Maybe there's no such thing as a random sequence

TL;DR

This paper explores the philosophical and mathematical question of whether truly random infinite binary sequences exist, considering the implications of definability and the potential absence of undefinable sets.

Contribution

It challenges the assumption that random sequences must exist by examining the role of definability and the possibility that no undefinable sets are present.

Findings

Questions the existence of random sequences under certain logical assumptions.

Highlights the impact of definability on the concept of randomness.

Suggests that the existence of randomness may depend on foundational set-theoretic considerations.

Abstract

An infinite binary sequence is deemed to be random if it has all definable properties that hold almost surely for the usual probability measure on the set of infinite binary sequences. There are only countably many such properties, so it would seem that the set of random sequences should have full measure. But in fact there might be no random sequences, because for all we know, there might be no undefinable sets.