Confused about truth in standard models vs non-standard models

Question

So let's say I have a set of sentences $T$, true in my standard model $\mathfrak{A}$. I construct the consequences of $T$, $Cn(T)$. For any non-standard model $\mathfrak{B}$, it has to be the case that $T$ and thus $Cn(T)$ are true in $\mathfrak{B}$, correct?

Answer 1

I think the key question here 'is standard model of what?' If it is a complete theory, e.g. $\operatorname{Th}(\mathbb N),$ then any collection of sentences true in $\mathbb N$ is true in any nonstandard model of $\operatorname{Th}(\mathbb N).$ If it is not a complete theory, e.g. PA, then this is no longer the case unless the sentences of $T$ are all consequences of PA. I guess that's the key point... all consequences of PA are true in all models of PA. Whereas any sentence that is consistent with, but not a consequence of PA is true in some models of PA but not others. (this is either tautological or a consequence of completeness/soundness theorem, depending on whether we mean semantic or syntactic consequence).

For instance, the Godel sentence $G$ and $\operatorname{Con}(PA)$ are both true in $\mathbb N,$ but there are nonstandard models of PA in which they are false. So $T=\{\operatorname{Con}(PA)\}$ is a theory whose consequences all hold in $\mathbb N$ (moreover its consequences modulo PA all hold), but there are nonstandard models of PA in which some of the consequences are not satisfied, namely $\operatorname{Con}(PA)$ itself.