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?
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.
