Descriptive properties of I2-embeddings

Abstract

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 $\prod_{i<ω}λ_i$ for a strictly increasing sequence $\langleλ_i ~ \vert ~ i<ω\rangle$ 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 $Σ_1$-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 $Σ_1$-definable sets with parameters in $V_λ\cup\{V_λ\}$, 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 $V_{λ+1}$ for which I2 is still not enough. The situation is similar for the Baire Property: under I2 all sets that are $Σ_1$-definable using elements of $V_λ$ and a cofinal sequence as parameters have the Baire property, but I2 is not enough for some parameter in $V_{λ+1}$. Finally, the existence of an I0-embedding implies that all sets that are $Σ^1_n$-definable with parameters in $V_{λ+1}$ have the Baire property.