# The Borel ordering on cardinal characteristics

Boise Set Theory Seminar, Boise, April 2013

*Abstract*: Many inequalities between cardinal characteristics of the continuum can be proved in a categorical manner. One simply has to exhibit a certain transformation between the two cardinal invariants called a *Tukey map*. In several applications, the existence of a Tukey map isn’t enough; one must also know the map can be chosen to be definable. For example, although it is easy to show the pseudo-intersection number $\mathfrak p$ lies below the (un)bounding number $\mathfrak b$, it is not clear if the two cardinals are connected by a definable map. In this talk I’ll introduce the core concept, give some examples, and then consider this slightly more challenging question.

This work was joint with Tamás Mátrai and Juris Steprāns.