Bounding by Canonical Functions, With Ch

Larson, Paul B.; Shelah, Saharon

doi:10.1142/s021906130300025x

Cited by 10 publications

(9 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(2) WRP([ω 2 ] ω ) implies-in fact is equivalent to-the non-existence of a costationary, local club subset of [ω 2 ] ω ; 12 and (3) If there is a special Aronszajn tree on ω 2 then there is a thin local club subset T of [ω 2 ] ω (Theorem 2.3 of Friedman-Krueger [8]); 13 and if CH fails then this T must be co-stationary in [ω 2 ] ω , by a result of Baumgartner-Taylor (see Theorem 2.7 of [8]).…”

Section: Some Remarks About Strong Chang's Conjecture Specialmentioning

confidence: 99%

See 1 more Smart Citation

Chang’s conjecture and semiproperness of nonreasonable posets

Cox

2018

Monatsh Math

View full text Add to dashboard Cite

Let Q denote the poset which adds a Cohen real then shoots a club through the complement of [ω 2 ] ω V with countable conditions. We prove that the version of Strong Chang's Conjecture from [19] implies semiproperness of Q, and that semiproperness of Q-in fact semiproperness of any poset which is sufficiently nonreasonable in the sense of Foreman-Magidor [5]-implies the version of Strong Chang's Conjecture from [24] and [18]. In particular, semiproperness of Q has large cardinal strength, which answers a question of Friedman-Krueger [8]. One corollary of our work is that the version of Strong Chang's Conjecture from [19] does not imply the existence of a precipitous ideal on ω 1 .

show abstract

Section: Some Remarks About Strong Chang's Conjecture Specialmentioning

confidence: 99%

“…12 A set T ⊆ [ω 2 ] ω is called local club iff T ∩ [β] ω contains a club for every β < ω 2 . 13 T is thin if for every β < ω 2 : |{a ∩ β | a ∈ T }| ≤ ω 1 .…”

Section: 2mentioning

confidence: 99%

Chang’s conjecture and semiproperness of nonreasonable posets

Cox

2018

Monatsh Math

View full text Add to dashboard Cite

show abstract

“…The stationary set defined in Lemma 3.10 below then forces that the image of g will take the value γ at δ. This contrasts with the situation when canonical function bounding (see [10], for instance) holds; then, no function in ω …”

Section: Proposition 39 In the Situation Of Theorem 32 Assuming ♦mentioning

confidence: 89%

Absoluteness for Universally Baire Sets and the Uncountable Ii

Farah¹,

Ketchersid²,

Larson³

et al. 2008

Computational Prospects of Infinity

Self Cite

View full text Add to dashboard Cite

Abstract. Using ♦ and large cardinals we extend results of Magidor-Malitz and Farah-Larson to obtain models correct for the existence of uncountable homogeneous sets for finite-dimensional partitions and universally Baire sets. Furthermore, we show that the constructions in this paper and its predecessor can be modified to produce a family of 2 ω 1 -many such models so that no two have a stationary, costationary subset of ω 1 in common. Finally, we extend a result of Steel to show that trees on reals of height ω 1 which are coded by universally Baire sets have either an uncountable path or an absolute impediment preventing one.In [3] it was shown (using large cardinals) that if a model of a theory T satisfying a certain second-order property P can be forced to exist, then a model of T satisfying P exists already. The properties P considered in [3] included the following.(1) Containing any specified set of ℵ 1 -many reals.(2) Correctness about NS ω 1 . (3) Correctness about any given universally Baire set of reals (with a predicate for this set added to the language). In this paper we add the following properties, all proved under the assumption of Jensen's ♦ principle. (1) and (7), we can obtain all of these properties simultaneously using the method (a) (with "some" being "all" for (5) and (6)). Aside from (1) and (4) we can prove all of these properties simultaneously using the method (b). Property (4)

show abstract

“…Corollary 5.2]. that an inaccessible limit of measurable cardinals suffices to force a model of club bounding: even though [21] is probably the first printed reference where this result appears, it is probably folklore and was known long before [21]. where a strengthening (its consistency with CH) is obtained.…”

Section: 21]mentioning

confidence: 99%