NS_ω1 saturated, Δ₁ ( \ ω₁ \ )-definable and a Δ¹₄-definable well-order of the reals

2025/07/01 by Stefan Hoffelner, Hoffelner, Stefan
Computer Science · Mathematics · #Advanced Banach Space Theory #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO)

paper · pdf · doi:10.48550/arxiv.2507.01183

openalex publication_date 2025/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Assuming M₁, the canonical inner model with one Woodin cardinal exists, we construct a model in which the nonstationary ideal on ω₁ is ℵ₂-saturated, Δ₁-definable with ω₁ as the only parameter and there is a Σ¹₄-definable well-order of the reals. This implies that contrary to the assumption that NS_ω1 is ℵ₁-dense, the assumption of NS_ω1 being saturated and Δ₁-definable does not imply any nice structural properties for the projective subsets of the reals.

Citations

• The complexity of non-stationary ideals
• Forcing Axioms and the Definabilty of the Nonstationary Ideal on ω₁
• Forcing the Π¹_n-Uniformization Property

Related

• Projective Wellorders and the Nonstationary Ideal
• Forcing Axioms and the Definabilty of the Nonstationary Ideal on ω₁
• A model with everything except for a well-ordering of the reals
• Σ₁(κ)-definable subsets of H(κ⁺)
• A Universe with a Δ¹_n-definable well-order of the reals, CH and Π¹_n-Uniformization