The happy coexistence of mad families and Laver measurability

TL;DR

The paper demonstrates that a Laver real $x$ over $L$ yields a model $L[x]$ in which a $Π^1_1$ mad family exists while all $Σ^1_2$ sets are Laver measurable, thereby showing that $Γ$-Laver measurability and $Γ$-uniformization do not preclude mad families in certain definable pointclasses. The core method is a Main Lemma: given a continuous $f$ on a Laver condition, one can refine to a $q$ with a continuous $ ilde f$ whose range is almost disjoint and contained in the original image, enabling a $\,Σ^1_2$-definable MAD family construction. The argument relies on a fusion-like analysis of Laver forcing, Miller-type dichotomies for analytic sets, and a carefully controlled tree-based construction to force almost-disjointness. This work highlights a nuanced contrast between Ramsey-type forcing properties and the coexistence of definable MAD families with rapid-growing forcing reals.

Abstract

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

Theorems & Definitions (21)

• Theorem 1.1
• Definition 1.2
• Definition 1.3
• Theorem 1.4
• Proposition 2.1
• proof : Sketch of proof
• Theorem 2.2: Miller, miller2012
• proof : Proof of Fact \ref{['f.miller']}
• Lemma 2.4: The Main lemma
• proof : Proof of Theorem \ref{['t.lavermad']}, given the Main Lemma