The Turing Degrees: On the Order Dimension of and Embeddings into the Turing Degrees

Abstract: In joint work with Higuchi, Raghavan and Stephan, we show that the order dimension of any locally countable partial ordering (P, <) of size κ+, for any κ of uncountable cofinality, is at most κ. In particular, this implies that it is consistent with ZFC that the dimension of the Turing degrees under partial ordering can be strictly less than the continuum. (Kumar and Raghavan have since shown that it can also be continuum, thus the order dimension of the Turing degrees is independent of ZFC.) This is closely related to an old question of Sacks from 1963 about whether the Turing degrees form a universal locally countable partial order of size continuum.

logic

Audience: researchers in the topic

Online logic seminar

Series comments: Description: Seminar on all areas of logic