Forcing With Copies of Countable Ordinals
arXiv:1304.7714
Abstract
Let αbe a countable ordinal and ¶(α) the collection of its subsets isomorphic to α. We show that the separative quotient of the set ¶(α) ordered by the inclusion is isomorphic to a forcing product of iterated reduced products of Boolean algebras of the form P(ω^γ)/I(ω^γ), where γis a limit ordinal or 1 and I(ω^γ) the corresponding ordinal ideal. Moreover, the poset ¶(α) is forcing equivalent to a two-step iteration P(ω)/Fin * π, where πis an ω_1-closed separative pre-order in each extension by P(ω)/Fin and, if the distributivity number is equal toω_1, to P(ω)/Fin. Also we analyze the quotients over ordinal ideals P(ω^δ)/I(ω^δ) and their distributivity and tower numbers.
arXiv comment15 pages