Reflection implies the SCH

Volume 198 / 2008

Saharon Shelah Fundamenta Mathematicae 198 (2008), 95-111 MSC: 03E04, 03E05. DOI: 10.4064/fm198-2-1


We prove that, e.g., if $\mu > \mathop{\rm cf}\nolimits(\mu) = \aleph_0$ and $\mu > 2^{\aleph_0}$ and every stationary family of countable subsets of $\mu^+$ reflects in some subset of $\mu^+$ of cardinality $\aleph_1$, then the SCH for $\mu^+$ holds (moreover, for $\mu^+$, any scale for $\mu^+$ has a bad stationary set of cofinality $\aleph_1$). This answers a question of Foreman and Todorčević who get such a conclusion from the simultaneous reflection of four stationary sets.


  • Saharon ShelahInstitute of Mathematics
    The Hebrew University
    Jerusalem, Israel
    Mathematics Department
    Rutgers University
    New Brunswick, NJ 08854, U.S.A.

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