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.
"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