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Dan Hathaway

Publications and source records attributed to Dan Hathaway.

15 recordsLinked to original sources
arXiv2025

On $(\Sigma^2_1)^{uB}$ Absoluteness Between V and HOD

We put together Woodin's $\Sigma^2_1$ basis theorem of AD$^+$ and Vop\v{e}nka's theorem to conclude the following: If there is a proper class of Woodin cardinals, then every $(\Sigma^2_1)^{\mbox{uB}}$ statement that is true in $V$ is true in $\mbox{HOD}$. Moreover, this is true even if we allow a parameter $C \subseteq \mathbb{R}$ such that $C$ and its complement have scales that are $\mbox{OD}$ and universally Baire. We also investigate whether $(\Sigma^2_1)^{\mbox{uB}}$ statements are upwards absolute from $\mbox{HOD}$ to $V$ under large cardinal hypotheses, observing that this is true if $\mbox{HOD}$ has a proper class of Woodin cardinals. Finally, we discuss $(\forall^{\mathbb{R}})\, (\Sigma^2_1)^{\mbox{uB}}$ absoluteness and conclude that this much absoluteness between $\mbox{HOD}$ and $V$ cannot be implied by any large cardinal axiom consistent with the axiom ``$V =$ Ultimate $L$''.

math.LO↗
arXiv2023

Applying Generic Coding with Help to Uniformizations

This is a follow up to a paper by the author where the disjointness relation for (the graphs of) definable functions from ${^ωω}$ to ${^ωω}$ is analyzed. In that paper, for each $a \in {^ωω}$ we defined a Baire class one function $f_a^{GC} : {^ωω} \to {^ωω}$ which encoded $a$ in a certain sense. Given $g : {^ωω} \to {^ωω}$, let $Ψ(g)$ be the statement that $g$ is disjoint from at most countably many of the functions $f_a^{GC}$. We show the consistency strength of $(\forall g)\, Ψ(g)$ is at most one inaccessible cardinal. We show that $\mbox{AD}^+$ implies $(\forall g)\, Ψ(g)$. Finally, we show that assuming large cardinals, $(\forall g)\, Ψ(g)$ holds in models of the form $L(\mathbb{R} [\mathcal{U}]$ where $\mathcal{U}$ is a selective ultrafilter on $ω$.

math.LO↗
arXiv2021

Generic Coding with Help and Amalgamation Failure

We show that if $M$ is a countable transitive model of ZF and if $a,b$ are reals not in $M$, then there is a $G$ generic over $M$ such that $b \in L[a,G]$. We then present several applications such as the following: if $J$ is any countable transitive model of ZFC and $M \not\subseteq J$ is another countable transitive model of ZFC of the same ordinal height $α$, then there is a forcing extension $N$ of $J$ such that $M \cup N$ is not included in any transitive model of ZFC of height $α$. Also, assuming $0^\#$ exists, letting $S$ be the set of reals generic over $L$, although $S$ is disjoint from the Turing cone above $0^\#$, we have that for any non-constructible real $a$, $\{ a \oplus s : s \in S \}$ is cofinal in the Turing degrees.

math.LO↗
arXiv2020

Perfect Tree Forcings for Singular Cardinals

We investigate forcing properties of perfect tree forcings defined by Prikry to answer a question of Solovay in the late 1960's regarding first failures of distributivity. Given a strictly increasing sequence of regular cardinals $\langle κ_n: n< ω\rangle$, Prikry defined the forcing $\mathbb{P}$ all perfect subtrees of $\prod_{n<ω}κ_n$, and proved that for $κ=\sup_{n<ω}κ_n$, assuming the necessary cardinal arithmetic, the Boolean completion $\mathbb{B}$ of $\mathbb{P}$ is $(ω,μ)$-distributive for all $μ<κ$ but $(ω,κ,δ)$-distributivity fails for all $δ<κ$, implying failure of the $(ω,κ)$-d.l. These hitherto unpublished results are included, setting the stage for the following recent results. $\mathbb{P}$ satisfies a Sacks-type property, implying that $\mathbb{B}$ is $(ω,\infty,<κ)$-distributive. The $(\mathfrak{h},2)$-d.l. and the $(\mathfrak{d},\infty,<κ)$-d.l. fail in $\mathbb{B}$. $\mathcal{P}(ω)/\mbox{Fin}$ completely embeds into $\mathbb{B}$. Also, $\mathbb{B}$ collapses $κ^ω$ to $\mathfrak{h}$. We further prove that if $κ$ is a limit of countably many measurable cardinals, then $\mathbb{B}$ adds a minimal degree of constructibility for new $ω$-sequences. Some of these results generalize to cardinals $κ$ with uncountable cofinality.

math.LO↗
arXiv2019

Forcing and the Halpern-Läuchli Theorem

We investigate the effects of various forcings on several forms of the Halpern-Läuchli Theorem. For inaccessible $κ$, we show they are preserved by forcings of size less than $κ$. Combining this with work of Zhang in \cite{Zhang17} yields that the polarized partition relations associated with finite products of the $κ$-rationals are preserved by all forcings of size less than $κ$ over models satisfying the Halpern-Läuchli Theorem at $κ$. We also show that the Halpern-Läuchli Theorem is preserved by ${{<}κ}$-closed forcings assuming $κ$ is measurable, following some observed reflection properties.

