More definable combinatorics around the first and second uncountable cardinals

Abstract

Assume ZF+AD. The following two continuity results for functions on certain subsets of P(ω1) and P(ω2) will be shown: For every < ω1 and function Φ : [ω1] → ω1, there is a club C ⊆ ω1 and a ζ < so that for all f, g ∈ [C] ∗, if f ζ = g ζ and sup(f) = sup(g), then Φ(f) = Φ(g). For every < ω2 and function Φ : [ω2] → ω2, there is an ω-club C ⊆ ω2 and a ζ < so that for all f, g ∈ [C] ∗, if f ζ = g ζ and sup(f) = sup(g), then Φ(f) = Φ(g). The previous two continuity results will be used to distinguish cardinals below P(ω2): |[ω1] | < |[ω1]1 |. |[ω2] | < |ω2]1 | < |[ω2]1 | < |[ω2]2 |. ¬(|[ω1]1 | ≤ [ω2] |). ¬(|[ω1]1 | ≤ ([ω2]1 |). [ω1] has the Jónsson property: That is, for every Φ : ([ω1]) → [ω1] , there is an X ⊆ [ω1] with |X| = |[ω1] | so that Φ[