Inner Models from Extended Logics

主讲：Prof. Jouko Vaananen

时间：2016年6月16日15:00-17:00

地点：中国科学院数学院南楼N820室

摘要：If we replace first order logic by second order logic in the original definition of Goedel's inner model L, we obtain HOD. In this talk 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, the cofinality quantifier, stationary logic or the Hartig-quantifier. We show that these extensions give rise to new robust inner models between L and HOD.  We show, among other things, that assuming a proper class of Woodin cardinals, the regular cardinals >aleph_1 of V are weakly compact in the inner model arising from the cofinality quantifier and the theory of that model is forcing absolute. Assuming a proper class of measurable Woodin cardinals the regular cardinals of V are measurable in the inner model arising from stationary logic and the theory of that model is again forcing absolute. We discuss the question, and present some results, concerning the question whether these inner models satisfy the Continuum Hypothesis, assuming large cardinals. This is joint work with Juliette Kennedy and Menachem Magidor.