On Analogues of the Church–turing Thesis in Algorithmic Randomness

Abstract

In this article, I consider the status of several statements analogous to the Church–Turing thesis that assert that some definition of algorithmic randomness captures the intuitive conception of randomness. I argue that we should not only reject the theses that have appeared in the algorithmic randomness literature, but more generally that we ought not evaluate the adequacy of a definition of randomness on the basis of whether it captures the so-called intuitive conception of randomness to begin with. Instead, I argue that a more promising alternative is to evaluate the adequacy of a definition of randomness on the basis of whether it captures what I refer to as a “notion of almost everywhere typicality.” In support of my main claims, I will appeal to recent work in showing the connection between of algorithmic randomness and certain “almost everywhere” theorems from classical mathematics.

Computable randomness and betting for computable probability spaces

“…Others, have argued the same for other different randomness notions. Porter [Por16], on the other hand, has argued against any one correct randomness notion. This survey, especially Section 4, supports this latter viewpoint.…”