A good example of that is how the Axiom of choice impacts the measure/probability theory. It imply the existence of some sets that cannot be Lebesgue measured

A good example of that is how the Axiom of choice impacts the measure/probability theory. It imply the existence of some sets that cannot be Lebesgue measured (which is an generalization of width, volume, etc for arbitrary sets, also generalization of probability for arbitrary sets)... but it's not possible to present a single example of those non measurable sets, only prove that they exist. And it's possible to construct an alternative theory with the axiom of determinacy, then any subset of R is measurable. * https://en.wikipedia.org/wiki/Axiom_of_choice https://en.wikipedia.org/wiki/Axiom_of_choice * https://en.wikipedia.org/wiki/Axiom_of_determinacy https://en.wikipedia.org/wiki/Axiom_of_determinacy * https://en.wikipedia.org/wiki/Lebesgue_measure https://en.wikipedia.org/wiki/Lebesgue_measure