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If 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 a
by rocqua 9y ago
If 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.