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The weakly compact reflection principle need not imply a high order of weak compactness

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Abstract

The weakly compact reflection principle\({\mathrm{Refl}}_{{\mathrm{wc}}}(\kappa )\) states that \(\kappa \) is a weakly compact cardinal and every weakly compact subset of \(\kappa \) has a weakly compact proper initial segment. The weakly compact reflection principle at \(\kappa \) implies that \(\kappa \) is an \(\omega \)-weakly compact cardinal. In this article we show that the weakly compact reflection principle does not imply that \(\kappa \) is \((\omega +1)\)-weakly compact. Moreover, we show that if the weakly compact reflection principle holds at \(\kappa \) then there is a forcing extension preserving this in which \(\kappa \) is the least \(\omega \)-weakly compact cardinal. Along the way we generalize the well-known result which states that if \(\kappa \) is a regular cardinal then in any forcing extension by \(\kappa \)-c.c. forcing the nonstationary ideal equals the ideal generated by the ground model nonstationary ideal; our generalization states that if \(\kappa \) is a weakly compact cardinal then after forcing with a ‘typical’ Easton-support iteration of length \(\kappa \) the weakly compact ideal equals the ideal generated by the ground model weakly compact ideal.

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Notes

  1. The second author proved that in L, a cardinal \(\kappa \) is \(\Pi ^1_{n+1}\)-indescribable if and only if the \(\Pi ^1_n\)-reflection principle holds. Independently, Bagaria-Magidor showed that in L, \(\kappa \) is a \(\Pi ^1_{n+1}\)-indescribable cardinal if and only if \(\kappa \) is what they call “n-stationary”; this is the version appearing in [2]. Thus, in L, a cardinal \(\kappa \) is \(\Pi ^1_{n+1}\)-indescribable if and only if \(\kappa \) is n-stationary if and only if the \(\Pi ^1_n\)-reflection principle holds.

  2. The first author would like to thank James Cummings for pointing this out.

  3. A set \(S\subseteq \kappa \) is called a non-reflecting\(\Pi ^1_n\)-indescribable subset of\(\kappa \) if S is a \(\Pi ^1_n\)-indescribable subset of \(\kappa \) and for every \(\gamma <\kappa \) the set \(S\cap \gamma \) is not a \(\Pi ^1_n\)-indescribable subset of \(\gamma \). See [4] for more on non-reflecting \(\Pi ^1_n\)-indescribable sets.

  4. It is not known whether or not there is a forcing which adds a non-reflecting \(\Pi ^1_1\)-indescribable subset to \(\kappa \) while preserving the great \(\Pi ^1_1\)-indescribability of \(\kappa \). See Sect. 5.

  5. This also follows from a result of Magidor [16] which states that \({\mathrm{Refl}}(\aleph _{\omega +1})\) is consistent relative to a supercompact cardinal. We emphasize Mekler and Shelah’s proof because our proof will be a generalization of theirs.

  6. It is not clear whether the methods in this article will provide an answer to this question for \(n>1\). For a more detailed discussion of this and other questions, see Sect. 5 below.

  7. Here \(\mathrm{ZFC}^-\) denotes the axioms of \(\mathrm{ZFC}\) without the powerset axiom and with the collection axiom instead of the replacement axiom [6].

  8. See Sect. 2.3 for our conventions on strategic closure terminology.

  9. Thanks to Sean Cox and Monroe Eskew for pointing this out.

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Cody, B., Sakai, H. The weakly compact reflection principle need not imply a high order of weak compactness. Arch. Math. Logic 59, 179–196 (2020). https://doi.org/10.1007/s00153-019-00686-7

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