Con(ZFC)

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The assertion Con(ZFC) is the assertion that the theory ZFC is consistent. This is an assertion with complexity $\Pi^0_1$ in arithmetic, since it is the assertion that every natural number is not the Gödel code of the proof of a contradiction from ZFC. Because of the G&odel completeness theorem, the assertion is equivalent to the assertion that the theory ZFC has a model $\langle M,\hat\in\rangle$. In general, one may not assume that $\hat\in$ is the actual set membership relation, since this would make the model a transitive model of ZFC, a stronger assertion.

The G&odel incompleteness theorem implies that if ZFC is consistent, then it does not prove Con(ZFC), and so the addition of this axiom is strictly stronger than ZFC alone.

Consistency hierarchy

The expression $\text{Con}^2(\text{ZFC})$ denotes the assertion $\text{Con}(\text{ZFC}+\text{Con}(\text{ZFC}))$, and iterating this more generally, one may consider the assertion $\text{Con}^\alpha(\text{ZFC})$ whenever $\alpha$ itself is expressible.