| Résume | If we replace first order logic by second order logic in the original definition of Gödel’s inner model L, we obtain HOD (result due to Myhill-Scott). In this talk after giving some historical background 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 ([21]), the cofinality quantifier ([32]), or stationary logic ([6]). Our first set of results show that both L and HOD manifest some amount of robustness 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 above \aleph_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. |
