Closed and Unbounded Classes and the Härtig Quantifier Model

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 and possess the same reals, satisfy the Generalised Continuum Hypothesis, and moreover are elementarily equivalent. Examples of such P are Card, the class of uncountable cardinals, I the uniform indiscernibles, or for any n the class ; moreover the theory of such models is invariant under ZFC-preserving extensions. They also all have a rich structure satisfying many of the usual combinatorial principles and a definable wellorder of the reals. The inner model constructed using definability in the language augmented by the Härtig quantifier is thus also characterized.