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That makes sense, thank you. However, at what point is the symbol "2" introduced in such a scheme? Is it before addition, or after? At some point it is given a
by mike741 8y ago
That makes sense, thank you. However, at what point is the symbol "2" introduced in such a scheme? Is it before addition, or after? At some point it is given a definition, like S(S(0)) where S(0)=1 and S(X)=X+1, at which point isnt it easy to substitute
2 = S(S(0))= S(0)+1 = 1+1
is this simple equivalence really not provable with sufficient axioms?
Obviously from the downvotes I am missing something fundamental, I would very much like to know what that is but am at a loss. Perhaps I am just missing mrleiter's original point.
- repsilat 8y agoNormally it's defined exactly as S(S(0)) like you said, so the above should constitute a proof that 1+1=2. Anyone disputing that sort of proof normally needs to take issue with one or more of, - The axioms, - The rules of inference. So someone might say, "In step one, you used an axiom of the form X=X. What basis do you have for assuming it's true?" It's worth looking up the Munchausen trilemma for a little more on that sort of thing, but I'm not sure how many people would seriously argue against the validity of that proof (or one just like it if I've made a mistake :-) As for why you've been downvoted, I have no idea. I think your question was perfectly fine, asked politely enough, and there are no doubt many perfectly reasonable logical systems in which 1+1=2 is trivially true (and not just "a short proof away.")