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Forcing With Copies of Countable Ordinals (1304.7714v1)

Published 29 Apr 2013 in math.LO

Abstract: 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.

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