Eccentricity, extendable choice and descending distributive forcing

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.