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

• The paper introduces a polynomial-based function whose measurability is independent of ZFC, questioning long-held assumptions in definability.
• It employs universal Diophantine equations to bridge effective descriptive set theory with measure theory, demonstrating nuanced complexity levels.
• The work highlights the impact of large cardinal axioms on real-valued functions, inviting further exploration of foundational set-theoretic questions.

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

This paper, authored by James E. Hanson, discusses intriguing results in the context of set theory, effective descriptive set theory, and functions that are ostensibly definable but potentially non-measurable. The paper systematically explores the intersection of model theory, large cardinals, and definability in the examination of mathematical functions, particularly in regard to their measurability within the framework of ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice).

Core Contributions

1. Non-Measurable Function Construction: The paper constructs a specific function defined by a polynomial of degree 7 with integer coefficients whose measurability is independent of ZFC. This illuminates how combinatorial and set-theoretical techniques can produce functions that challenge the assumption that explicitly definable functions are necessarily measurable.
2. Universal Polynomial Utilization: Utilizing Matiyasevich's resolution of Hilbert's tenth problem, the paper employs universal Diophantine equations to articulate the definability of complex projective sets that correspond to ostensibly computable or explicit functions. These polynomials serve as the starting point for constructing functions whose status regarding measurability and other regularity properties remains undetermined within ZFC alone.
3. Complexity and Definability: Hanson explores defining functions with varying complexity levels, demonstrating that even for low-degree polynomials, the transition between definability and non-measurability is nuanced and constrained by set-theoretic independence results.
4. Interplay with Large Cardinals: The exploration extends to the implications of large cardinal axioms. The existence of large cardinals impacts the measurability of certain sets derived from real functions. These implications suggest that under strong large cardinal assumptions, functions that are expressible can universally maintain regular measure-theoretic properties.

Technical Analysis

• Effective Descriptive Set Theory: The paper employs effective descriptive set theory extensively to delineate between lightface sets (which are more computationally restrictive) and boldface sets, underpinning a detailed discussion on hierarchical structures of definability and measurability.
• Complexity Measures: The analysis of functions includes a sophisticated assessment of projective and arithmetic hierarchies. The paper demonstrates that certain definable sets and functions are independent of ordinary set-theoretic axioms due to the complexity of their construction involving universal polynomials and logic operations.
• Specific Set Constructions: A significant portion of the paper is dedicated to constructing and analyzing specific non-measurable sets, including those analogous to the Vitali set and the Banach–Tarski paradox within the context of the continuum. These constructions illustrate the limits of measurability imposed by foundational set theoretic assumptions.

Implications and Future Directions

• Theoretical Implications: The paper contributes to theoretical mathematics by showing that measurability is not guaranteed for all definable functions, emphasizing independence from ZFC. It enhances our understanding of hierarchical complexities within set theory and serves as a cornerstone for combining real analysis with logical definability.
• Measure Theory and Real Analysis: For fields relying on measure theory, Hanson's results encourage re-evaluation of which functions can be assumed to automatically possess traditional properties like measurability or Baire property when derived in different axiomatic systems.
• Continuation of Set-Theoretic Research: Future investigations might further explore the boundaries between definability, computation, and measure theory, potentially leading to more sophisticated understanding of foundational questions in mathematics. Moreover, exploring the implications of other large cardinal axioms on real-valued functions beyond projective sets might yield new insights.

James E. Hanson's paper establishes a profound link between logic, algebra, and measure theory, charting a sophisticated yet theoretically accessible path for continuing exploration in mathematical logic and foundational studies in mathematics.