# 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 Gödel completeness theorem, the assertion is equivalent to the assertion that the theory ZFC has a model $\langle M,\hat\in\rangle$. One such model is the Henkin model, built in the syntactic procedure from any complete consistent Henkin theory extending ZFC. 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, whose existence is a strictly stronger assertion than Con(ZFC).

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.

## Every model of ZFC contains a model of ZFC as an element

Every model $M$ of ZFC has an element $N$, which it believes to be a first-order structure in the language of set theory, which is a model of ZFC, as viewed externally from $M$. This is clear in the case where $M$ is an $\omega$-model of ZFC, since in this case $M$ agrees that ZFC is consistent and can therefore build a Henkin model of ZFC. In the remaining case, $M$ has nonstandard natural numbers. By the reflection theorem applied in $M$, we know that the $\Sigma_n$ fragment of ZFC is true in models of the form $V_\beta^M$, for every standard natural number $n$. Since $M$ cannot identify its standard cut, it follows that there must be some nonstandard $n$ for which $M$ thinks some $V_\beta^M$ satisfies the (nonstandard) $\Sigma_n$ fragment of ZFC. Since $n$ is nonstandard, this includes the full standard theory of ZFC, as desired.

The fact mentioned in the previous paragraph is occasionally found to be surprising by some beginning set-theorists, perhaps because naively the conclusion seems to contradict the fact that there can be models of $\text{ZFC}+\neg\text{Con}(\text{ZFC})$. The paradox is resolved, however, by realizing that although the model $N$ inside $M$ is actually a model of full ZFC, the model $M$ need not agree that it is a model of ZFC, in the case that $M$ has nonstandard natural numbers and hence nonstandard length axioms of ZFC.