On (Σ²₁)^uB Absoluteness Between V and HOD

2025/05/02 by Gabriel Goldberg, Goldberg, Gabriel, Dan Hathaway +1
Computer Science · Mathematics · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Mathematical and Theoretical Analysis

paper · pdf · doi:10.48550/arxiv.2505.01393

openalex publication_date 2025/05/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29

Abstract

We put together Woodin's Σ²₁ basis theorem of AD⁺ and Vopěnka's theorem to conclude the following: If there is a proper class of Woodin cardinals, then every (Σ²₁)^uB statement that is true in V is true in HOD. Moreover, this is true even if we allow a parameter C ⊆ ℝ such that C and its complement have scales that are OD and universally Baire. We also investigate whether (Σ²₁)^uB statements are upwards absolute from HOD to V under large cardinal hypotheses, observing that this is true if HOD has a proper class of Woodin cardinals. Finally, we discuss (∀^ℝ) (Σ²₁)^uB absoluteness and conclude that this much absoluteness between HOD and V cannot be implied by any large cardinal axiom consistent with the axiom ``V = Ultimate L''.

Citations

• The HOD Dichotomy

Related

• The extender algebra and Σ²₁-absoluteness
• Large cardinals, structural reflection, and the HOD Conjecture
• Strongly compact cardinals and ordinal definability
• On the optimality of the HOD dichotomy
• A note on Woodin's HOD dichotomy