This talk is the first of a series of two talks. The second talk will be delivered by Menachem Magidor. For any extension L* of first order logic, e.g. by generalized quantifiers, we associate an inner model C(L*) obtained as the inner model L of constructible sets but using the logic L* in place of first order logic. For many but not all logics L* the inner model C(L*) is simply L. Interesting examples are the generalized quantifier which says that a definable linear order has countable cofinality giving rise to an inner model we denote by C*. Another interesting example is the second order quantifier which says that a formula is satisfied by a club of countable subsets of the domain of the model, giving rise to an inner model we call C(aa). Clearly C* is a submodel of C(aa). Assuming a proper class of Woodin cardinals, C* has a proper class of weakly compact cardinals and its theory is forcing absolute. Under the same assumption, C(aa) has a proper class of measurable cardinals, it satisfies GCH and its theory is again forcing absolute. In this two part talk we compare the models C* and C(aa) as well as some of their cousins to the extender models of inner model theory and their mice. It turns out that our inner models have a closer relationship with extender models and their mice than meets the eye. In many cases one can determine a particular mouse that projects to omega which is associated with a given C(L*). It is the minimal mouse (in the mice order) which is not in C(L*). It can be considered as a gauge for the strength of the logic L*. In these two talks we shall survey some of the ideas which are used in the analysis of inner models of the form C(L*) and their relations to the usual fine structural models.