Functional Projection and Complexity Reduction

Abstract

Törnquist (arXiv:1310.3250) proved that if there exists a $\boldsymbolΣ^1_2$ maximal almost disjoint (MAD) family, then there exists a $\boldsymbolΠ^1_1$ MAD family. Since then, several other "complexity reduction" proofs have been found for other combinatorial objects, such as maximal independent families or ultrafilters. All of these proofs have relied on $\boldsymbolΠ^1_1$-uniformization, so they did not easily lift to higher projective levels. In this paper, we introduce a principle weaker than Uniformization which we call "Functional Projection", and show that this principle is sufficient for most complexity reduction proofs. On the other hand, we prove that in $L[U]$, the canonical inner model for a measurable cardinal, Functional Projection holds for $\boldsymbolΠ^1_2$ relations. As an immediate corollary, we obtain that in $L[U]$ there are $\boldsymbolΠ^1_2$ witnesses for many combinatorial families.