Ramsey-like cardinals II

Smallish large cardinals κ are often characterized by the existence of a collection of filters on κ, each of which is an ultrafilter on the subsets of κ of some transitive ZFC − -model of size κ. 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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“…Thus, M = n<ω M n and U = n<ω U n are in V . Finally, Lemma 3.8 of [GW11] implies that U is α-iterable.…”

Section: Strongly Ramsey and Super Ramsey Cardinals Most Of The Work Inmentioning

confidence: 94%

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

Carmody¹,

Gitman

Habič

2020

Fund. Math.

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“…In this section, we consider properties that weakly measurable cardinals may or may not already necessarily possess in an attempt to better understand the possibilities along these lines. The first property that we consider for a weakly measurable cardinal κ is one of weak amenability which strengthens the (weak embedding characterization of a) weakly measurable cardinal in an analogous way to how a weakly Ramsey cardinal (See 3) strengthens the corresponding characterization of weak compactness.…”

Section: Toward Stronger Notions and Open Questionsmentioning

confidence: 97%

“…Having only one such ZFC − model M containing V κ and one weakly amenable embedding \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}${j: M \longrightarrow N}$\end{document} with critical point κ would have been sufficient. In fact, from 3, we know that if the implication were to hold, then every weakly measurable cardinal would be weakly ineffable and more.…”

Section: Toward Stronger Notions and Open Questionsmentioning

confidence: 99%

Weakly measurable cardinals

Schanker¹

2011

Mathematical Logic Quarterly

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Key words Weakly measurable, consistency of a measurable cardinal, least weakly compact cardinal, first failure of the GCH, surgery method, Silver iteration. MSC (2010) 03E35, 03E45, 03E55In this article, we introduce the notion of weakly measurable cardinal, a new large cardinal concept obtained by weakening the familiar concept of a measurable cardinal. Specifically, a cardinal κ is weakly measurable if for any collection A containing at most κ + many subsets of κ, there exists a nonprincipal κ-complete filter on κ measuring all sets in A. Every measurable cardinal is weakly measurable, but a weakly measurable cardinal need not be measurable. Moreover, while the GCH cannot fail first at a measurable cardinal, I will show that it can fail first at a weakly measurable cardinal. More generally, if κ is measurable, then we can make its weak measurability indestructible by the forcing Add(κ, η) for any η while forcing the GCH to hold below κ. Nevertheless, I shall prove that weakly measurable cardinals and measurable cardinals are equiconsistent.In this article, we introduce the notion of weakly measurable cardinal, a new large cardinal concept obtained by weakening the familiar notion of a measurable cardinal, and we show some of the features that can be exhibited by it but not by measurable cardinals. We know that a cardinal κ is measurable if there exists a nonprincipal κ-complete (ultra)filter on κ measuring all of its subsets. We propose a weakening of this notion by insisting only that for any A ⊆ P(κ) of size at most κ + , we can find a nonprincipal κ-complete filter on κ measuring all subsets in A. Of course, if 2 κ = κ + , then these large cardinal concepts are equivalent assertions for κ because we can set A = P(κ). However, our analysis reveals the fact that a measurable cardinal κ can become nonmeasurable and yet still be weakly measurable in a forcing extension (where the GCH fails first at κ).Main Definition A cardinal κ is weakly measurable if for every collection A ⊆ P(κ) containing at most κ + many subsets of κ, there exists a nonprincipal κ-complete filter on κ measuring every set in A. (i.e., for every subset A ∈ A, either A or κ \ A is in the filter.) Main Theorem(1) If κ is measurable, then there exists a forcing extension where κ is weakly measurable, but not measurable.(2) In fact, if κ is measurable, then there exists a forcing extension where κ is weakly measurable (but not measurable), and the GCH fails first at κ.(3) Indeed, if κ is measurable then there exists a forcing extension where the GCH holds, the measurability of κ is preserved, and the weak measurability of κ is indestructible by further forcing Add(κ, η) for any η.(4) Finally, if κ is weakly measurable, then κ is measurable in an inner model, and so the two concepts of measurability are equiconsistent.While the consistency strength of weakly measurable cardinals is settled by statement (4) of the Main Theorem, many open questions remain. If we had only required such filters for A containing at most κ many subsets of κ, then the ...

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“…Weak amenability allows for the iterated ultrapower construction to proceed but it does not guarantee well-foundedness of the iterated ultrapowers. Weakly amenable M -ultrafilters with wellfounded ultrapowers span the full spectrum of iterability; the possibilities range from having exactly one well-founded ultrapower, to having exactly α-many well-founded iterated ultrapowers for any α < ω 1 , to being fully iterable [13]. 11 Kunen showed in [17] that being ω 1 -intersecting is a sufficient condition for a weakly amenable M -ultrafilter to be fully iterable, which leads us into the elementary embeddings characterization of Ramsey cardinals.…”

Section: Ramsey and Strongly Ramsey Cardinalsmentioning

confidence: 99%