[Paper Review] Madness and regularity properties

Haim Horowitz, Saharon Shelah|arXiv (Cornell University)|Apr 26, 2017

Advanced Topology and Set Theory6 references3 citations

TL;DR

This paper constructs a model of ZF+DC where a maximal almost disjoint (mad) family exists, yet all sets of reals are measurable with respect to $ω^\omega$-bounding, sufficiently absolute forcing notions—specifically, Lebesgue measurable. Using a novel amalgamation argument over $ω^\omega$-bounding generics, the authors show that Lebesgue measurability does not imply the non-existence of mad families, resolving a key independence question in set theory.

ABSTRACT

Starting from an inaccessible cardinal, we construct a model of $ZF+DC$ where there exists a mad family and all sets of reals are $\mathbb Q$-measurable for $ω^ω$-bounding sufficiently absolute forcing notions $\mathbb Q$.

Motivation & Objective

• To investigate the independence between the existence of mad families and Lebesgue measurability in models of ZF+DC.
• To demonstrate that the existence of a mad family is consistent with all sets of reals being Lebesgue measurable.
• To show that $ω^\omega$-bounding, sufficiently absolute forcing notions do not imply the non-existence of mad families.
• To construct a model from an inaccessible cardinal where all sets of reals are $ω^\omega$-bounding forcing-measurable and a mad family exists.
• To resolve the independence of the Weak Continuum Hypothesis (WCH) from ZF+DC + 'all sets of reals are Lebesgue measurable'.

Proposed method

• Introduces a partial order $AP$ consisting of pairs $(\mathbb{P}, \Gamma)$, where $\mathbb{P}$ is a forcing notion from $H(\kappa)$ and $\Gamma$ is an approximation of a mad family.
• Defines $AP$ with conditions ensuring that finite unions of names in $\Gamma$ are not dominated by reals from the ground model, preserving almost disjointness.
• Employs an amalgamation argument over $\mathbb{Q}$-generic reals for an $\omega^\omega$-bounding, sufficiently absolute forcing notion $\mathbb{Q}$, adapting Solovay’s method.
• Uses a forcing iteration combining $AP$ with a generic extension to construct a model where all sets of reals are $\mathbb{Q}$-measurable.
• Applies a symmetry argument via automorphisms induced by complete embeddings to derive a contradiction from the existence of an injection from $\mathbb{R}$ to a mad family.
• Relies on the existence of an inaccessible cardinal to ensure the consistency of the construction.

Experimental results

Research questions

• RQ1 Does Lebesgue measurability for all sets of reals imply the non-existence of mad families in models of ZF+DC?
• RQ2 Can a mad family exist in a model where all sets of reals are measurable under $\omega^\omega$-bounding forcing notions?
• RQ3 Is the Weak Continuum Hypothesis (WCH) independent of ZF+DC + 'all sets of reals are Lebesgue measurable'?
• RQ4 Can the existence of a mad family be consistently preserved in a model where all sets of reals are $\mathbb{Q}$-measurable for $\mathbb{Q}$ being $\omega^\omega$-bounding and sufficiently absolute?
• RQ5 Is the existence of a mad family compatible with full Lebesgue measurability under minimal large cardinal assumptions?

Key findings

• The paper constructs a model of ZF+DC where a mad family exists and all sets of reals are $\mathbb{Q}$-measurable for $\omega^\omega$-bounding, sufficiently absolute forcing notions $\mathbb{Q}$.
• It proves that Lebesgue measurability does not imply the non-existence of mad families, as Random real forcing is $\omega^\omega$-bounding and thus compatible with mad families.
• The model satisfies $\neg WCH$, showing that the Weak Continuum Hypothesis is independent of ZF+DC + 'all sets of reals are Lebesgue measurable'.
• The construction uses a novel amalgamation argument over $\mathbb{Q}$-generic reals to preserve the almost disjointness of the family through forcing.
• The proof shows that assuming an injection from $\mathbb{R}$ to a mad family leads to a contradiction via automorphisms induced by complete embeddings, confirming the family's maximality is preserved.
• The result is consistent relative to an inaccessible cardinal, improving on prior constructions that required a proper class of Woodin cardinals.