A note on the eightfold way

Gilton, Thomas; Krueger, John

doi:10.1090/proc/14771

Cited by 7 publications

(9 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…After this article was completed, I. Neeman discovered a shorter proof of the consistency of stationary set reflection at ω 2 together with an arbitrarily large continuum. Specifically, starting with a model in which stationary set reflection holds, adding any number of Cohen reals preserves stationary set reflection (see [2,Theorem 3.1]). This new proof is, however, somewhat limited in its applications.…”

Section: Postscriptmentioning

confidence: 99%

“…(3) p p( ) ∩Ṡ = ∅. By Lemma 1.12, (2) implies that p( ) is a P -name for a closed and bounded subset of 2 . So by (3)…”

mentioning

confidence: 96%

“…Since in that case not all reals would be added by the iteration collapsing κ to become 2 , extending the elementary embedding becomes more difficult. Indeed, in the model referred to in the previous paragraph, a stronger stationary set reflection principle holds, namely WRP 2, which asserts that any stationary subset of [ 2 ] reflects to [ ] for some uncountable < 2 , and by a result of Todorcević, WRP( 2 ) implies 2 ≤ 2 (see [5,Lemma 2.9]).…”

mentioning

confidence: 99%

See 2 more Smart Citations

The Harrington–shelah Model With Large Continuum

Gilton¹,

Krueger²

2019

J. symb. log.

Self Cite

View full text Add to dashboard Cite

We prove from the existence of a Mahlo cardinal the consistency of the statement that 2 ω = ω 3 holds and every stationary subset of ω 2 ∩ cof(ω) reflects to an ordinal less than ω 2 with cofinality ω 1 .Let us say that stationary set reflection holds at ω 2 if for any stationary set S ⊆ ω 2 ∩ cof(ω) there is an ordinal α ∈ ω 2 ∩ cof(ω 1 ) such that S ∩ α is stationary in α (that is, S reflects to α). In a classic forcing construction, Harrington and Shelah [3] proved the equiconsistency of stationary set reflection at ω 2 with the existence of a Mahlo cardinal. Specifically, if stationary set reflection holds at ω 2 , then ω1 fails, and hence ω 2 is a Mahlo cardinal in L. Conversely, if κ is a Mahlo cardinal, then the generic extension obtained by Lévy collapsing κ to become ω 2 and then iterating to kill the stationarity of nonreflecting sets satisfies stationary set reflection at ω 2 . The Harrington-Shelah argument is notable because the majority of stationary set reflection principles are derived by extending large cardinal elementary embeddings, and thus use large cardinal principles much stronger than the existence of a Mahlo cardinal.The original Harrington-Shelah model satisfies the generalized continuum hypothesis, and in particular, that 2 ω = ω 1 . Suppose we would like to obtain a model of stationary set reflection at ω 2 together with 2 ω = ω 2 . A natural construction would be to iterate forcing with countable support of length a weakly compact cardinal κ, alternating between adding reals and collapsing ω 2 to have size ω 1 . Such an iteration P would be proper, κ-c.c., collapse κ to become ω 2 , and satisfy that 2 ω = ω 2 . The fact that stationary set reflection holds in any generic extension V [G] by P follows from the ability to extend an elementary embedding j with critical point κ after forcing with the proper forcing j(P)/G over V [G].Consider the problem of obtaining a model satisfying stationary set reflection at ω 2 together with 2 ω > ω 2 . Since in that case not all reals would be added by the iteration collapsing κ to become ω 2 , extending the elementary embedding becomes more difficult. Indeed, in the model referred to in the previous paragraph, a stronger stationary set reflection principle holds, namely WRP(ω 2 ), which asserts that any stationary subset of [ω 2 ] ω reflects to [β] ω for some uncountable β < ω 2 , and by a result of Todorcević, WRP(ω 2 ) implies 2 ω ≤ ω 2 (see [5, Lemma 2.9]).In this paper we demonstrate that the cardinality of the continuum provides a natural separation between ordinary stationary set reflection and higher order

show abstract

Section: Postscriptmentioning

confidence: 99%

“…(3) p p( ) ∩Ṡ = ∅. By Lemma 1.12, (2) implies that p( ) is a P -name for a closed and bounded subset of 2 . So by (3)…”

mentioning

confidence: 96%

mentioning

confidence: 99%

See 1 more Smart Citation

The Harrington–shelah Model With Large Continuum

Gilton¹,

Krueger²

2019

J. symb. log.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Remark 4.13. Some preservation theorems for stationary reflection were known before: it was known that the Prikry-style forcings at κ preserve SR(κ ++ ) due to their Prikry property (see for instance [3]) and that in general κ + -cc forcings of size < κ ++ preserve SR(κ ++ ) and the Cohen forcing at ω of any length preserves SR(ω 2 ) (attributed to Neeman in [9]). Proof.…”

Section: Stationary Reflectionmentioning

confidence: 99%

Small u(kappa) at singular kappa with compactness at kappa++

Honzík¹,

Stejskalová²

2019

Preprint

View full text Add to dashboard Cite

We show that the tree property, stationary reflection and the failure of approachability at κ ++ are consistent with u(κ) = κ + < 2 κ , where κ is a singular strong limit cardinal with the countable or uncountable cofinality. As a by-product, we show that if λ is a regular cardinal, then stationary reflection at λ + is indestructible under all λ-cc forcings (out of general interest, we also state a related result for the preservation of club stationary reflection). Contents1. Introduction 1 2. Preliminaries 3 2.1. Compactness properties 3 2.2. Branch lemmas 4 2.3. Some pcf notions 5 2.4. Prikry and Magidor forcing 5 3. A review of the arguments for a small u(κ) 6 4. Countable cofinality 9 4.1. Definition of forcing 9 4.2. Small ultrafilter number 11 4.3. Tree property 11 4.4. Stationary reflection 14 4.5. Failure of approachability 17 5. Uncountable cofinalities 17 6. Open questions 18 References 19

show abstract

“…The appearance of Add(τ ) after the initial component, together with the preservation properties of the quotient Q, allowed Krueger's new arguments to go through various complicated constructions. Mixed support iterations have found several applications since [5], particularly in regard to guessing models [16].…”

Section: Introductionmentioning

confidence: 99%

On disjoint stationary sequences

Levine¹

2023

Preprint

View full text Add to dashboard Cite

We answer a question of Krueger by obtaining disjoint stationary sequences on successive cardinals. The main idea is an alternative presentation of a mixed support iteration, using it even more explicitly as a variant of Mitchell forcing. We also use a Mahlo cardinal to obtain a model in which ℵ 2 / ∈ I[ℵ 2 ] and there is no disjoint stationary sequence on ℵ 2 , answering a question of Gilton.

show abstract

A note on the eightfold way

Abstract: Assuming the existence of a Mahlo cardinal, we construct a model in which there exists an ω 2 -Aronszajn tree, the ω 1 -approachability property fails, and every stationary subset of ω 2 ∩ cof(ω) reflects. This solves an open problem of [1].

Cited by 7 publications

References 6 publications

The Harrington–shelah Model With Large Continuum

The Harrington–shelah Model With Large Continuum

Small u(kappa) at singular kappa with compactness at kappa++

On disjoint stationary sequences

Contact Info

Product

Resources

About