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.
- 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. (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.)