[Paper Review] Definable discrete sets with large continuum

David Schrittesser|arXiv (Cornell University)|Oct 11, 2016

Advanced Topology and Set Theory9 references3 citations

TL;DR

This paper establishes that in the iterated Sacks forcing extension of Gödel's constructible universe L, every Σ¹₁ binary relation admits a Δ¹₂ maximal discrete set, even when the continuum is large (e.g., 2^ℵ₀ = ℵ₂). This result is applied to construct a Π¹₁ maximal orthogonal family of Borel probability measures, demonstrating the existence of definable maximal discrete sets under forcing extensions with large continuum, using a new Ramsey-theoretic principle for iterated Sacks forcing.

ABSTRACT

Let $\mathcal R$ be a $Σ^1_1$ binary relation and call a set $\mathcal R$-discrete iff no two distinct of its elements are $\mathcal R$-related. We show that in the extension of $\mathbf{L}$ by iterated Sacks forcing, there is a $Δ^1_2$ maximal $\mathcal R$-discrete set, and thus the existence of such sets is compatible with the negation of the continuum hypothesis. As an application we find a $Π^1_1$ maximal orthogonal family of Borel probability measures in said extension. The basis of this is a new Ramsey theoretic result.

Motivation & Objective

• To investigate the existence of definable maximal discrete sets for Σ¹₁ binary relations in forcing extensions with large continuum.
• To extend Galvin's theorem on homogeneous sets in Sacks forcing to iterated Sacks forcing.
• To construct a Π¹₁ maximal orthogonal family of Borel probability measures in the iterated Sacks extension of L.
• To demonstrate that the existence of such definable maximal discrete sets is consistent with the negation of the continuum hypothesis.
• To explore the limits of definability in forcing extensions by analyzing the role of projective definability and universality in Ramsey-theoretic principles.

Proposed method

• Using iterated Sacks forcing of length ω₂ over L, the paper constructs a Δ¹₂ maximal R-discrete set for any Σ¹₁ relation R on an effectively presented Polish space.
• The key technical tool is a new Ramsey-theoretic result: for any C-universally Baire coloring on the branch space of an iterated Sacks condition, there exists a stronger condition whose branch set supports a homogeneous coloring on a partitioned domain.
• The proof leverages topologically determined conditions in the iteration, defining a branch space [p̄] ⊆ λ(ω2) for limit λ, and analyzing colorings based on the least coordinate where two branches differ.
• A forcing argument shows that a generic condition q̄ forces the existence of a maximal discrete set via a fusion process that preserves Δ¹₂ definability.
• The construction uses a Galvin witness at each stage to ensure homogeneity on initial segments of the tree, ensuring the final set is R-discrete and maximal.
• The application to orthogonal families of measures relies on the fact that P*(X), the space of atomless Borel probability measures on a perfect Polish space X, is itself an effective Polish space with a Borel orthogonality relation.

Experimental results

Research questions

• RQ1 Does Galvin’s theorem on homogeneous sets for Baire measurable colorings extend from Sacks forcing to iterated Sacks forcing?
• RQ2 Can Σ¹₁ relations in forcing extensions with large continuum still admit projective maximal discrete sets?
• RQ3 Is there a Π¹₁ maximal orthogonal family of Borel probability measures in the iterated Sacks extension of L?
• RQ4 What is the consistency strength of the statement that every projective binary relation on a Polish space has a projective transversal?
• RQ5 Can maximal orthogonal families of measures be preserved across forcing extensions, or are they always destroyed in outer models?

Key findings

• In the iterated Sacks extension of L of length ω₂, every Σ¹₁ binary relation admits a Δ¹₂ maximal R-discrete set, even when the continuum is larger than ℵ₁.
• The existence of such definable maximal discrete sets is consistent with the negation of the continuum hypothesis.
• A Π¹₁ maximal orthogonal family of Borel probability measures exists in the same forcing extension, as a consequence of the main theorem and a result from prior work.
• The paper establishes a new Ramsey-theoretic principle for iterated Sacks forcing: for any C-universally Baire coloring on the branch space of a condition, there is a stronger condition whose branch set supports a homogeneous coloring on a partitioned domain.
• The construction shows that the maximal discrete set obtained is Δ¹₂-definable without parameters in the forcing extension.
• The result is optimal in the sense that the obstruction from colorings based on the first differing coordinate (cξ) cannot be overcome, and thus the partitioning into Δξ is necessary.