Weak Version: Is there a 2nd 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

1. $T$ is not $C$-recursive (meaning the set of $C$-codes of the sentences in T is not recursive),

2. every $C$-recursive subtheory of $T$ is satisfiable, and

3. every subtheory of $T$ that isn’t $C$-recursive is unsatisfiable?

Strong Version: Is there a 2nd order language $L$ (with only countably-many formulas) such that there is a theory $T$ of $L$ where

4. $T$ is not recursively axiomatizable,

5. every recursively axiomatizable subtheory of $T$ is satisfiable, and

6. every subtheory of $T$ that isn’t recursively axiomatizable is unsatisfiable?

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.

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

1. $T$ is not satisfiable,

2. every $C$-recursive proper subtheory of $T$ (meaning the set of $C$-codes of the sentences in the subtheory is not recursive) is satisfiable, and

3. every proper 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

4. $T$ is not satisfiable,

5. every recursively axiomatizable proper subtheory of $T$ is satisfiable, and

6. every proper subtheory of $T$ that isn’t recursively axiomatizable is unsatisfiable?

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.

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

1. $T$ is not $C$-recursive (meaning the set of $C$-codes of the sentences in T is not recursive),

2. every $C$-recursive subtheory of $T$ is satisfiable, and

3. 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

4. $T$ is not recursively axiomatizable,

5. every recursively axiomatizable subtheory of $T$ is satisfiable, and

6. 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.

Weak Version: Is there a 2nd 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

1. $T$ is not $C$-recursive (meaning the set of $C$-codes of the sentences in T is not recursive),

2. every $C$-recursive subtheory of $T$ is satisfiable, and

3. every subtheory of $T$ that isn’t $C$-recursive is unsatisfiable?

Strong Version: Is there a 2nd order language $L$ (with only countably-many formulas) such that there is a theory $T$ of $L$ where

4. $T$ is not recursively axiomatizable,

5. every recursively axiomatizable subtheory of $T$ is satisfiable, and

6. every subtheory of $T$ that isn’t recursively axiomatizable is unsatisfiable?

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.