3 ms·
Axioms shouldn't be provable (other than from themselves) - if the fifth axiom of geometry had turned out to be provable from something else, then it would be h
by doubleunplussed 7y ago
Axioms shouldn't be provable (other than from themselves) - if the fifth axiom of geometry had turned out to be provable from something else, then it would be have just been renamed a theorem instead of an axiom.
- fao_ 7y agoThat's the wrong way to think about it, I think. It's more correct to think of axioms as being preconditions. We say, "Here is what happens if these things are true", and then specify what happens, but that doesn't deprive us from independently checking whether those conditions are true. Just because something has been proven does not stop it from being an axiom. It's contextural.