Completely determined Borel sets and measurability

2020/01/07 by Linda Brown Westrick, Westrick, Linda
Computer Science · Mathematics · #03B30 #03D32 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Rough Sets and Fuzzy Logic

paper · pdf · doi:10.48550/arxiv.2001.01881

openalex publication_date 2020/01/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the reverse math strength of the statement C\text-DM:"Every completely determined Borel set is measurable." Over WWKL₀, we obtain the following results analogous to the previously studied category case. C\text-DM lies strictly between ATR₀ and L_ω1,ω\text-CA. Whenever M⊆ 2^ω is the second-order part of an ω-model of C\text-DM, then for every Z ∈ M, there is a R ∈ M such that R is Δ¹₁-random relative to Z. On the other hand, without WWKL₀, all sets have measure zero (as measured according to C\text-DM), and it follows vacuously that ¬ WWKL₀ implies C\text-DM over RCA₀.

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