Working in Zermelo-Fraenkel Set Theory with Atoms over an $\omega$-categorical $\omega$-stable structure, we show how \emph{infinite} constructions over definable sets can be encoded as \emph{finite} constructions over the Stone-\v{C}ech compactification of the sets. In particular, we show that for a definable set $X$ with its Stone-\v{C}ech 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-\v{C}ech compactification of a definable set is still definable, which allows us to obtain some results about equivalence of certain formalizations of register machines.

翻译：在具有原子的Zermelo-Fraenkel集合论框架下，针对ω-范畴的ω-稳定结构，我们展示了如何将可定义集上的无穷构造编码为该集合Stone-Čech紧化上的有穷构造。特别地，对于可定义集X及其Stone-Čech紧化X̄，我们证明以下命题成立：a) X的幂集P(X)同构于X̄的有穷幂集P_fin(X̄)；b) 域K上的向量空间K^X是X̄上关于K的自由向量空间F_K(X̄)；c) X上的每个测度等价于X̄上的离散测度。此外，我们证明了可定义集的Stone-Čech紧化仍然是可定义的，这使我们能够获得关于寄存器机器某些形式化等价的若干结论。