Disjoint Infinity-Borel Functions

This is a followup to a paper by the author where the disjointness relation for definable functions from ${^\omega \omega}$ to ${^\omega \omega}$ is analyzed. In that paper, for each $a \in {^\omega \omega}$ we defined a Baire class one function $f_a : {^\omega \omega} \to {^\omega \omega}$ which encoded $a$ in a certain sense. Given $g : {^\omega \omega} \to {^\omega \omega}$, let $\Psi(g)$ be the statement that $g$ is disjoint from at most countably many of the functions $f_a$. We show the consistency strength of $(\forall g)\, \Psi(g)$ is that of an inaccessible cardinal. We show that $\textrm{AD}^+$ implies $(\forall g)\, \Psi(g)$. Finally, we show that assuming large cardinals, $(\forall g)\, \Psi(g)$ holds in models of the form $L(\mathbb{R})[\mathcal{U}]$ where $\mathcal{U}$ is a selective ultrafilter on $\omega$.Comment: 16 page

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 $\lambda$ is an infinite cardinal and $\mathbb{B}$ is weakly $(\lambda^\omega, \omega)$-distributive, then $\mathbb{B}$ is $(\lambda, 2)$-distributive. Using a parallel result, we show that if $\kappa$ is a weakly compact cardinal such that $\mathbb{B}$ is weakly $(2^\kappa, \kappa)$-distributive and $\mathbb{B}$ is $(\alpha, 2)$-distributive for each $\alpha < \kappa$, then $\mathbb{B}$ is $(\kappa, 2)$-distributive.Comment: 12 page

Combinatorics of reductions between equivalence relations

We discuss combinatorial conditions for the existence of various types of reductions between equivalence relations, and in particular identify necessary and sufficient conditions for the existence of injective reductions.Comment: 7 pages, 2 figure

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.Comment: 16 page