math.LO↗
arXiv2019

Sacks Forcing and the Shrink Wrapping Property

We consider a property stronger than the Sacks property, called the shrink wrapping property, which holds between the ground model and each Sacks forcing extension. Unlike the Sacks property, the shrink wrapping property does not hold between the ground model and a Silver forcing extension. We also show an application of the shrink wrapping property.

math.LO↗
arXiv2018

The Halpern-Läuchli Theorem at a Measurable Cardinal

Several variants of the Halpern-Läuchli Theorem for trees of uncountable height are investigated. For $κ$ weakly compact, we prove that the various statements are all equivalent. We show that the strong tree version holds for one tree on any infinite cardinal. For any finite $d \ge 2$, we prove the consistency of the Halpern-Läuchli Theorem on $d$ many $κ$-trees at a measurable cardinal $κ$, given the consistency of a $κ+d$-strong cardinal. This follows from a more general consistency result at measurable $κ$, which includes the possibility of infinitely many trees, assuming partition relations which hold in models of AD.

math.LO↗
arXiv2018

Ramsey Theory on Generalized Baire Space

We show that although the Galvin-Prikry Theorem does not hold on generalized Baire space with the standard topology, there are similar theorems which do hold on generalized Baire space with certain coarser topologies.

math.LO↗
arXiv2017

Disjoint Borel Functions

For each $a \in \mathbb{R}$, we define a Borel function $f_a : \mathbb{R} \to \mathbb{R}$ which encodes $a$ in a certain sense. We show that for each Borel $g : \mathbb{R} \to \mathbb{R}$, $f_a \cap g = \emptyset$ implies $a \in Δ^1_1(c)$ where $c$ is any code for $g$. We generalize this theorem for $g$ in larger pointclasses $Γ$. Specifically, if $Γ= \mathbfΔ^1_2$, then $a \in L[c]$. Also for all $n \in ω$, if $Γ= \mathbfΔ^1_{3 + n}$, then $a \in \mathcal{M}_{1 + n}(c)$.

math.LO↗
arXiv2016

Weak Distributivity Implying Distributivity

Let $\mathbb{B}$ be a complete Boolean algebra. We show, as an application of a previous result of the author, that if $λ$ is an infinite cardinal and $\mathbb{B}$ is weakly $(λ^ω, ω)$-distributive, then $\mathbb{B}$ is $(λ, 2)$-distributive. Using a parallel result, we show that if $κ$ is a weakly compact cardinal such that $\mathbb{B}$ is weakly $(2^κ, κ)$-distributive and $\mathbb{B}$ is $(α, 2)$-distributive for each $α< κ$, then $\mathbb{B}$ is $(κ, 2)$-distributive.

math.LO↗
arXiv2016

Bounding 2D Functions by Products of 1D Functions

Given sets $X,Y$ and a regular cardinal $\mu$, let $\Phi(X,Y,\mu)$ be the statement that for any function $f : X \times Y \to \mu$, there are functions $g_1 : X \to \mu$ and $g_2 : Y \to \mu$ such that or all $(x,y) \in X \times Y$, $$f(x,y) \le \max \{ g_1(x), g_2(y) \}.$$ In ZFC, the statement $\Phi(\omega_1, \omega_1, \omega)$ is false. However, we show the theory ZF + ``the club filter on $\omega_1$ is normal'' + $\Phi(\omega_1, \omega_1, \omega)$ (which is implied by ZF + AD) implies that for every $\alpha < \omega_1$ there is a $\kappa \in (\alpha,\omega_1)$ such that in some inner model, $\kappa$ is measurable with Mitchell order $\ge \alpha$. There was an error in Welch's paper ``Characterizing Subsets of $\omega_1$ Constructible From a Real'', which he has retracted in a personal communication. Our paper originally referenced that paper. In this version of our paper, we are not using that result. Our consistency strength upper bound has changed accordingly.

math.LO↗
arXiv2015

No Unwanted Universally Baire Morphisms

We show that the usual proof that there are no morphisms (in the sense of cardinal characteristics), whose constituent maps are Borel, between certain challenge-response relations generalizes to show that there are no morphisms whose constituent maps are universally Baire.

math.LO↗
arXiv2014

A Lower Bound for Generalized Dominating Numbers

We show a new proof for the fact that when $κ$ and $λ$ are infinite cardinals satisfying $λ^ κ= λ$, the cofinality of the set of all functions from $λ$ to $κ$ ordered by everywhere domination is $2^λ$. An earlier proof was a consequence of a result about independent families of functions. The new proof follows directly from the main theorem we present: for every $A \subseteq λ$ there is a function $f: {^κλ} \to κ$ such that whenever $M$ is a transitive model of $\textrm{ZF}$ such that ${^κλ} \subseteq M$ and some $g: {^κλ} \to κ$ in $M$ dominates $f$, then $A \in M$. That is, "constructibility can be reduced to domination".

math.LO↗
arXiv2013

A simple C*-algebra with finite nuclear dimension which is not Z-stable

We construct a simple C*-algebra with nuclear dimension zero that is not isomorphic to its tensor product with the Jiang-Su algebra Z, and a hyperfinite II_1 factor not isomorphic to its tensor product with the separable hyperfinite II_1 factor R. The proofs use a weakening of the Continuum Hypothesis.

math.OA↗