# Con(ZFC)

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.