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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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Yes: $V_\omega$, the set of hereditarily finite sets, is a model of $(\mathrm{ZFC}-\mathrm{Inf})+\lnot\mathrm{Inf}$.
Brian M. Scott
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2You mean of course (ZFC-Inf)$+\neg$ Inf. – Hagen von Eitzen Oct 20 '12 at 13:02
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@Hagen: I do indeed; thanks. – Brian M. Scott Oct 20 '12 at 13:03