A Mitchell-like order for Ramsey and Ramsey-like cardinals
Erin Carmody, Victoria Gitman, Miha E. Habi\v{c}

TL;DR
This paper introduces a Mitchell-like order for Ramsey and Ramsey-like cardinals, establishing a new hierarchy with desirable properties and analyzing its behavior under forcing extensions.
Contribution
It defines a novel Mitchell-like order for small large cardinals and demonstrates its properties and robustness under forcing, extending the understanding of Ramsey-like cardinals.
Findings
The Mitchell-like order has properties similar to the classical Mitchell order.
Forcing extensions with cover and approximation properties do not increase the rank.
Techniques for soft killing of large-cardinal degrees are applied to Ramsey-like cardinals.
Abstract
Smallish large cardinals are often characterized by the existence of a collection of filters on , each of which is an ultrafilter on the subsets of of some transitive -model of size . 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…
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