Notre Dame Journal of Formal Logic
- Notre Dame J. Formal Logic
- Volume 60, Number 3 (2019), 503-521.
Adding a Nonreflecting Weakly Compact Set
For , we say that the-reflection principle holds at and write if and only if is a -indescribable cardinal and every -indescribable subset of has a -indescribable proper initial segment. The -reflection principle generalizes a certain stationary reflection principle and implies that is -indescribable of order . We define a forcing which shows that the converse of this implication can be false in the case ; that is, we show that being -indescribable of order need not imply . Moreover, we prove that if is -weakly compact where , then there is a forcing extension in which there is a weakly compact set having no weakly compact proper initial segment, the class of weakly compact cardinals is preserved and remains -weakly compact. We also formulate several open problems and highlight places in which standard arguments seem to break down.
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
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
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. https://projecteuclid.org/euclid.ndjfl/1560218426