Overgeneration in the Higher Infinite

Abstract

The Overgeneration Argument is a prominent objection against the model-theoretic account of logical consequence for second-order languages. In previouswork we have offered a reconstruction of this argument which locates its source inthe conflict between the neutrality of second-order logic and its alleged entanglement with mathematics. Some cases of this conflict concern small large cardinals. In this article, we show that in these cases the conflict can be resolved by moving from a set-theoretic implementation of the model-theoretic account to one which uses higher-order resources