Cichon's diagram for computability theory

29 May 2014
Andrew Brooke-Taylor
<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.&nbsp; Cichon's diagram is a standard diagram laying out all of the ZFC-provable inequalities between the most familiar cardinal characteristics of the continuum.&nbsp; 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.&nbsp; 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>