The Undecidable Function

Any function you can write down is measurable — right? The intuition is that measurability failures (like indicator functions of Vitali sets) require the axiom of choice to construct and therefore cannot be “explicitly” defined. Concrete, written-down functions should be safe.

The paper constructs a polynomial of degree 7 whose Lebesgue measurability is undecidable in ZFC. The function is as explicit as mathematics gets — a finite polynomial with integer coefficients. Whether it is measurable depends on set-theoretic axioms beyond ZFC. The function is not exotic; the question is.

The construction connects measurability to Diophantine equations via Hilbert’s tenth problem. The polynomial encodes a number-theoretic statement whose truth value is independent of ZFC. If the statement is true (in a given model), the polynomial’s zero set has one measure-theoretic property. If false, another. The measurability of the function inherits the undecidability of the encoded statement.

Three variants are constructed: one whose measurability is independent of ZFC and stronger theories; one whose universal measurability (across all parameter values) has the consistency strength of large cardinals; one that could represent a Banach-Tarski paradox. All are explicitly written polynomials.

The structural point: measurability is not a property of the function’s complexity but of the axioms used to evaluate it. The same written-down function is measurable in one model of set theory and non-measurable in another. The question “is this function measurable?” is not about the function — it is about the mathematical universe in which you ask.