Stationary logic is the extension of first-order logic with a quantifier expressing that “almost all” countable subsets of a structure have a given property. This paper studies \(C(aa)\) , the smallest model of ZF containing the ordinal numbers and closed under the satisfaction predicate for stationary logic. This model was constructed by Kennedy et al. (2024) as a generalization of Gödel’s constructible universe \(L\) , obtained by iterating definability in stationary logic rather than first-order logic. Unlike \(L\) , however, \(C(aa)\) can contain large cardinals far beyond a measurable cardinal. We show in this paper that nevertheless, assuming large cardinals in \(V\) , the inner model \(C(aa)\) shares many of the nice properties of \(L\) . In particular, we prove that \(C(aa)\) satisfies the Generalized Continuum Hypothesis, the Ultrapower Axiom, and the axiom \(V = \text {HOD}\) .
