Closed Unbounded classes and the Haertig Quantifier Model

Abstract

We show that assuming modest large cardinals, there is a definable class of ordinals, closed and unbounded beneath every uncountable cardinal, so that for any closed and unbounded subclasses $P,Q$, $⟨L[P],\in ,P⟩$ and $⟨L[Q],\in ,Q⟩$ possess the same reals, satisfy the Generalised Continuum Hypothesis, and moreover are elementarily equivalent. The theory of such models is thus invariant under set forcing. They also all have a rich structure satisfying many of the usual combinatorial principles and a definable wellorder of the reals. One outcome is that we can characterize the inner model constructed using definability in the language augmented by the Härtig quantifier when such a $P$ is itself $Card$.