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Gödel's theorem is a theorem, not an axiom.
by stared 5y ago
Gödel's theorem is a theorem, not an axiom.
- VirusNewbie 5y agooops in my original post I meant to say meta axioms and meta theorems, how embarrassing. As to your reply above, regardless of what you call axioms, semantically speaking there is an isomorphic subset in common for all useful axiomatic system is what I'm postulating. Can you think of an axiomatic system that doesn't have something iso to identity and equivalence for instance?
- stared 5y agoI never went deep into formal logic, so I can misname a few things as well. 1. When it comes to the Gödel's incompleteness theorem - for non-recursive systems you may you can get without that, vide https://math.stackexchange.com/questions/3359878/what-is-some-easy-illustrative-example-of-an-non-recursive-formal-system-and-rec https://math.stackexchange.com/questions/3359878/what-is-som... 2. In general, you can do a lot without equivalence. For example, in set theory (ZF), you get only trivial equivalence (sets having the same elements are the same set). But you can still do a lot with "contains". A similar thing works with partially ordered sets, in which you have the "<=" operator.