4 ms·
It's better to think of axioms as definitions. If the axioms are inconsistent, it just means nothing satisfies the definition, which isn't the end of the world
by xamuel 9y ago
It's better to think of axioms as definitions. If the axioms are inconsistent, it just means nothing satisfies the definition, which isn't the end of the world. People have spent time studying hypothetical properties of polynomial-time factorization algorithms, knowing full well there may be no such things if P<>NP.
There are models of ZF where the axiom of choice holds, and there are models of ZF where the axiom of choice doesn't hold. To "assume" the axiom of choice, is nothing more than to restrict attention to the former. No more controversial than, say, deciding to write a paper specifically about golden retrievers, rather than a paper about all dogs.
- rocqua 9y agoIf your axioms are inconsistent, you can use reductio ad absurdum to prove any statement. I'd say that it is rather problematic when I can prove any statement and it's converse. Of course, just because the axiom of choice isn't obvious doesn't mean that ZFC is inconsistent. It just means that the argument for ZFC being inconsistent is more hand-wavy.
- xamuel 9y agoOf course you can prove any statement. The statements only hold for things which satisfy the axioms. "If X is a model of ZF where 1=0, then X also satisfies anything else you want." (Fine, there is no such X so who cares.) You're getting confused because of the shorthand where statements within the model are stated as if they are statements about the world we live in. That's just a shorthand.
- Ericson2314 9y agoYes we must take saxioms on faith alone. Embrace pluralism and move on. https://m.youtube.com/watch?v=O45LaFsaqMA https://m.youtube.com/watch?v=O45LaFsaqMA by the late great Voedvodvosky is more interesting.