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I recently saw a paper on ``condensed mathematics'', in which I found the following quote interesting (see Condensed Mathematics: The internal Hom of condensed sets and condensed abelian groups and a prismatic construction of the real numbers):

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Here [CS20] refers to ``Dustin Clausen and Peter Scholze, Masterclass in condensed mathematics''. This lead me to the following natural question:

Question. Are there any other results which are proved to be independent from ZFC, but are solvable in the sense of condensed mathematics?

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    $\begingroup$ The condensed extension group "knows" much more than the classical extension group when the abelian groups in question are infinite, therefore the vanishing of the condensed extension group is much stronger than the vanishing of the classical extension group. A toy analogue is the following: let $k$ be a field and $V$ a $k$-vector space. I am not sure whether one could recover $V$ from its dual $\operatorname{Hom}(V,k)$. However, if we endow the dual with the compact open topology, then the continuous double dual recovers $V$. $\endgroup$
    – Z. M
    Commented Sep 27, 2021 at 18:40

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