Applications of infinity-Borel codes to definability and definable cardinals

Assume ZF + AD + V = L(P(R)). If H ⊆ R has the property that there is a nonempty OD set of reals K so that H is ODz for any z ∈ K, then H is OD. Assume ZF + AD + ¬ADR + V = L(P(R)). Then there is a cardinal strictly between |[ω1]1 | and |[ω1]1 | = |P(ω1)|. Assume ZF + AD. S1 = {f ∈ [ω1]1 : sup(f) = ω ] 1 } does not inject into ωON, the class of ω-sequences of ordinals. This implies |R| < |S1| and |[ω1] | < |[ω1]1 |. Assuming ZF + AD. Let X be a surjective image of R and let Pω1 (X) = {A ⊆ X : |A| < ω1}. If ω1 ≤ |Pω1 (X)|, then ω1 ≤ |X|. If |P(ω1)| = |[ω1]1 | ≤ |Pω1 (X)|, then |R t ω1| ≤ |X|. ZF + ADR implies that the uncountable cardinals below |R × ω1| are ω1, |R|, |R t ω1|, and |R × ω1|. An elaborate structure of cardinals below |R× ω1| will be described under the assumption of ZF + AD + ¬ADR + V = L(P(R)).