We prove that, e.g., if μ > cf(μ) = ℵ₀ and and every stationary family of countable subsets of μ⁺ reflects in some subset of μ⁺ of cardinality ℵ₁, then the SCH for μ⁺ holds (moreover, for μ⁺, any scale for μ⁺ has a bad stationary set of cofinality ℵ₁). This answers a question of Foreman and Todorčević who get such a conclusion from the simultaneous reflection of four stationary sets.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm198-2-1,
author = {Saharon Shelah},
title = {Reflection implies the SCH},
journal = {Fundamenta Mathematicae},
volume = {201},
year = {2008},
pages = {95-111},
zbl = {1147.03027},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm198-2-1}
}
Saharon Shelah. Reflection implies the SCH. Fundamenta Mathematicae, Tome 201 (2008) pp. 95-111. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm198-2-1/