Definable Davies' Theorem

TL;DR

This paper establishes a set-theoretic condition under which every definable function of two real variables can be decomposed into a sum of simpler definable functions, extending Davies' classical theorem.

Contribution

It proves an analogue of Davies' theorem for $oldsymbol{ ext{Σ}}^1_2$ functions, linking the decomposition property to the constructibility of all reals.

Findings

Such a decomposition exists if and only if all reals are constructible.

The result characterizes the set-theoretic assumptions needed for definable function decompositions.

It extends classical theorems to the realm of definable and projective functions.

Abstract

We prove the following analogue of a Theorem of R.O. Davies: Every $\Sigma^1_2$ function $f:\R\times\R\to\R$ can be represented as a sum of rectangular $\Sigma^1_2$ functions if and only if all reals are constructible.