On some variants of the club principle
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On some variants of the club principle Ashutosh Kumar1 · Saharon Shelah1,2
Received: 25 February 2018 / Revised: 7 June 2020 / Accepted: 10 July 2020 © Springer Nature Switzerland AG 2020
Abstract We study some asymptotic variants of the club principle. Along the way, we construct some forcings and use them to separate several of these principles. Keywords Club principle · Forcing Mathematics Subject Classification 03E05 · 03E35
1 Introduction For a regular uncountable cardinal κ and a stationary S ⊆ Lim(κ), the club principle ♣ S says the following: There exists A = Aδ : δ ∈ S where each Aδ is an unbounded subset of δ of order type cf (δ) such that for every A ∈ [κ]κ, there exists some (equivalently, stationary many) δ ∈ S such that Aδ ⊆ A. We say that A is a ♣ S witnessing sequence. If κ = ω1 and S = Lim(ω1 ) is the set of all countable limit ordinals, we drop the S and write ♣. The principle ♣ was introduced by Andrzej Ostaszewski in [6] where he used ♣+CH (equivalently, ♦) to construct an Ostaszewski space. Several variants of this principle have since been studied [1,2]. For example, in [1], it was shown that ♣1 does not imply
AK supported by a Postdoctoral Fellowship at the Einstein Institute of Mathematics funded by European Research Council grant 338821. SS partially supported by European Research Council grant 338821; Publication no. 1136.
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Ashutosh Kumar [email protected] Saharon Shelah [email protected]
1
Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Edmond J Safra Campus, Givat Ram, Jerusalem 91904, Israel
2
Department of Mathematics, Rutgers, The State University of New Jersey, Hill Center-Busch Campus, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA
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A. Kumar, S. Shelah
♣ where ♣1 is the following statement: There exists A = Aδ : δ ∈ Lim(ω1 ) where each Aδ is an unbounded subset of δ of order type ω such that for every A ∈ [ω1 ]ℵ1, there exists δ such that Aδ \ A is finite. In this work, we mostly study asymptotic versions of the club principle where the requirement Aδ ⊆ A is replaced by Aδ ∩ A is a “large” subset of Aδ . Some of these principles have previously appeared in [3,4]. As a motivating example, suppose we start with a model of ♣ and add ℵ2 Cohen reals. Then it is easy to see that ♣1 and therefore ♣ are destroyed. But the following continues to hold (see Remark 7.2): There exists Aδ : δ ∈ Lim(ω1 ) where each Aδ = {αδ,n : n < ω} where αδ,n ’s are increasing cofinal in δ and for every A ∈ [ω1 ]ℵ1, here exists δ such that {n < ω : αδ,n ∈ A} has upper asymptotic density 1. It follows that ¬ ♣1 ∧ ♣sup 1 is consistent. Definition 1.1 xxx For a ∈ (0, 1] and a stationary set S ⊆ Lim(ω1 ), the principle a says the following: There exists A = Aδ : δ ∈ S such that ♣inf S (a) each Aδ = {αδ,n : n < ω} and αδ,n ’s are increasing cofinal in δ, and (b) for every A ∈ [ω1 ]ℵ1, there exists δ ∈ S such that lim inf n
|{k < n : αδ,k ∈ A}| n
a.
If S = Lim(ω1 ), we write ♣inf a. By ♣lim , we mean ♣inf 1. It is clear that ♣1 implies ♣l
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