An abelian group $A$ is said to be $\mathit{Whitehead}$ if every short exact sequence $0 \to \mathbb{Z} \to B \to A \to 0$ splits. Free abelian groups are Whitehead and whether the converse holds is known as the Whitehead Problem. Shelah proved that the Whitehead Problem is independent of ZFC by showing that it has a positive answer if $V=L$ and a negative answer assuming $\mathrm{MA}_{\aleph_1}$. This talk will give an exposition of this result based on a survey of Paul Ekloff and will focus on the $V=L$ side of the solution.