Notre Dame Journal of Formal Logic 60 (3):503-521 (2019)

For n<ω, we say that theΠn1-reflection principle holds at κ and write Refln 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 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. Moreover, we prove that if κ is -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 -weakly compact. We also formulate several open problems and highlight places in which standard arguments seem to break down.
Keywords Easton-support iteration   indescribability   reflection  weak compactness
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DOI 10.1215/00294527-2019-0014
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References found in this work BETA

Saturated Ideals.Kenneth Kunen - 1978 - Journal of Symbolic Logic 43 (1):65-76.
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Indescribable Cardinals and Elementary Embeddings.Kai Hauser - 1991 - Journal of Symbolic Logic 56 (2):439-457.

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Characterizations of the Weakly Compact Ideal on Pλ.Brent Cody - 2020 - Annals of Pure and Applied Logic 171 (6):102791.

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