A note on encoding infinity in ZFA with applications to register automata
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Working in Zermelo-Fraenkel Set Theory with Atoms over an ω-categorical ω-stable structure, we show how infinite constructions over definable sets can be encoded as 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 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 over 𝒦, c) every measure on X is tantamount to a discrete measure on 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.
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