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A note on forcing with ideals and Hechler forcing. (English) Zbl 1052.03032

Let \(\kappa\) be an infinite cardinal, \({\mathcal B}\) a Boolean algebra. \({\mathcal B}\) is \(\kappa\)-representable if there is a \(\kappa\)-complete ideal \(I\) on \(\kappa\) such that \({\mathcal B}\cong{\mathcal P}(\kappa)/{\mathcal I}\).
Gitik and Shelah have shown that the Hechler algebra \({\mathcal H}\) (the Boolen algebra of regular open sets of Hechler forcing) is not representable. Here the author gives a simpler proof of this fact using some observation concerning Hechler forcing.

MSC:

03E40 Other aspects of forcing and Boolean-valued models
06E10 Chain conditions, complete algebras