Adding a Nonreflecting Weakly Compact Set

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 W⊆κ 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.