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This paper is about naturally occurring objects in computability theory, the area inside mathematical logic that studies the complexity of infinite countable objects. It is about the structure that emerges when we look at almosteverywhere behavior with respect to a measure called Martin’s measure. Posed in the 1970s, Martin’s conjecture was the first indication of this hidden structure. Martin’s conjecture is currently one of themain open questions in computability theory and is of great foundational importance. Progress has been slow, and only recently we have started to appreciate the extent to which Martin’s measure can be used to understand the behavior of naturally occurring objects in computability theory.

Martin's Conjecture: A Classification of the Naturally Occurring Turing Degrees