Eccentricity, extendable choice and descending distributive forcing

TL;DR

This paper introduces the concept of descending distributive forcing, explores eccentric sets, and develops the axiom of extendable choice to analyze models with small violations of choice.

Contribution

It defines descending distributive forcing, refines the understanding of eccentric sets, and connects these to the axiom of extendable choice, providing new tools for set-theoretic constructions.

Findings

Descending distributive forcing preserves cofinalities and does not add new functions on ta.

Explicit calculations of Hartogs and Lindenbaum numbers in eccentric models.

Construction of symmetric extensions controlling the validity of the axiom of extendable choice.

Abstract

We introduce the forcing property of descending distributivity. A forcing $\mathbb{P}$ is $\kappa$-descending distributive if for all decreasing sequences $(D_\alpha)_{\alpha<\kappa}$ of open dense sets, $\bigcap_\alpha D_\alpha$ is open dense. This generalises the informal idea that $\mathbb{P}$ doesn't affect much on the scale of $\kappa$, such as if $\mathbb{P}$ is $\kappa$-distributive or if $\kappa > \lvert \mathbb{P} \rvert$. For example, a $\kappa$-descending distributive forcing will not change the cofinality of $\kappa$ or introduce fresh functions on $\kappa$. Using this, we investigate the phenomenon of eccentric sets, those sets $X$ such that, for some ordinal $\alpha$, $X$ surjects onto $\alpha$, but $\alpha$ does not inject into $X$. We refine prior works of the author by giving explicit calculations for the Hartogs and Lindenbaum numbers in eccentric constructions and providing a sharper description of the Hartogs-Lindenbaum spectra of models of small violations of choice. To do so we further develop an axiom (scheme) introduced by Levy that we call the axiom of extendable choice. For an ordinal $\alpha$, $\mathsf{EC}_\alpha$ asserts that if $\emptyset \notin A = \{A_\gamma \mid \gamma < \alpha\}$ and, for all $\beta < \alpha$, $\{ A_\gamma \mid \gamma < \beta\}$ has a choice function, then $A$ has a choice function. This is closely tied to the presence of eccentric sets, and we construct symmetric extensions that give fine control over the $\alpha$ for which $\mathsf{EC}_\alpha$ holds by using descending distributivity.