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arXiv · 1609.07645

A Mitchell-like order for Ramsey and Ramsey-like cardinals

Erin Carmody·Victoria Gitman·Miha E. Habič
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Abstract

Smallish large cardinals $\kappa$ are often characterized by the existence of a collection of filters on $\kappa$, each of which is an ultrafilter on the subsets of $\kappa$ of some transitive $\mathrm{ZFC}^-$-model of size $ \kappa$. We introduce a Mitchell-like order for Ramsey and Ramsey-like cardinals, ordering such collections of small filters. We show that the Mitchell-like order and the resulting notion of rank have all the desirable properties of the Mitchell order on normal measures on a measurable cardinal. The Mitchell-like order behaves robustly with respect to forcing constructions. We show that extensions with cover and approximation properties cannot increase the rank of a Ramsey or Ramsey-like cardinal. We use the results about extensions with cover and approximation properties together with recently developed techniques about soft killing of large-cardinal degrees by forcing to softly kill the ranks of Ramsey and Ramsey-like cardinals.

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Erin Carmody, Victoria Gitman, Miha E. Habič. 2016-09-24. A Mitchell-like order for Ramsey and Ramsey-like cardinals. https://doi.org/10.4064/fm701-3-2019

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