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I believe that's a misinterpretation of Goedel; it's an incompleteness theorem, not an unsoundness theorem. Incompleteness: there exist true statements that la
by wk_end 8y ago
I believe that's a misinterpretation of Goedel; it's an incompleteness theorem, not an unsoundness theorem.
Incompleteness: there exist true statements that lack proofs
Unsoundness: there exist false statements with proofs
If I understand all this correctly, what Goedel demonstrated was that a sufficiently complex logical system could self-reference, and from self-reference unprovable (not contradictory) true statements fall out.
By reductio ad absurdum any self-contradictory logical system is useless as a logic because everything is provable in it, so one wonders if such systems should even be considered "logical systems".
- raverbashing 8y agoThe inconsistency part is the 2nd incompleteness theory. You don't RAA it because you can't prove the system is consistent (because you can't prove the counterexample)
- wk_end 8y ago"Gödel's second incompleteness theorem states no consistent axiomatic system which includes Peano arithmetic can prove its own consistency. Stated more colloquially, any formal system that is interesting enough to formulate its own consistency can prove its own consistency iff it is inconsistent." [1] I believe your position is that "sufficiently complex" == "able to formulate and prove its own consistency"; my position is that such a system would be worthless as a logic, because in such a system you can prove anything. Your second paragraph is inscrutable to me. Are you trying to claim that you can't prove everything with a known inconsistent logic? Are you referring to paraconsistent logics? [1] http://mathworld.wolfram.com/GoedelsSecondIncompletenessTheorem.html http://mathworld.wolfram.com/GoedelsSecondIncompletenessTheo...
- raverbashing 8y ago> "sufficiently complex" Meaning, not "trivially simple". The description in your link is good enough, the description in the Wikipedia is more detailed > any formal system that is interesting enough to formulate its own consistency can prove its own consistency iff it is inconsistent. That's what "sufficiently complex" means and why you can't Reduction Ad Absurdum it.