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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?
2026-04-03 00:22:14.1775175734
What will happen when Ax.Inf is replaced to its negation in ZFC?
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Yes: $V_\omega$, the set of hereditarily finite sets, is a model of $(\mathrm{ZFC}-\mathrm{Inf})+\lnot\mathrm{Inf}$.