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 $$f(x) = \inf_{ y \in \mathbb{ R}}\sup_{ z \in \mathbb{ R}}\inf_{ n \in \mathbb{ N}}\sup_{ \bar{ k} \in \mathbb{ N}^{ 70}}p(x,y,z,n,\bar{ k})$$ 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 Vop\v{ e}nka'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.
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