Forcing With Copies of Countable Ordinals

TL;DR

This paper investigates the structure of forcing notions derived from countable ordinals, revealing their decomposition into products and iterations of Boolean algebras, and analyzing their forcing properties and related invariants.

Contribution

It provides a detailed structural analysis of forcing notions associated with countable ordinals, including their decomposition into products and iterations, and examines their forcing and combinatorial properties.

Findings

The separative quotient of P(α) is isomorphic to a product of iterated reduced products of Boolean algebras.

P(α) is forcing equivalent to a two-step iteration involving P(ω)/Fin.

Analysis of quotients over ordinal ideals and their distributivity and tower numbers.

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.