DESCRIPTIVE PROPERTIES OF I2-EMBEDDINGS

We contribute to the study of generalizations of the Perfect Set Property and the Baire Property to subsets of spaces of higher cardinalities, like the power set P(λ) of a singular cardinal λ of countable cofinality or products ∏i<ωλi for a strictly increasing sequence ⟨λi | i<ω⟩ of cardinals. We consider the question under which large cardinal hypotheses classes of definable subsets of these spaces possess such regularity properties, focusing on rank-into-rank axioms and classes of sets definable by Σ1-formulas with parameters from various collections of sets. We prove that ω-many measurable cardinals, while sufficient to prove the Perfect Set Property of all Σ1-definable sets with parameters in Vλ∪{Vλ}, are not enough to prove it if there is a cofinal sequence in λ in the parameters. For this conclusion, the existence of an I2-embedding is enough, but there are parameters in Vλ+1 for which I2 is still not enough. The situation is similar for the Baire Property: under I2 all sets that are Σ1-definable using elements of Vλ and a cofinal sequence as parameters have the Baire property, but I2 is not enough for some parameter in Vλ+1. Finally, the existence of an I0-embedding implies that all sets that are Σ1n-definable with parameters in Vλ+1 have the Baire property.