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Keywords:
partially ordered set; pcf theory
partially ordered set; pcf theory
Summary:
Shelah's pcf theory describes a certain structure which must exist if $\aleph _{\omega }$ is strong limit and $2^{\aleph _\omega }>\aleph _{\omega _1}$ holds. Jech and Shelah proved the surprising result that this structure exists in ZFC. They first give a forcing extension in which the structure exists then argue that by some absoluteness results it must exist anyway. We reformulate the statement to the existence of a certain partially ordered set, and then we show by a straightforward, elementary (i.e., non-metamathematical) argument that such partially ordered sets exist.
References:
[1] Jech, T., Shelah, S.: Possible pcf algebras. J. Symb. Log. 61 (1996), 313-317. DOI 10.2307/2275613 | MR 1380692 | Zbl 0878.03036
[2] Shelah, S., Laflamme, C., Hart, B.: Models with second order properties V: A general principle. Ann. Pure Appl. Logic 64 (1993), 169-194. DOI 10.1016/0168-0072(93)90033-A | MR 1241253 | Zbl 0788.03046
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