Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2304.09986 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Table of Contents:
- 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.