Uncovering the Link Between Randomness, Set Theory, and Reverse Mathematics 🔍

Abstract : I will discuss two recent interactions of the field called randomness via algorithmic tests. With Yokoyama and Triplett, I study the reverse mathematical strength of two results of analysis. (1) The Jordan decomposition theorem says that every function of bounded variation is the difference of two nondecreasing functions. This is equivalent to ACA or to WKL, depending on the formalisation. (2) A theorem of Lebesgue states that each function of bounded variation is differentiable almost everywhere. This turns out to be equivalent WWKL (with some fine work left to be done on the amount of induction needed). The Gamma operator maps Turing degrees to real numbers; a smaller value means a higher complexity. This operator has an analog in the field of cardinal characteristics along the lines of the Rupprecht correspondence [4]; also see [1]. Given a real p between 0 and 1/2, d(p) is the least size of a set G so that for each set x of natural numbers, there is a set y in G such that x and y agree on asymptotically more than p of the bits. Clearly, d is monotonic. Based on Monin's recent solution to the Gamma question (see [3] for background, and the post in [2] for a sketch), I will discuss the result with J. Brendle that the cardinal d(p) doesn't depend on p. Remaining open questions in computability (is weakly Schnorr engulfing equivalent to "Gamma = 0"?) nicely match open questions about these cardinal characteristics.

Recording during the thematic meeting : "Computability, randomness and applications" the June 23, 2016 at the Centre International de Rencontres Mathématiques (Marseille, France)

Filmmaker: Guillaume Hennenfent

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