This is a follow-up question on this one, where I asked for an example for a finitely axiomatizable consistent second-order theory without a model. It was pointed out that this can not be answered objectively without me stating a concrete deductive system for second-order logic (there is no standard one like in FOL).
In this answer (which I also mentioned in my previous question), Asaf gave a consistent second-order theory without a model, but with an axiom schema, hence not finitely axiomized. But he also never mentioned any deductive system. Instead he proved (implicitely) that his example is consistent w.r.t. any reasonable deductive system. Reasonable means:
- compact: proves are finite, hence can only use finitely many axioms.
- sound: we can only deduce true sentences.
So here is an update to my former question:
Is there an example of a finitely axiomatized second-order theory without a model that is consistent w.r.t. any reasonable (i.e. compact + sound) deductive system?