The happy coexistence of mad families and Laver measurability

TL;DR

This paper demonstrates the existence of a $ ext{Pi}^1_1$ mad family in a model where $ ext{Pi}^1_1$ and $ ext{Sigma}^1_2$ sets are Laver measurable, challenging previous assumptions about their properties.

Contribution

It constructs a model with a $ ext{Pi}^1_1$ mad family where these sets are Laver measurable, showing a surprising coexistence contrary to earlier expectations.

Findings

Existence of a $ ext{Pi}^1_1$ mad family in $L[x]$

$ ext{Pi}^1_1$ and $ ext{Sigma}^1_2$ sets are Laver measurable in this model

Contradicts the idea that uniformization and Ramsey properties prevent mad families in these classes.

Abstract

Let $x$ denote a Laver real over $L$. We prove that in $L[x]$ there is a $\Pi^1_1$ infinite mad family. Since $\Pi^1_1$ and $\Sigma^1_2$ sets are Laver measurable in $L[x]$, this shows that there are examples of well-behaved classical pointclasses $\Gamma$, namely $\Gamma=\Pi^1_1$ and $\Gamma=\Sigma^1_2$, where $\Gamma$-uniformization and ``all sets in $\Gamma$ are Laver measurable'' hold, but there is a mad family in $\Gamma$. This result stands in contrast to that for reasonable pointclasses, the $\Gamma$-Ramsey property together with uniformization implies that there are no mad families in $\Gamma$.