Weak Version: Is there a 1st order language $L$ (with only countably-many formulas) such that for each recursive coding $C$ of the formulas of $L$, there is a theory $T$ of $L$ where
$T$ is not $C$-recursive (meaning the set of $C$-codes of the sentences in T is not recursive),
every $C$-recursive subtheory of $T$ is satisfiable, and
every subtheory of $T$ that isn’t $C$-recursive is unsatisfiable?
Strong Version: Is there a 1st order language $L$ (with only countably-many formulas) such that there is a theory $T$ of $L$ where
$T$ is not recursively axiomatizable,
every recursively axiomatizable subtheory of $T$ is satisfiable, and
every subtheory of $T$ that isn’t recursively axiomatizable is unsatisfiable?
What about examples of 2nd order languages (with only countably-many formulas) that have these properties?
Restricting to languages with only countably-many formulas excludes uncountable theories (which trivially fail to be recursively axiomatizable), but this restriction could be relaxed if we require that the theory T also be countable.