The typical Turing degree
Seminar Room 1, Newton Institute
AbstractSince Turing degrees are tailsets, it follows from Kolmogorov's 0-1 law that for any property which may or may not be satisfied by any given Turing degree, the satisfying class will either be of Lebesgue measure 0 or 1, so long as it is measurable. So either the typical degree satisfies the property, or else the typical degree satisfies its negation. Further, there is then some level of randomness sufficient to ensure typicality in this regard. I shall describe a number of results in a largely new programme of research which aims to establish the (order theoretically) definable properties of the typical Turing degree, and the level of randomness required in order to guarantee typicality.
A similar analysis can be made in terms of Baire category, where a standard form of genericity now plays the role that randomness plays in the context of measure. This case has been fairly extensively examined in the previous literature. I shall analyse how our new results for the measure theoretic case contrast with existing results for Baire category, and also provide some new results for the category theoretic analysis.
This is joint work with George Barmpalias and Adam Day.