# Closed Unbounded classes and the Haertig Quantifier Model

## TL;DR

This paper demonstrates that under certain large cardinal assumptions, there exists a definable class of ordinals with properties that make models built from them invariant under set forcing, sharing many structural features.

## Contribution

It introduces a new class of models based on the Haertig quantifier, showing their invariance under forcing and rich combinatorial structure, advancing inner model theory.

## Key findings

- Models satisfy GCH and are elementarily equivalent
- Models have a definable wellorder of the reals
- Models exhibit many combinatorial principles

## 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 $P, Q$, $\langle L[P],\in ,P \rangle$ and $\langle L[Q],\in ,Q \rangle$ 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 H\"artig quantifier when such a $P$ is itself $Card$.