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analyticsophy comments on Induction, Deduction, and the Collatz Conjecture: the Decidedly Undecidable Propositions. - Less Wrong
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Comments (69)
"1+1=giraffe" is meaningless "Godel sentence" is undecidable
"1+1=giraffe" isn't meaningless. It means that if you add one and one, you get a giraffe. Everyone knows that's where giraffes come from.
If there is a difference between undecidable and meaningless, and a statement can be shown to be undecidable, then we need to accept that not every meaningful statement is true or false at least in the case of the natural numbers.
So does that mean that we should reject the principle of excluded middle? If so, that means that our standard logics are useless for dealing with mathematical (if not all) reasoning. Intuitionistic logic might be better suited at dealing with these sorts of issues, but it seems strange that some meaningful statements might be neither true nor false.