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2019 Layered Posets and Kunen’s Universal Collapse
Sean Cox
Notre Dame J. Formal Logic 60(1): 27-60 (2019). DOI: 10.1215/00294527-2018-0022

Abstract

We develop the theory of layered posets and use the notion of layering to prove a new iteration theorem (Theorem 6: if κ is weakly compact, then any universal Kunen iteration of κ-cc posets (each possibly of size κ) is κ-cc, as long as direct limits are used sufficiently often. This iteration theorem simplifies and generalizes the various chain condition arguments for universal Kunen iterations in the literature on saturated ideals, especially in situations where finite support iterations are not possible. We also provide two applications:

1 For any n1, a wide variety of <ωn1-closed, ωn+1-cc posets of size ωn+1 can consistently be absorbed (as regular suborders) by quotients of saturated ideals on ωn (see Theorem 7 and Corollary 8).

2 For any nω, the tree property at ωn+3 is consistent with Chang’s conjecture (ωn+3,ωn+1)(ωn+1,ωn) (Theorem 9).

Citation

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Sean Cox. "Layered Posets and Kunen’s Universal Collapse." Notre Dame J. Formal Logic 60 (1) 27 - 60, 2019. https://doi.org/10.1215/00294527-2018-0022

Information

Received: 30 May 2015; Accepted: 28 November 2016; Published: 2019
First available in Project Euclid: 25 January 2019

zbMATH: 07060307
MathSciNet: MR3911105
Digital Object Identifier: 10.1215/00294527-2018-0022

Subjects:
Primary: 03E05
Secondary: 03E35, 03E55

Rights: Copyright © 2019 University of Notre Dame

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Vol.60 • No. 1 • 2019
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