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Corey Bacal Switzer

Publications and source records attributed to Corey Bacal Switzer.

24 records · Page 2Linked to original sources
arXiv2022

Destructibility and Axiomatizability of Kaufmann Models

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 this question is independent of $\mathrm{ZFC}$ and closely related to similar questions about Aronszajn trees. As an application of these methods we also show that it is independent of $\mathrm{ZFC}$ whether or not Kaufmann models can be axiomatized in the logic $L_{ω_1, ω} (Q)$ where $Q$ is the quantifier "there exists uncountably many".

math.LO↗
arXiv2021

Higher Dimensional Cardinal Characteristics for Sets of Functions

Much recent work in cardinal characteristics has focused on generalizing results about $ω$ to uncountable cardinals by studying analogues of classical cardinal characteristics on the generalized Baire and Cantor spaces $κ^κ$ and $2^κ$. In this note I look at generalizations to other function spaces, focusing particularly on the space of functions $f:ω^ω\to ω^ω$. By considering classical cardinal invariants on Baire space in this setting I derive a number of "higher dimensional" analogues of such cardinals, ultimately introducing 18 new cardinal invariants, alongside a framework that allows for numerous others. These 18 form two separate diagrams consisting of 6 and 12 cardinals respectively, each resembling versions of the Cichoń diagram. These ZFC-inequalities are the first main result of the paper. I then consider other relations between these cardinals, as well as the cardinal $\mathfrak{c}^+$ and show that these results rely on additional assumptions about cardinal characteristics on $ω$. Finally, using variations of Cohen, Hechler, and localization forcing I prove a number of consistency results for possible values of the new cardinals.

math.LO↗
arXiv2021

Independence in Arithmetic: The Method of $(\mathcal L, n)$-Models

I develop in depth the machinery of $(\mathcal L, n)$-models originally introduced by Shelah and, independently in a slightly different form by Kripke. This machinery allows fairly routine constructions of true but unprovable sentences in $\mathsf{PA}$. I give two applications: 1. Shelah's alternative proof of the Paris-Harrington theorem, and 2. The independence over $\mathsf{PA}$ of a new $Π^0_1$ Ramsey theoretic statement about colorings of finite sequences of structures.

math.LO↗
arXiv2021

The Structure of $κ$-Maximal Cofinitary Groups

We study $κ$-maximal cofinitary groups for $κ$ regular uncountable, $κ= κ^{<κ}$. Revisiting earlier work of Kastermans and building upon a recently obtained higher analogue of Bell's theorem, we show that: 1. Any $κ$-maximal cofinitary group has ${<}κ$ many orbits under the natural group action of $S(κ)$ on $κ$. 2. If $\mathfrak{p}(κ) = 2^κ$ then any partition of $κ$ into less than $κ$ many sets can be realized as the orbits of a $κ$-maximal cofinitary group. 3. For any regular $λ> κ$ it is consistent that there is a $κ$-maximal cofinitary group which is universal for groups of size ${<}2^κ= λ$. If we only require the group to be universal for groups of size $κ$ then this follows from $\mathfrak{p}(κ) = 2^κ$.

math.LO↗
arXiv2020

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 $f:ω^ω\to ω^ω$. I prove that these cardinals can be organized into two diagrams analogous to the standard Cichoń 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 $ω_1$ with no branch can be embedded into an $ω_1$-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.

math.LO↗