# Difference between revisions of "Con ZFC"

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− | {{DISPLAYTITLE: Con(ZFC)}} | + | {{DISPLAYTITLE: $\text{Con(ZFC)}$}} |

+ | 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 $\text{Con(ZFC)}$. | ||

− | The | + | 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 == | == 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. | 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 of $\text{ZFC}$ contains a model of $\text{ZFC}$ 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 | $\omega$-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. | + | Every model $M$ of $\text{ZFC}$ has an element $N$, which it believes to be a first-order structure in the language of set theory, which is a model of $\text{ZFC}$, 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. |

− | 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. | + | 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 $\text{ZFC}$, 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 $\text{ZFC}$. |

## Revision as of 12:59, 11 November 2017

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 model of $\text{ZFC}$, whose existence is a strictly stronger assertion than $\text{Con(ZFC)}$.

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.

## 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 $\text{ZFC}$ contains a model of $\text{ZFC}$ as an element

Every model $M$ of $\text{ZFC}$ has an element $N$, which it believes to be a first-order structure in the language of set theory, which is a model of $\text{ZFC}$, as viewed externally from $M$. This is clear in the case where $M$ is an $\omega$-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.

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 $\text{ZFC}$, 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 $\text{ZFC}$.