**Computability theory beyond the countable**

Computability is very well-understood in the context of the natural numbers: essentially every

reasonable model of computation gives rise to the same notion. This in turn provides a

framework for studying the role of (non)computability throughout “countable mathematics.” In

particular, we can study the complexity of specific structures of interest (“computable structure

theory”) or the types of constructions needed for proofs of various theorems (“reverse

mathematics”).

However, things become much more difficult when we look beyond the countable; for example,

it is not at all clear how to talk about the computability-theoretic complexity of the field of real

numbers or of various forms of the axiom of choice. In this talk I’ll present work on extending

the computability-theoretic analysis of mathematics to situations far from the natural numbers:

uncountable computable structure theory and higher-order reverse mathematics.