2 adjusted statement and proof of statement in erroneous proof example/slight tweaking of explanation

There is often a lot of confusion surrounding the differences between relativizing individual formulas to models and the expression of "is a model of" through coding the satisfaction relation with Gödel operations. I think part of this can be attributed to the common preference for using formulas over codings. For example, a standard proof showing that $V_{\kappa} \models ZFC$ for $\kappa$ inaccessible will appeal to the fact that all of the ZFC axioms relativized to $V_{\kappa}$ are true. But then one learns about the Lévy Reflection Theorem scheme which allows every (finite) conjunction of formulas to be reflected to some $V_{\alpha}$. Perhaps this knowledge is followed by a question of whether the Compactness theorem can be used to contradict Gödel's Second Incompleteness Theorem.

Specifically, consider the following erroneous proof that ZFC + CON(ZFC) proves its own consistency:

Introduce a new constant $M$ into the language of set theory and add to the axioms of ZFC all of its axioms $\varphi$ \varphi_n$relativized to$M$, denoted$\varphi^M$. \varphi_n^M$. Provided that ZFC is consistent, every finite collection of this theory is consistent by the Lévy Reflection Theorem whereby the Compactness Theorem tells us that the entire theory ZFC + $M "$M \models ZFC$'' ZFC$" will be consistent. Consequently, this theory has a (ZFC) model $N$ so in this model, there exists a model $M$ of ZFC. To summarize then, arguing in ZFC + CON(ZFC), we've seemingly proven that we have a ZFC model $N$ will model modeling the consistency of ZFC by virtue of it having the model $M$.M$(i.e., seemingly$N \models ZFC + CON(ZFC)$so we would have a proof of CON(ZFC + CON(ZFC)). The misstep in this proof is of course a misuse of the conclusion of the Compactness theorem, mainly the assumption that such an$N$will think that$M$is a ZFC model. With some enumeration of the formulas of the axioms$\{\varphi_n| n \in \mathbb{N}\}$of ZFC, it is clear that$N$will certainly think that$M \models \varphi_n$for any particular$n \in \mathbb{N}$analogous to how a nonstandard model of Peano arithmetic thinks that we have a fixed has an element$c$satisfying$c > n$for any particular$n \in \mathbb{N}$. The problem of course in the case of$N$is that there may be formulas with nonstandard indices not accounted for just as there will definitely be nonstandard numbers greater than$c$in the PA example. If one were to carry out the same proof with the more tedious arithmetization of syntax, then this link may be more apparent. To a lesser extent, there may also be confusion with the fact that$0^{\sharp}$provides us with a proper class of$L(\alpha) \preceq L$. This may lead to the question of whether$L$has its own truth predicate, contradicting Tarski's Theorem. But of course$L$will only realize that each of these$\varphi^{L(\alpha)}$is true for any ZFC axiom$\varphi$, and if one attempts to appeal to the arithmetization of syntax, one can begin to see the problem that these$\alpha$may not (and of course will not) be definable (without parameters) in the constructible universe L. Since these types of misconceptions can be common among logicians and non-logicians alike, I thought I would ask the highly intelligent mathematicians who have worked through such problems or helped illuminate them to others if they would do so here as well. I think compiling a collection of tidbits of wisdom in this area from the collective perspectives of the MO Community can be illuminating to all. As such, my question is as follows: What insights can you share regarding the questions of formalizing "is a model of ZFC" in ZFC and the various "paradoxes" that arise? For example, maybe you can show a related seemingly paradoxical problem and resolve it, or simply share your thoughts on how to avoid such traps of logic. 1 # Clearing misconceptions: Defining "is a model of ZFC" in ZFC There is often a lot of confusion surrounding the differences between relativizing individual formulas to models and the expression of "is a model of" through coding the satisfaction relation with Gödel operations. I think part of this can be attributed to the common preference for using formulas over codings. For example, a standard proof showing that$V_{\kappa} \models ZFC$for$\kappa$inaccessible will appeal to the fact that all of the ZFC axioms relativized to$V_{\kappa}$are true. But then one learns about the Lévy Reflection Theorem scheme which allows every (finite) conjunction of formulas to be reflected to some$V_{\alpha}$. Perhaps this knowledge is followed by a question of whether the Compactness theorem can be used to contradict Gödel's Second Incompleteness Theorem. Specifically, consider the following erroneous proof that ZFC proves its own consistency: Introduce a new constant$M$into the language of set theory and add to the axioms of ZFC all of its axioms$\varphi$relativized to$M$,$\varphi^M$. Provided that ZFC is consistent, every finite collection of this theory is consistent by the Lévy Reflection Theorem whereby the Compactness Theorem tells us that the entire theory ZFC + $M \models ZFC$'' will be consistent. Consequently, this theory has a (ZFC) model$N$so in this model, there exists a model$M$of ZFC. To summarize then, arguing in ZFC, we've seemingly proven that a ZFC model$N$will model the consistency of ZFC by virtue of it having the model$M$. The misstep in this proof is of course a misuse of the conclusion of the Compactness theorem, mainly the assumption that such an$N$will think that$M$is a ZFC model. With some enumeration of the formulas of the axioms$\{\varphi_n| n \in \mathbb{N}\}$of ZFC, it is clear that$N$will certainly think that$M \models \varphi_n$analogous to how a nonstandard model of Peano arithmetic thinks that we have a fixed$c > n$for any$n \in \mathbb{N}$. The problem of course in the case of$N$is that there may be formulas with nonstandard indices not accounted for just as there will definitely be numbers greater than$c$in the PA example. If one were to carry out the same proof with the more tedious arithmetization of syntax, then this link may be more apparent. To a lesser extent, there may also be confusion with the fact that$0^{\sharp}$provides us with a proper class of$L(\alpha) \preceq L$. This may lead to the question of whether$L$has its own truth predicate, contradicting Tarski's Theorem. But of course$L$will only realize that each of these$\varphi^{L(\alpha)}$is true for any ZFC axiom$\varphi$, and if one attempts to appeal to the arithmetization of syntax, one can begin to see the problem that these$\alpha\$ may not (and of course will not) be definable (without parameters) in the constructible universe L.

Since these types of misconceptions can be common among logicians and non-logicians alike, I thought I would ask the highly intelligent mathematicians who have worked through such problems or helped illuminate them to others if they would do so here as well. I think compiling a collection of tidbits of wisdom in this area from the collective perspectives of the MO Community can be illuminating to all. As such, my question is as follows:

What insights can you share regarding the questions of formalizing "is a model of ZFC" in ZFC and the various "paradoxes" that arise?

For example, maybe you can show a related seemingly paradoxical problem and resolve it, or simply share your thoughts on how to avoid such traps of logic.