The discovery of [the Banach-Tarski paradox](https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox) was of course a great thing in mathematics but raises the issue of the relation between mathematics and reality. Empirically there are good reasons for faith in mathematical proofs: well derived mathematical theories and formulas, used in an accurate manner in science, really bring observable results. I don't know any scientific failures where mathematical results was blamed.

Naturally constructive proofs are preferable but there shouldn't be any problem with [non-constructive proofs](https://en.wikipedia.org/wiki/Constructive_proof#Non-constructive_proofs): the mathematical-logical machinery should guaranty the expected success. With or without the use of the law of the excluded middle!

So how to explain the non-intuitive theorem of Banach-Tarski? Which are the "weak pillars" or combinations of them? Personally I can't see the axiom of choice as a problem, since it only is a condition of what's to be called a set. Is the theorem perhaps a sign of lack of continuity in reality?

> 
> I really would like a big list of suggestions what "axioms" could be
>  changed to avoid the "possibility" of the the theorem.
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I have been re-considering. Suppose that a new discrete theory about space and time was developed and was found to be consistent with observations. How long would it take before a corresponding theory was developed from a pure geometrical perspective? Compare with the corresponding theories of Heisenberg and Schrödinger!

What was explained in one way in the discrete theory maybe was described in a totally differnt way in the continuous theory? And who knows, maybe similar constructions as in the paradoxes of Banach-Tarski and Sierpinski-Mazurkiewicz, would be used to explain the expansion of what is now called dark energy? And even "worse", to explain the result of high energy proton- proton collisions?

It strikes me that the relative terms of consistency as is used in mathematics is all what is needed for the mathematical material to be used in scientific theories. Physicists do sometimes use mathematics in extraordinary ways and get good results.

But that does not make it less interesting to ponder the pillars of the "paradoxes".

Also, thanks for reopen and for great answers!