Forcing with copies of countable ordinals

Abstract

Let α be a countable ordinal and P(α) the collection of its subsets isomorphic to α. We show that the separative quotient of the poset P(α), ⊂ is isomorphic to a forcing product of iterated reduced products of Boolean algebras of the form P (ω γ )/I ω γ , where γ ∈ Lim ∪{1} and I ω γ is the corresponding ordinal ideal. Moreover, the poset P(α), ⊂ is forcing equivalent to a twostep iteration of the form (P (ω)/ Fin) + * π, where [ω] "π is an ω 1 -closed separative pre-order" and, if h = ω 1 , to (P (ω)/ Fin) + . Also we analyze the quotients over ordinal ideals P (ω δ )/I ω δ and the corresponding cardinal invariants h ω δ and t ω δ .

Classes of Barren Extensions

“…If α B is infinite, then for any 1 ≤ k < , the projection of G B to its first k coordinates reproduces G k . We point out that the forcing properties of a related hierarchy was studied by Kurilić in [24]; his hierarchy agrees with the one here for k finite, but differs for α ≥ .…”

Isomorphic and strongly connected components

“…So, for example, for the structures from column D of Diagram 1 of [6] the corresponding posets are forcing equivalent to an atomless ω 1closed poset and, consistently, to P (ω)/Fin. This class of structures includes all scattered linear orders [9] (in particular, all countable ordinals [8]), all structures with maximally embeddable components [7] (in particular, all countable equivalence relations and all disjoint unions of countable ordinals) and in this paper we show that it contains a large class of ultrahomogeneous structures.…”

Posets of copies of countable scattered linear orders

“…L sq P(L), ⊂ is sq P(L), ⊂ is ZFC ⊢ sq P(L), ⊂ isomorphic to is h-distributive ω (P (ω)/ Fin) + t-closed yes ω + ω (P (ω)/ Fin) + × (P (ω)/ Fin) + t-closed no ω · ω (P (ω × ω)/(Fin × Fin)) + ω1 but not ω2-closed no Remark 7.3 Concerning Theorem 1.2 we note that for countable ordinals we have more information. Namely, by [6], if α = ω γn+rn s n + . .…”