Finding bases of uncountable free abelian groups is usually difficult

TL;DR

This paper explores the computational complexity of identifying free abelian groups and constructing bases in uncountable cases, revealing that such problems are often computationally hard, especially under certain set-theoretic assumptions.

Contribution

It demonstrates the difficulty of recognizing free abelian groups and constructing bases in uncountable cases, including the existence of definable groups without definable bases under $V=L$.

Findings

Identifying free abelian groups is computationally hard depending on cardinality.

Constructing bases for uncountable free abelian groups can be impossible to do definably.

Under $V=L$, there exist free abelian groups with no first-order definable bases.

Abstract

We investigate effective properties of uncountable free abelian groups. We show that identifying free abelian groups and constructing bases for such groups is often computationally hard, depending on the cardinality. For example, we show, under the assumption $V=L$, that there is a first-order definable free abelian group with no first-order definable basis.