A strong law of computationally weak subsets

TL;DR

This paper proves that within a fair-coin measure on natural numbers, most random sets contain infinite subsets that cannot compute any random set, revealing a form of computational weakness.

Contribution

It establishes a strong law showing that sufficiently random sets almost surely contain infinite subsets with limited computational power, a novel insight in algorithmic randomness.

Findings

Almost sure existence of infinite subsets with limited computational power

Random sets contain subsets that cannot compute any random set

New connections between randomness and computational weakness

Abstract

We show that in the setting of fair-coin measure on the power set of the natural numbers, each sufficiently random set has an infinite subset that computes no random set. That is, there is an almost sure event $\mathcal A$ such that if $X\in\mathcal A$ then $X$ has an infinite subset $Y$ such that no element of $\mathcal A$ is Turing computable from $Y$.