Inner Models from Extended Logics: Part 1

Abstract

If we replace first order logic by second order logic in the original definition of Gödel's inner model L, we obtain HOD ([32]). In this paper we consider inner models that arise if we replace first order logic by a logic that has some, but not all, of the strength of second order logic. Typical examples are the extensions of first order logic by generalized quantifiers, such as the Magidor-Malitz quantifier ([23]), the cofinality quantifier ([34]), or stationary logic ([6]). Our first set of results show that both L and HOD manifest some amount of formalism freeness in the sense that they are not very sensitive to the choice of the underlying logic. Our second set of results shows that the cofinality quantifier gives rise to a new robust inner model between L and HOD. We show, among other things, that assuming a proper class of Woodin cardinals the regular cardinals > ℵ 1 of V are weakly compact in the inner model arising from the cofinality quantifier and the theory of that model is (set) forcing absolute and independent of the cofinality in question. * The authors would like to thank the Isaac Newton Institute for Mathematical Sciences for its hospitality during the programme Mathematical, Foundational and Computational Aspects of the Higher Infinite supported by EPSRC Grant Number EP/K032208/1. The authors are grateful to John Steel, Philip Welch and Hugh Woodin for comments on the results presented here.