Let us work in Kelly-morse set theory, so we can talk about $V$. For some model $M=(\mathbb N, \in\_M)$ that is elementary equivalent $(V, \in)$, we can have an oracle that corresponds to $(\mathbb N, \in\_M)$'s truth predicate. We will say that $M$ is a true countable model of set theory, and we will call the oracle $O\_M$.


My question is, is there an oracle that can compute a function iff every oracle $O\_M$ for ever true countable model of set theory can compute the function?


Such an oracle would at least be able to determine if a first order sentence without free variables is true in $(V, \in)$. My conjecture is it does exist, and is defined by the previous statement.


Some notes:


* For any definable set, there is a Turing machine with oracle $O\_M$ that outputs a number corresponding to that set.
* Any $O\_M$ (for true countable M) can calculate [Rayo's function](https://googology.fandom.com/wiki/Rayo%27s_number). (Another question to ask would be if a Turing machine equipped with an oracle for Rayo's function let's you determine truth of statements in $(V, \in)$ without free variables.)



Is there an oracle that can compute something iff it is computable in every countable model that is equivalent to $(V, \in)$?
Yes - this real is exactly the parameter-free theory of $V$ (or anything Turing-equivalent to it).


One direction is immediate: if $M$ is elementarily equivalent to $V$, then from the elementary diagram of $M$ we can compute the theory of $V$ (just look at the parameter-free sentences). It's the other direction that is interesting, and the key is that *Henkinization is effective*: from any complete consistent first-order theory $T$ we can uniformly compute a model of $T$ together with its elementary diagram.


Note that this applies to arbitrary structures, not just $V$: the reals computable from the elementary diagram of any structure $\mathcal{N}$ elementarily equivalent to $\mathcal{M}$ are exactly the reals computable from $Th(\mathcal{M})$.




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Incidentally, a very similar-sounding question has a very different answer in general. Given a structure $\mathcal{A}$, we can look at its *copies* - those structures $\mathcal{B}$ with domain $\mathbb{N}$ which are isomorphic to $\mathcal{A}$. We can ask what reals are computable from (the atomic diagram of) every copy of $\mathcal{A}$. Interestingly, even very complicated structures can have very *little* computing power in this sense; e.g. Linda Jean Richter showed in *[Degrees of Structures](https://doi.org/10.2307/2273222)* ([JSTOR](https://www.jstor.org/stable/2273222)) that if $\mathcal{L}$ is a linear order, it has *no* computing power - for every noncomputable real $x$ there is some copy $\mathcal{J}$ of $\mathcal{L}$ whose atomic diagram doesn't compute $x$. If you're interested in this sort of question, Ash and Knight's book *[Computable Structures and the Hyperarithmetical Hierarchy](https://www.sciencedirect.com/bookseries/studies-in-logic-and-the-foundations-of-mathematics/vol/144/suppl/C)* is the standard reference.


It's also worth noting that for any computable theory $T$ and any noncomputable set $x$, there is a model of $T$ whose atomic diagram does not compute $x$; moreover, any such theory has a model with a [low](https://en.wikipedia.org/wiki/Low_(computability)) atomic (or even elementary) diagram - even a really complicated theory like ZFC.


