Comment by JoeCamel

I would argue that trying to describe our physical world and reality has only one "axiom" which is: your description has to be consistent with our observations of the world. Mathematics doesn't have this limitation so I don't think your example is valid. If you don't like LEM, fine, maybe there is something interesting in systems without LEM, but most math is done with LEM. I don't think is a matter of "right" and "wrong".