2019/03/06 by Welch, Philip
#03E45 03E55 #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.1903.02663
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],∈ ,P ⟩ and ⟨ L[Q],∈ ,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.