Iterating the construction of inner models from extended logics, part 1

One way of generalizing Goedel's constructible universe L, is to replace the notion of definability used at successor stage, and take all subsets of the last stage which are definable using a logic L* extending first order logic. This will result in a model of ZF, denoted C(L*). In some cases it will also be a model of AC. As in the case of L, we can formulate the axiom "V=C(L*)", but unlike L, it is not always the case that C(L*) |= "V=C(L*)", that is, when we construct C(L*) inside C(L*) we might get a smaller model. If that is the case, we can iterate this construction, and if it doesn't stabilize, continue transfinitely with intersections at limit stages.
In this series of lectures we'll present some of the results concerning these sequences of iterated constructible models, focusing mainly on the case of the logic obtained by adding the cofinality-omega quantifier, and the case of Stationary logic. We'll show that in the first case the possibilities are rather limited, and that in the second case we can get almost everything we want.