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 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 Gödel 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.