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−  {{DISPLAYTITLE: $\text{Con(ZFC)}$}}
 +  #REDIRECT [[ZFC#Consistency of ZFC]] 
−  The assertion $\text{Con(ZFC)}$ is the assertion that the theory $\text{ZFC}$ 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 $\text{ZFC}$. Because of the Gödel completeness theorem, the assertion is equivalent to the assertion that the theory $\text{ZFC}$ 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 $\text{ZFC}$. In general, one may not assume that $\hat\in$ is the actual set membership relation, since this would make the model a [[transitive ZFC model  transitive model of $\text{ZFC}$]], whose existence is a strictly stronger assertion than $\text{Con(ZFC)}$.
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−  The Gödel incompleteness theorem implies that if $\text{ZFC}$ is consistent, then it does not prove $\text{Con(ZFC)}$, and so the addition of this axiom is strictly stronger than $\text{ZFC}$ alone.
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−  == Consistency hierarchy ==
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−  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.
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−  == Every model of $\text{ZFC}$ contains a model of $\text{ZFC}$ as an element ==
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−  Every model $M$ of $\text{ZFC}$ has an element $N$, which it believes to be a firstorder structure in the language of set theory, which is a model of $\text{ZFC}$, as viewed externally from $M$. This is clear in the case where $M$ is an [[omega model  $\omega$model]] of $\text{ZFC}$, since in this case $M$ agrees that $\text{ZFC}$ is consistent and can therefore build a Henkin model of $\text{ZFC}$. In the remaining case, $M$ has nonstandard natural numbers. By the [[reflection theorem]] applied in $M$, we know that the $\Sigma_n$ fragment of $\text{ZFC}$ 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 $\text{ZFC}$. Since $n$ is nonstandard, this includes the full standard theory of $\text{ZFC}$, as desired.
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−  The fact mentioned in the previous paragraph is occasionally found to be surprising by some beginning settheorists, 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 $\text{ZFC}$, the model $M$ need not agree that it is a model of $\text{ZFC}$, in the case that $M$ has nonstandard natural numbers and hence nonstandard length axioms of $\text{ZFC}$.
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