[Ragnarork]

Joke aside, is there a field (or sub-fields) of mathematics that just... studies what breaking some axioms would do and where would it lead? This seems both completely stupid but also potentially fascinating at the same time.

[mafuy]

Yes. Prominent example is the axiom of choice. There was a LOT of discussion about it by mathematicians. https://plato.stanford.edu/entries/axiom-choice/

For instance, with it, you can double the volume of a body by looking at it from a weird angle.

[SetTheorist]

That's a very misleading description of Banach-Tarski. You need to break up the body into a few (very weird) pieces and maneuver them (via only rigid motions - rotations and translations).

[debo_]

The mathematics equivalent of the any% speedrun community?

[monkeyelite]

It’s not sub-field. It’s a technique. You relax or change axioms defining your thing and see what happens.

[SetTheorist]

Paraconsistent logic looks into logical systems that allow contradictions.

A lot of set-theory research is looking into the consequences of various axiomatic assumptions.