# Finitely axiomatized second-order theory without a model, which is consistent for reasonable deductive systems

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?

No, though perhaps only because your criteria for "reasonable" are too weak. Consider the following "deductive system". Given a set of axioms $\Gamma$, a sentence $\sigma$ can be deduced from $\Gamma$ if there exists a finite subset $\Gamma_0$ such that $\sigma$ is true in every model of $\Gamma_0$. This is compact and sound, but clearly a finite set of axioms is consistent with respect to this system iff it has a model.
More generally, even if you strengthen your definition of reasonable somehow, consider the following argument. Suppose you had a finite set of axioms $\Gamma$ which has no model but which is consistent with respect to any "reasonable" deductive system. Now take your favorite "reasonable" deductive system and strengthen it by adding a new inference rule that allows you to deduce a contradiction from $\Gamma$. This new deductive system breaks your example $\Gamma$, so it cannot be "reasonable". So if you want such a $\Gamma$ to exist, "reasonable" deductive systems must not be closed under adding one additional sound inference rule of this sort.