Title: The complexity of free abelian groups

Slides

Abstract:
Vector spaces are defined by a set of first-order axioms; the theorem every vector space has a basis is second-order, but is a consequence of these axioms.

In contrast, free abelian groups are typically defined by having a basis.

It is then natural to wonder if free abelian groups could also be defined by some set of first-order axioms.

A negative answer might take the form of a theorem that the index set of computable free abelian groups is not arithmetic.

Unfortunately, this theorem is false, at least in standard computability theory.

It can be rescued, however, by moving to uncountable computability theory.