of ${\rm MM}$ with $\aleph\_ω$ not being Jonsson which answers a question raised during Oberwolfach 2005.
Since being isolated by Viale and Weiss in 2009, the Guessing Model Property has emerged as a particularly prominent and powerful consequence of the Proper Forcing Axiom. In this paper, we investigate connections between variations of the Guessing Model Property and cardinal arithmetic, broadly construed. We improve upon results of Viale and Krueger by proving that a weakening of the Guessing Model Property implies Shelah's Strong Hypothesis. We also prove that, though the Guessing Model Property is known not to put an upper bound on the size of the continuum, it does imply that $2^{ω_1}$ is a…
We prove two general results about the preservation of extendible and $C^{(n)}$-extendible cardinals under a wide class of forcing iterations (Theorems 5.4 and 7.5). As applications we give new proofs of the preservation of Vopěnka's Principle and $C^{(n)}$-extendible cardinals under Jensen's iteration for forcing the GCH, previously obtained by Brooke-Taylor and Tsaprounis, res\-pectively. We prove that $C^{(n)}$-extendible cardinals are preserved by forcing with standard Easton-support iterations for any possible $Δ_2$-definable behaviour of the power-set function on regular cardinals. We sh…
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 …
I survey an array of topics in set theory in the context of a novel class of forcing notions: subcomplete forcing. Subcompleteness was originally defined by Ronald Jensen. I have attempted to make the subject somewhat more approachable to set theorists, while showing various properties of subcomplete forcing which one might desire of a forcing class, drawing comparisons between subcomplete forcing and countably closed forcing. In particular, I look at the interaction between subcomplete forcing and $ω_1$-trees, preservation properties of subcomplete forcing, the subcomplete maximality principl…
