10 - The countable support iteration ideals

Summary

The countable support iteration is a central operation on forcings since the pioneering work of Shelah (1998). It corresponds to a certain operation on σ-ideals, the transfinite Fubini power, as shown in Zapletal (2008, section 5.1). The natural question then arises whether this operation preserves natural canonization properties of the σ-ideals in question.

The question may be natural, but it also seems to be very hard, even for iterations of length two in the general case. We will prove a rather restrictive canonization result for iterations of a certain broad class of ideals, the best possible canonization result for the special case of iterations of Sacks forcing, and an anticanonization theorem for some other ideals. For a σ-ideal I on a Polish space X and a countable ordinal α ∈ ω₁ write I^α for the transfinite Fubini power of I. This is a σ-ideal on X^α defined in the next section. There are certain obvious obstacles for canonization on Iα. For ordinal β ∈ α let id^β be the equivalence relation on Xβ connecting sequences x, y ∈ Xα if x ↾ β = y ↾ β. For an equivalence relation F on X, write id_β × F × ev for the equivalence relation connecting x, y ∈ Xα if x ↾ β = y ↾ β, and x(β) F y(β). It turns out that these are the only obstacles in a certain context.