An analogy between cardinal characteristics and highness properties of oracles

Abstract

We present an analogy between cardinal characteristics from set theory and highness properties from computability theory, which specify a sense in which a Turing oracle is computationally strong. While this analogy was first studied explicitly by Rupprecht in his PhD thesis, many prior results can be viewed from this perspective. After a comprehensive survey of the analogy for characteristics from Cichon's diagram, we extend it to Kurtz randomness and the analogue of the Specker-Eda number.

28 pages; version 2 post refereeing, in particular with due credit to Rupprecht for results from his thesis of which we were previously unaware