Forcing With Copies of Countable Ordinals

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.