TITLE: Necessary use of large cardinals in mathematics
QUESTION [12 upvotes]: There are some statements, whose consistency (or the consistency of their negation) require the existence of large cardinals (in the sense that if the statement (or its negation) is consistent, then it is consistent that there are some large cardinals). I know many of such examples inside set theory.

Question. What are examples of  statements in  mathematics (other than set theory), where it is known we need large cardinals to prove their consistency 
  (or the consistency of their negation)? 

For each example, giving what kind of large cardinals are sufficient and  what kind of large cardinals are necessary is appreciated. Please also give references.

REPLY [19 votes]: The dual of an abelian group $A$ is defined to be the group $\text{Hom}(A,\mathbb Z)$ of homomorphisms to the infinite cyclic group. As usual with such dualities, there's a canonical homomorphism from $A$ to its double dual
$$
A\to A^{**}:a\mapsto(h\mapsto h(a)).
$$
If this is an isomorphism, $A$ is said to be reflexive. 
Question: Are all free abelian groups reflexive?
Answer: Yes if and only if there are no measurable cardinals.