Another proof that Condensed Whitehead's problem has a positive answer
As a part of their "Masterclass in Condensed Mathematics", Clausen and Scholze defined a Condensed version of Whitehead's problem and showed that it has a positive answer. More recently, Bergfalk, Lambie-Hanson and Saroch gave a set theoretic proof. We present a proof of their theorem which draws parallels with the proof that under the sufficiently many instances of the set theoretic principle "weak diamond" all Whitehead groups are free. An initial step in the proof isolates the combinatorial content of what it means for a group to be Condensed Whitehead. Taking this step as a blackbox essentially no knowledge of Condensed Mathematics is required. The fundamental contribution of set theory is to associate to each nonfree Abelian group a relevant site with an analysis of its Ext. Intuition from forcing is present in the arguments, but forcing is not required to understand the proof. The main background required for the talk is some general topology. This is joint work with Jerry Wei.

