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Abstract

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.