I have a few questions about absoluteness in set theory; I think I'm largely fine with absoluteness regarding formulas, but then I see absoluteness applied to non-formulas, and that's confusing me.
Fix some transitive classes $M, N$ with $M \subseteq N$ and maybe $N = V$ for convenience. So absoluteness of a formula $\varphi(x_1, \cdots, x_n)$ with $a_1, \cdots, a_n \in M$, simply means that $\varphi^M(a_1, \cdots, a_n) \iff \varphi^N(a_1, \cdots, a_n)$. Basically it's trying to say $M \prec_{\varphi} N$ except since it's a proper class you can't quite say that.
But then, given sets $A$, I've also seen people talk about the set $A^M \in M$. $A$ is a set, not a formula, so what does $A^M$ mean?
My interpretation has been that if $A$ is defined by a formula $\varphi$ in $V$, then $A^M$ is the unique set that satisfies $\varphi^M$. Is that correct? Couching it in model theory terms, $A$ is the unique thing such that $V \vDash \varphi(A)$ and $A^M$ is the unique thing such that $M \vDash \varphi(A^M)$. (Ignoring issues with defining $\vDash$ when talking about classes due to Tarski Indefinability shenanigans).
So if $\varphi$ is absolute (and uses parameters from $M$), does this mean that $A^M = A \cap M$?
I feel like this is largely the right interpretation. I'm even more confused about the absoluteness of functions. Given a (possibly class) function $F : V \rightarrow V$, what does the absoluteness of $F$ mean exactly, and what does $F^M$ refer to?
Is it right to say that if $\varphi$ defines $F$ (with parameters from $M$), meaning that for every $x$, $\varphi(F(x), x)$, then $\varphi$ is absolute? I believe there is also an additional restriction that $F(x) \in M$ for every $x \in M$, which makes sense. So is it fine to say that in this case, $F^M = F \cap M$ as sets ($F$ is conflated with the graph of $F$)? Appealing to model theory, can I imagine expanding the language/model with $F$, and that $M$ forms a submodel in this new language (i.e, basically $F^M = F |_{M})$?
Absoluteness of functions is something I'm confused with, and in particular how to relate absoluteness of the set with the absoluteness of the (graph of) the function.
Finally, I would just like to cement some general intuition about absoluteness. Initially I was thinking about $\varphi$ being absolute to $M, N$ as simply saying "$\varphi$ is true or false in both $M$ and $N$". But now I'm thinking that perhaps it's more accurate to say that $M$ and $N$ "think" that $\varphi$ is true or false at the same time? that is, even if $\varphi$ is absolute to $M$ and $V$, this doesn't necessarily mean $\varphi$ is 'followed' in $M$, it's just that $M$ thinks it is followed. Is that an accurate assessment?
Best Answer
Your confusion is a good one: when people say e.g. "$\omega$ is absolute between transitive models of $\mathsf{ZFC}$," what they really mean is "the formula $\varphi_\omega$ is absolute between transitive models of $\mathsf{ZFC}$" where $\varphi_\omega$ is the usual formula defining $\omega$ in $V$.
Note the bolded clause, however - there are multiple ways to define $\omega$ in $V$, and not all of them are absolute! For example, suppose $\mathsf{CH}$ holds in $V$, and consider the formula $$\psi(x)\equiv (\mathsf{CH}\rightarrow \varphi_\omega(x))\wedge(\neg\mathsf{CH}\rightarrow \forall y(\neg y\in x)).$$ This still defines $\omega$ in $V$, but is clearly not absolute between transitive models of $\mathsf{ZFC}$: if $M\models\mathsf{ZFC+\neg CH}$ is transitive then $\psi^M=\emptyset$.
So actually there's a huge potential for nonsense here, and this conflation of an object with some fixed definition of it is a really annoying abuse of notation. Fortunately it's rarely an issue, since generally there's only one reasonable choice of definition and it's clear from context, but it's still very much an abuse. (And every so often it does matter - less in terms of proving results than in developing intuitions, but still.)
The same is true for (class) functions: we often conflate an object with some fixed formula defining it. And again, this depends crucially on what definition we pick.
Meanwhile I don't really understand your last paragraph: it sounds like you're asserting a difference between "$\varphi$ is true in $M$" and "$M$ thinks $\varphi$ is followed," but I don't see what that difference would be.
(Note that, following the above abuse of notation, we do have $V^M=M$: the "standard" formula defining $V$ is $x=x$, and $M\models\forall x(x=x)$. So in fact if $M\models\theta$ then $M$ thinks $\theta$ is true in $V$ - since $M$ thinks it is $V$. Maybe that's what you're getting at?)