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I am not a fan of the axiom of choice, and open-minded about truth values beyond true and false. So I'm really curious about the progress of constructive mathematics, regarding proving known results with non-ZNF axioms, and maybe establishing some more "intuitively satisfying" (whatever that may mean) results where ZNF gives subjectively bogus results like Banach-Tarski?

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A lot of people blame Banach-Tarski on the axiom of choice. I disagree. I think it's in essence just a variant of the Hilbert hotel, with the axiom of choice merely serving as part of the machinery for this manifestation. In other words, it's just a counterintuitive but true consequence of the nature of infinity. As such, I think any mathematical system that acknowledges infinity, is going to have such counterintuitive results.

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