A Mitchell-like order for Ramsey and Ramsey-like cardinals
| Autor: | Erin Carmody, Miha E. Habič, Victoria Gitman |
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| Rok vydání: | 2020 |
| Předmět: |
Transitive relation
Mathematics::Combinatorics Algebra and Number Theory Forcing (recursion theory) Rank (linear algebra) 010102 general mathematics Ultrafilter Mathematics::General Topology Measurable cardinal Order (ring theory) Mitchell order Mathematics - Logic 01 natural sciences Combinatorics Mathematics::Logic 03E55 Cover (topology) FOS: Mathematics 0101 mathematics Logic (math.LO) Mathematics |
| Zdroj: | Fundamenta Mathematicae. 248:1-32 |
| ISSN: | 1730-6329 0016-2736 |
| Popis: | 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. Comment: 23 pages |
| Databáze: | OpenAIRE |
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