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 $\mathrm{ZFC}$ (and much stronger theories) whether the function $f\left(x\right)=\underset{y\in R}{inf}\underset{z\in R}{sup}\underset{n\in N}{inf}\underset{\bar{k}\in N^{70}}{sup}p(x,y,z,n,\bar{k})$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 $r\in R$” has large cardinal consistency strength (and in particular implies the consistency of $\mathrm{ZFC}$) 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–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 $f\left(x\right)$, $g(x,y)$, and $h(m,x,y,z)$) are universally measurable.