The happy coexistence of mad families and Laver measurability

2025/10/24 by Tornquist, Asger, Schrittesser, David
#03E15 #05D10 #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2510.21374

Abstract

Let x denote a Laver real over L. We prove that in L[x] there is a Π¹₁ infinite mad family. Since Π¹₁ and Σ¹₂ sets are Laver measurable in L[x], this shows that there are examples of well-behaved classical pointclasses Γ, namely Γ=Π¹₁ and Γ=Σ¹₂, 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.

Citations

• Ramsey regularity implies no MAD families without uniformization
• Combinatorial properties of MAD families
• The Ramsey property and higher dimensional mad families
• Compact Sets of Baire Class One Functions and Maximal Almost Disjoint Families
• Definability and almost disjoint families
• Solovay-type characterizations for forcing-algebras

Related

• Ramsey regularity implies no MAD families without uniformization
• MAD Families and SANE Player
• The consistency strength of projective uniformization, revisited
• P-points, MAD families and Cardinal Invariants
• The Ramsey property and higher dimensional mad families