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 Forcing With Copies of Countable Ordinals


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Let \alpha be a countable ordinal and \P(\alpha) the collection of its subsets isomorphic to \alpha. We show that the separative quotient of the set \P (\alpha) ordered by the inclusion is isomorphic to a forcing product of iterated reduced products of Boolean algebras of the form P(\omega ^\gamma)-I(\omega ^\gamma), where \gamma is a limit ordinal or 1 and I(\omega ^\gamma) the corresponding ordinal ideal. Moreover, the poset \P(\alpha) is forcing equivalent to a two-step iteration P(\omega)-Fin * \pi, where \pi is an \omega 1-closed separative pre-order in each extension by P(\omega)-Fin and, if the distributivity number is equal to\omega 1, to P(\omega)-Fin. Also we analyze the quotients over ordinal ideals P(\omega ^\delta)-I(\omega ^\delta) and their distributivity and tower numbers.



Author: Milos Kurilic

Source: https://archive.org/







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