The majority of the talk won't assume much background beyond basic logic.
In this talk, after making sense of the first paragraph of this abstract, we will show that the interplay between hypergraphs and flat pregeometries runs deeper, and make some reckless conjectures.
An ongoing project of mine is to understand the general behavior of Medvedev reducibility on the ordinals. In this talk, I'll present one particular aspect of the problem: Given an ordinal α, how does the supremum of the ordinals Medvedev reducible to α (= Med(α)) compare with other notions of "least non-α-computable ordinal?" Specifically, I'll look at two such points emerging naturally from generalized computability theory, and show that in general Med(α) lies strictly between these two points. One of these inequalities is straightforward computability theory; the other requires a more set-theoretic argument, using forcing.
If time remains, I'll also outline a result on the degree structure of Medvedev reducibility on the ordinals; I'll show that it has "large antichains," the key difficulty being proving this result inside ZFC.
This is a joint work with Calvert, Frolov, Harizanov, Knight, McCoy, and Vatev.
Archived 2016-17 Logic Seminar Schedule