Some remarks on Stone-Čech compactification in ZFA

Working in Zermelo-Fraenkel Set Theory with Atoms over an \omega-categorical \omega-stable structure, we show how some infinite constructions over definable sets can be encoded as finite constructions over Stone-Čech compactification of the sets. In particular, we show that for a definable set X with its Stone-Čech compactification X* the following holds: a) the powerset P(X) of X is isomorphic to the finite-powerset P_fin(X*) of X*, b) the vector space K^X over a field K is the free vector space F_K(X*) on X* generators over K, c) every measure on X is tantamount to a discrete measure on X^*. Moreover, we prove that 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.