The countable support iteration ideals

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 ∈ ω 1 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 I α. For ordinal β ∈ α let id β be the equivalence relation on X β connecting sequences x, y ∈ X α if x ↾ β = y ↾ β. For an equivalence relation F on X , write id β × F × ev for the equivalence relation connecting x, y ∈ X α if x ↾ β = y ↾ β, and x (β) F y (β). It turns out that these are the only obstacles in a certain context.