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What are the consequences if Axiom of Infinity is negated?

In ZFC, if we replace Ax.Inf to such a statement that every set is finite, then does this theory satisfiable?

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marked as duplicate by Asaf Karagila, rschwieb, Chris Eagle, Ross Millikan, Thomas Oct 20 '12 at 14:47

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Yes: $V_\omega$, the set of hereditarily finite sets, is a model of $(\mathrm{ZFC}-\mathrm{Inf})+\lnot\mathrm{Inf}$.

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You mean of course (ZFC-Inf)$+\neg$ Inf. –  Hagen von Eitzen Oct 20 '12 at 13:02
@Hagen: I do indeed; thanks. –  Brian M. Scott Oct 20 '12 at 13:03

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