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Godel constructed statements S that are neither true nor false (more precisely, you could add an axiom saying "S is true", or you could add an axiom saying "S i
by obastani 12y ago
Godel constructed statements S that are neither true nor false (more precisely, you could add an axiom saying "S is true", or you could add an axiom saying "S is false", and you'd get a consistent logical system either way).
It turns out that if you can prove something is both true and false, then EVERYTHING is both true and false. For example, let's say you want to prove X. Note that Y is true. But then you obtain a contradiction (since Y is false)! Hence X is true.
- guygurari 12y agoYou assume that proof by contradiction always works, but is that necessarily the case?
- edanm 12y ago"Godel constructed statements S that are neither true nor false" No, that's not what he did. He constructed a statement that is true, but that can't be proved in a certain logical system. The statement was basically "This statement is unprovable with these axioms", which can either be proved - meaning you've proved something false, or can't be proved, meaning the statement is true, but is unprovable. Note that this uses two notions of "true" - provable (can be derived from axioms), and actually true.
- ot 12y ago> Note that this uses two notions of "true" - provable (can be derived from axioms), and actually true. Even the second notion of truth actually just semantic consequence of the second-order theory of the naturals, which is a mathematical formal concept, not quite the same as actual (ontological) truth.