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August 2019 Adding a Nonreflecting Weakly Compact Set
Brent Cody
Notre Dame J. Formal Logic 60(3): 503-521 (August 2019). DOI: 10.1215/00294527-2019-0014

Abstract

For n<ω, we say that theΠn1-reflection principle holds at κ and write Refln(κ) if and only if κ is a Πn1-indescribable cardinal and every Πn1-indescribable subset of κ has a Πn1-indescribable proper initial segment. The Πn1-reflection principle Refln(κ) generalizes a certain stationary reflection principle and implies that κ is Πn1-indescribable of order ω. We define a forcing which shows that the converse of this implication can be false in the case n=1; that is, we show that κ being Π11-indescribable of order ω need not imply Refl1(κ). Moreover, we prove that if κ is (α+1)-weakly compact where α<κ+, then there is a forcing extension in which there is a weakly compact set Wκ having no weakly compact proper initial segment, the class of weakly compact cardinals is preserved and κ remains (α+1)-weakly compact. We also formulate several open problems and highlight places in which standard arguments seem to break down.

Citation

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Brent Cody. "Adding a Nonreflecting Weakly Compact Set." Notre Dame J. Formal Logic 60 (3) 503 - 521, August 2019. https://doi.org/10.1215/00294527-2019-0014

Information

Received: 10 January 2017; Accepted: 7 November 2017; Published: August 2019
First available in Project Euclid: 11 June 2019

zbMATH: 07120753
MathSciNet: MR3985624
Digital Object Identifier: 10.1215/00294527-2019-0014

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

Rights: Copyright © 2019 University of Notre Dame

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Vol.60 • No. 3 • August 2019
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