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The best/worst part is that there are useful branches of mathematics which assume the Axiom of Choice is true AND there are useful branches of mathematics which
by pvillano 2y ago
The best/worst part is that there are useful branches of mathematics which assume the Axiom of Choice is true AND there are useful branches of mathematics which assume the Axiom of Choice is false. That's one reason I take the view that mathematics does not exist: if math did exist, there would only be one set of axioms consistent with nature.
- 9991 2y agoYou conclude the problem is with math. I conclude the problem is with reality.
- alkyon 2y agoIn physical reality there is no infinity of anything* so Axiom of Choice is irrelevant for it. * Universe could be itself inifinite but locally everything is finite.
- 9991 2y ago> locally everything is finite. You don't know that.
- ColinWright 2y ago>> Universe could be itself infinite but locally everything is finite. > You don't know that. I think they were saying that it could be that the Universe is infinite with everything still being locally finite. I do not think they were asserting that everything is locally finite, I think it was included in the "it could be" part. But certainly you're right that we don't know that.
- alkyon 2y agoAC is an axiom (something assumed without proof) and not the result that would render mathematics itself inconsistent. Banach Tarski is not about AC, it is about existence of some pathological sets that have no measure. Edit: AC is generally assumed to be true because rejecting it leads to even stranger paradoxes than that of Banach-Tarski.