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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2304.09986 |
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| _version_ | 1866929240968003584 |
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| author | Przybyłek, Michał R. |
| author_facet | Przybyłek, Michał R. |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_09986 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A note on Stone-Čech compactification in ZFA Przybyłek, Michał R. Logic in Computer Science Logic 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. |
| title | A note on Stone-Čech compactification in ZFA |
| topic | Logic in Computer Science Logic |
| url | https://arxiv.org/abs/2304.09986 |