Small models, large cardinals, and induced ideals

Holy, Peter; Lücke, Philipp

doi:10.1016/j.apal.2020.102889

Cited by 11 publications

(43 citation statements)

References 21 publications

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“…Since α = α + m + 1 < ot ( 1 \ 0 )• it follows that 0 +α +2(m+1) < 1 +α +2m+1. 17 Now by Theorem 7.2, we see that Π 1 1 +α+2m+1 (κ) ⊆ R α (Π 1 1 (κ)), and thus, the set…”

Section: Suppose (κ ∈ A) |= ∀Xmentioning

confidence: 88%

A Refinement of the Ramsey Hierarchy via Indescribability

Cody¹

2020

J. symb. log.

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We study large cardinal properties associated with Ramseyness in which homogeneous sets are demanded to satisfy various transfinite degrees of indescribability. Sharpe and Welch [25], and independently Bagaria [1], extended the notion of -indescribability where to that of -indescribability where . By iterating Feng’s Ramsey operator [12] on the various -indescribability ideals, we obtain new large cardinal hierarchies and corresponding nonlinear increasing hierarchies of normal ideals. We provide a complete account of the containment relationships between the resulting ideals and show that the corresponding large cardinal properties yield a strict linear refinement of Feng’s original Ramsey hierarchy. We isolate Ramsey properties which provide strictly increasing hierarchies between Feng’s -Ramsey and -Ramsey cardinals for all odd and for all . We also show that, given any ordinals the increasing chains of ideals obtained by iterating the Ramsey operator on the -indescribability ideal and the -indescribability ideal respectively, are eventually equal; moreover, we identify the least degree of Ramseyness at which this equality occurs. As an application of our results we show that one can characterize our new large cardinal notions and the corresponding ideals in terms of generic elementary embeddings; as a special case this yields generic embedding characterizations of -indescribability and Ramseyness.

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“…Since α = α + m + 1 < ot ( 1 \ 0 )• it follows that 0 +α +2(m+1) < 1 +α +2m+1. 17 Now by Theorem 7.2, we see that Π 1 1 +α+2m+1 (κ) ⊆ R α (Π 1 1 (κ)), and thus, the set…”

Section: Suppose (κ ∈ A) |= ∀Xmentioning

confidence: 88%

A Refinement of the Ramsey Hierarchy via Indescribability

Cody¹

2020

J. symb. log.

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“…Let us show that NSub κ I([κ] <κ ) when κ is almost ineffable (see Corollary 3.22 below). Similar arguments, which are left to the reader, establish the remaining proper containments (for a more general result see [29,Theorem 1.5]).…”

Section: 3mentioning

confidence: 61%

Large cardinal ideals

Cody

2021

Preprint

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Building on work of Holy, Lücke and Njegomir [30] on small embedding characterizations of large cardinals, we use some classical results of Baumgartner (see [7] and [9]), to give characterizations of several well-known large cardinal ideals, including the Ramsey ideal, in terms of generic elementary embeddings; we also point out some seemingly inherent differences between small embedding and generic embedding characterizations of subtle cardinals. Additionally, we present a simple and uniform proof which shows that, when κ is weakly compact, many large cardinal ideals on κ are nowhere κ-saturated. Lastly, we survey some recent consistency results concerning the weakly compact ideal as well as some recent results on the subtle, ineffable and Π 1 1 -indescribable ideals on Pκλ, and we close with a list of open questions. Contents1. Introduction 2. Preliminaries 2.1. Basic terminology and facts about ideals 2.2. Small embeddings 2.3. Embedding characterizations of stationarity 3. Large cardinal ideals and elementary embeddings 3.1. The Π m n -indescribability ideals 3.2. The subtle ideal 3.3. The almost ineffable and ineffable ideals 3.4. The Ramsey ideal 4. Splitting positive sets assuming weak compactness 5. Consistency results 5.1. n-clubs and indescribability embeddings 5.2. A theorem of Hellsten 5.3. The weakly compact reflection principle 5.4. A (κ)-like principle consistent with weak compactness 6. Large cardinal ideals on P κ λ 6.1. Stationary vs. strongly stationary subsets of P κ λ 6.2. Indescribable subsets of P κ λ 6.3. Subtle, strongly subtle and ineffable subsets of P κ λ 6.4. Generic embedding characterizations of large cardinal ideals on P κ λ 7. Questions

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“…In this section, we review a framework for large cardinal operators that was introduced by the second author in [Hol], which in particular fits the ineffability operator, the Ramsey operator, and the strongly Ramsey subset operator. This framework builds on statements about the existence of certain ultrafilters for small models of set theory, and is based on a framework for the characterization of large cardinal ideals that was introduced in [HL21]. In the present paper, we apply this framework, verifying a number of results on the relationship between higher indescribability and large cardinal operators in a uniform way.…”

Section: A Framework For Large Cardinal Operatorsmentioning

confidence: 95%

“…It is easy to see that κ is strongly Ramsey (as defined in [Git11]) if and only if κ ∈ S([κ] <κ ) + , and furthermore, when κ ∈ S(I) + , it follows that S(I) is a nontrivial normal ideal on κ. If κ is strongly Ramsey, then S([κ] <κ ) is the strongly Ramsey ideal on κ, as introduced in [HL21].…”

Section: Introductionmentioning

confidence: 99%