29 May 2014

17:15

Andrew Brooke-Taylor

Abstract

<p>Cardinal characteristics of the continuum are (definitions for)
cardinals that are provably uncountable and at
most the cardinality c of the reals, but which (if the continuum
hypothesis fails) may be strictly less than c. Cichon's diagram is a
standard diagram laying out all of the ZFC-provable inequalities between
the most familiar cardinal characteristics of the continuum. There is a
natural analogy that can be drawn between these cardinal
characteristics and highness properties of Turing oracles in
computability theory, with implications taking the place of
inequalities. The diagram in this context is mostly the same with a few
extra equivalences: many of the implications were trivial or already
known, but there remained gaps, which in joint work with Brendle, Ng and
Nies we have filled in.</p>