Any function I can actually write down is measurable, right?

Abstract

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 $p(x,y,z,n,k_1,\dots ,k_{70})$ of degree $7$ with integer coefficients such that it is independent of $\mathsf{ZFC}$ (and much stronger theories) whether the function $$f(x)=\underset{y\in \mathbb{R}}{\inf }\underset{z\in \mathbb{R}}{\sup }\underset{n\in \mathbb{N}}{\inf }\underset{\bar{k}\in {\mathbb{N}}^{70}}{\sup }p(x,y,z,n,\bar{k})$$ is Lebesgue measurable. We also give similarly defined $g(x,y)$ with the property that the statement "$x\mapsto g(x,r)$ is measurable for every $r\in \mathbb{R}$" has large cardinal consistency strength (and in particular implies the consistency of $\mathsf{ZFC}$) and $h(m,x,y,z)$ such that $h(1,x,y,z),\dots ,h(16,x,y,z)$ can consistently be the indicator functions of a Banach$\unicode{x2013}$Tarski 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.

29 pages, 7 figures, 1 table