I was reading Andrej Bauer's paper First Steps in Synthetic Computability Theory. In the conclusion he notes that
Our axiomatization has its limit: it cannot prove any results in computability theory that fail to relativize to oracle computations. This is so because the theory can be interpreted in a variant of the effective topos built from partial recursive functions with access to an oracle.
This made me wonder about non-relativizing results in computability. All results I know from computability theory do relativize to computation with oracles.
Are there results in computability theory that do not relativize? I.e. results which hold for computability but do not hold for computability relative to some oracle?
By result I mean a known theorem in computability theory, not some cooked up statement. If the notion of relativization doesn't make sense for the result then it is not what I am looking for.
It is also interesting to know if the result can be stated in the language of Synthetic Computability Theory or not.