A note on Stone-Čech compactification in ZFA
Michał R. Przybyłek
TL;DR
This work analyzes Stone-Čech compactification within ZFA over ω-categorical ω-stable structures, showing that infinite constructions on definable sets can be faithfully encoded by finite constructions on their compactifications $\overline{X}$. Key results include $\mathcal{P}(X) \cong \mathcal{P}_{fin}(\overline{X})$, $\mathcal{K}^X \cong F_{\mathcal{K}}(\overline{X})$, and that measures on $X$ reduce to discrete measures on $\overline{X}$, with $\overline{X}$ itself definable. The ultrafilter monad framework yields a robust internal Stone-Čech theory, enabling a monoidal-closed category of vector spaces on definable bases and connecting definable weighted automata with linear monoids; this also supports novel probabilistic register-machine models via Keisler-type measures. Together, these results provide a bridge from infinite definable-set constructions to finite, well-behaved algebraic and probabilistic structures, with applications to register machines, weighted automata, and probabilistic semantics in definable contexts. The work highlights the definability of compactifications, characterizes measures as finite ultrafilter mixtures in ω-stable settings, and opens directions for extending these methods to NIP and beyond.
Abstract
Working in Zermelo-Fraenkel Set Theory with Atoms over an $ω$-categorical $ω$-stable structure, we show how \emph{infinite} constructions over definable sets can be encoded as \emph{finite} constructions over the Stone-Čech compactification of the sets. In particular, we show that for a definable set $X$ with its Stone-Čech compactification $\overline{X}$ the following holds: a) the powerset $\mathcal{P}(X)$ of $X$ is isomorphic to the finite-powerset $\mathcal{P}_{\textit{fin}}(\overline{X})$ of $\overline{X}$, b) the vector space $\mathcal{K}^X$ over a field $\mathcal{K}$ is the free vector space $F_{\mathcal{K}}(\overline{X})$ on $\overline{X}$ over $\mathcal{K}$, c) every measure on $X$ is tantamount to a \emph{discrete} measure on $\overline{X}$. Moreover, we prove that the Stone-Čech compactification of a definable set is still definable, which allows us to obtain some results about equivalence of certain formalizations of register machines.