2025/01/06 by Hanson, James E. · 1 citation
Mathematics · #Advanced Topology and Set Theory #Advanced Banach Space Theory #Mathematical and Theoretical Analysis
paper · pdf · doi:10.48550/arxiv.2501.02693
In this expository paper aimed at a general mathematical audience, we discuss how to combine certain classic theorems of set-theoretic inner model theory and effective descriptive set theory with work on Hilbert's tenth problem and universal Diophantine equations to produce the following surprising result: There is a specific polynomial p(x,y,z,n,k1,…,k70) of degree 7 with integer coefficients such that it is independent of ZFC (and much stronger theories) whether the function f(x) = infy ∈ ℝsupz ∈ ℝinfn ∈ ℕsup_k ∈ ℕ70p(x,y,z,n,k) is Lebesgue measurable. We also give similarly defined g(x,y) with the property that the statement "x ↦ g(x,r) is measurable for every r ∈ ℝ" has large cardinal consistency strength (and in particular implies the consistency of ZFC) and h(m,x,y,z) such that h(1,x,y,z),…,h(16,x,y,z) can consistently be the indicator functions of a Banach\unicodex2013Tarski paradoxical decomposition of the sphere. Finally, we discuss some situations in which measurability of analogously defined functions can be concluded by inspection, which touches on model-theoretic o-minimality and the fact that sufficiently strong large cardinal hypotheses (such as Vopěnka's principle and much weaker assumptions) imply that all 'reasonably definable' functions (including the above f(x), g(x,y), and h(m,x,y,z)) are universally measurable.