I'm reading the book Mathematical Logic by Cori and Lascar, in which Gödel's second incompleteness theorem is stated as follows:
Let $T$ be a consistent, recursive theory that extends $\mathcal{P}$. Then $Con(T)$ is not derivable in $T$.
Why is this theorem interpreted as "a theory cannot prove its own consistency"? Since $Con(T)$ is not exactly the same as the consistency of $T$. They are the same notion for $\mathbb{N}$, but not for some non-standard models of $\mathcal{P}$.
Edit: So is the theorem still worth the existential crisis?
Best Answer
I don't agree with the general idea that $\mathrm{Con}(T)$ isn't meaningful in non-$\omega$-models of theories like $\mathsf{PA}$ and $\mathsf{ZFC}$. This is alluded to in some of the discussion of Artemov's paper on FOM, but I think it's worth an answer here, specifically to your question, 'Why is this theorem interpreted as "a theory cannot prove its own consistency"?' The short answer is that $M \models \mathrm{Con}(T)$ really does mean that $T$ is consistent 'internally' inside $M$.
The way in which $\mathrm{Con}(T)$ is meaningful is easier to see in the context of $\mathsf{ZFC}$, but a version of this is true in $\mathsf{PA}$ as well. A really important result in mathematical logic is Gödel's completeness theorem which says that a first-order theory has a model if and only if it is consistent in the sense of a syntactic proof calculus. This fact is of course provable in $\mathsf{ZFC}$, that is to say that $\mathsf{ZFC}$ proves $$\forall T (\mathsf{Con}(T) \leftrightarrow \exists M \models T).$$ What's important here is that any model $M$ of $\mathsf{ZFC}$ is correct about satisfying individual formulas. That is to say, if $A$ is some structure in $M\models \mathsf{ZFC}$, the for any standard formula $\varphi(\bar{x})$ and any $\bar{a} \in A$, $M$ 'thinks' $A \models \varphi(\bar{a})$ if and only if $A \models \varphi(\bar{a})$. For infinite theories the issue of course is that there are now non-standard formulas, but you still get one direction, specifically, if $M \models \mathsf{ZFC}$ thinks that $A$ is a model of a computably enumerable theory $T$, then $A$ is actually a model of that theory.
$\mathsf{PA}$ and $\mathsf{ZFC}$ have the special property that they non-uniformly prove the consistency of any finite fragment of themselves. By a slightly tricky argument, this implies that every model of $\mathsf{ZFC}$ contains what is externally a model of $\mathsf{ZFC}$. In a model of $\mathsf{ZFC}+\neg\mathrm{Con}(\mathsf{ZFC})$, though, there is no way to definably separate these models from models of arbitrarily large finite fragments of $\mathsf{ZFC}$.
With finitely axiomatizable set theories like $\mathsf{NBG}$ or $\mathsf{NFU}$, Gödel's second incompleteness theorem has a far more striking implication. There are, for instance, models of $\mathsf{NFU}$ which contain no models of $\mathsf{NFU}$. $\mathsf{ZFC}$ has a similar phenomenon, which is that there are models of $\mathsf{ZFC}$ which contain no transitive models of $\mathsf{ZFC}$.
The analogous phenomenon with $\mathsf{PA}$ involves the fact that a certain form of the completeness theorem is provable in $\mathsf{PA}$: If a model $N$ of $\mathsf{PA}$ satisfies $\mathrm{Con}(T)$, then $N$ actually interprets the full elementary diagram of a model of $T$. The converse is true as well, so this means that a model $N$ of $\mathsf{PA} + \neg \mathrm{Con}(\mathsf{PA})$ does not interpret the full elementary diagram of a model of $\mathsf{PA}$.