Abstract
Successors of singular cardinals are a peculiar—although they are successor cardinals, they can still exhibit some of the behaviors typically associated with large cardinals. In this chapter, we examine the combinatorics of successors of singular cardinals in detail. We use stationary reflection as our point of entry into the subject, and we sketch Magidor’s proof that it is consistent that all stationary subsets of such a cardinal reflect. Further consideration of Magidor’s proof brings us to Shelah’s ideal I[λ] and the related Approachability Property (AP); we give a fairly comprehensive treatment of these topics. Building on this, we then turn to squares, scales, and the influence these objects exert on questions of pertaining to reflection phenomena. The chapter concludes with a brief look at square-brackets partition relations and their relation to club-guessing principles.
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The author acknowledges support from NSF grant DMS-0506063 during the preparation of this chapter.
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Eisworth, T. (2010). Successors of Singular Cardinals. In: Foreman, M., Kanamori, A. (eds) Handbook of Set Theory. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-5764-9_16
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