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I think that the most celebrated result in this direction is Shelah's famous work on Whitehead's Problem:

Is every abelian group $A$ such that $Ext^1(A, \mathbb{Z})=0$ free?

This is known to be indepedent of ZFC (cf. Shelah, Whitehead groups may not be free, even assuming CH I and II), and it seems that there is an interesting connection between certain questions of ZFC and abelian group theory (cf. Eklof The affinity of set theory and abelian group theory).

Another famous work relating algebra and 'higher' set theory is B. L. Osofsky Homological dimension and the continuum hypothesis and subsequent works.

Independence results, Martin's axiom, Continuum Hypothesis and large cardinal axioms have proven to be interesting to 'pratical' questions in many areas of mathematics (point-set topology, functional analysis, order theory, etc).

Is there any current research trend in using sistematically Independence results and other aspects of 'higher' set theory to understand problems of algebra, such as in the above mentioned mathematical disciplines?

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    $\begingroup$ The following survey by Scholze talks about cardinals, so potentially it could lead to some independence!! math.uni-bonn.de/people/scholze/Condensed.pdf $\endgroup$
    – Rahman. M
    Commented Jul 12, 2020 at 10:27
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    $\begingroup$ The other one which comes to the stage again, is strong homology, for example you can consult the following paper arxiv.org/abs/1509.09267 There are also some new papers in this direction, in particular of Jeff himself. This may not be directly related to algebra, but... $\endgroup$
    – Rahman. M
    Commented Jul 12, 2020 at 10:29
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    $\begingroup$ @Rahman.M thank you very much! $\endgroup$
    – jg1896
    Commented Jul 12, 2020 at 13:00
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    $\begingroup$ Have you looked at Eklof and Mekler "almost free modules: set theoretic methods"? You might find something interesting in there. A simple example "there is a nonreflexive free abelian group iff there is a measurable cardinal" $\endgroup$
    – Alessandro Codenotti
    Commented Jul 13, 2020 at 19:03
  • $\begingroup$ @AlessandroCodenotti this example is impressive. I will check it! $\endgroup$
    – jg1896
    Commented Jul 13, 2020 at 20:14

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