Title: Inner models from extended logics



Abstract: Given some logic L* extending first-order logic, we can construct an inner model C(L*) in an L-like fashion - at each level of the construction, we add all the subsets of the current level, which are L*-defineable instead of merely first order definable. This gives us a plethora of inner models, with various properties. In my research I focused on models constructed from the logics obtained by adding cofinality-quantifiers, and the so-called stationary-logic. In this talk I will review some of the results concerning models from cofinality-quantifiers, and then focus on the question of iterating the construction: unlike the case of L, it may be the case that C(L*) does not satisfy V=C(L*), in which case we can repeat the construction inside C(L*), obtaining a descending sequence of models, and at some cases even proceed transfinitely by taking intersections at limit stages. We will show, as time permits, how we can obtain models with descending sequences of the stationary-logic constructable model C(aa). For that purpose we will present shortly the mechanism of iterating club shooting forcings of specific kinds, which we use as a coding mechanism.



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