Iterability for (transfinite) stacks

Schlutzenberg Farmer

Zusammenfassung

We establish natural criteria under which normally iterable premice are iterable for stacks of normal trees. Let Ω be a regular uncountable cardinal. Let m < ω and M be an m-sound premouse and Σ be an (m,Ω+1)-iteration strategy for M (roughly, a normal (Ω+1)-strategy). We define a natural condensation property for iteration strategies, inflation condensation. We show that if Σ has inflation condensation then M is (m,Ω,Ω+1)^∗-iterable (roughly, M is iterable for length ≤ Ω stacks of normal trees each of length < Ω), and moreover, we define a specific such strategy Σ^st and a reduction of stacks via Σ^st to normal trees via Σ. If Σ has the Dodd-Jensen property and card(M) < Ω then Σ has inflation condensation. We also apply some of the techniques developed to prove that if Σ has strong hull condensation (a slight strengthening of inflation condensation) and G is V-generic for an Ω-cc forcing, then Σ extends to an (m,Ω+1)-strategy Σ^+ for M with strong hull condensation, in the sense of V[G]. Moreover, this extension is unique. We deduce that if G is V-generic for a ccc forcing then V and V[G] have the same ω-sound, (ω,Ω+1)-iterable premice which project to ω.