Notre Dame Journal of Formal Logic

Adding a Nonreflecting Weakly Compact Set

Brent Cody

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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.

Article information

Notre Dame J. Formal Logic, Volume 60, Number 3 (2019), 503-521.

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 03E35: Consistency and independence results
Secondary: 03E05: Other combinatorial set theory 03E55: Large cardinals

weak compactness reflection indescribability Easton-support iteration


Cody, Brent. Adding a Nonreflecting Weakly Compact Set. Notre Dame J. Formal Logic 60 (2019), no. 3, 503--521. doi:10.1215/00294527-2019-0014.

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