4 ms·
The thing is, that Godel's theorem is provable. So by your definition, it is true. The usual definition of false is not "I can't prove it's true" as that is pr
by Chirono 16y ago
The thing is, that Godel's theorem is provable. So by your definition, it is true.
The usual definition of false is not "I can't prove it's true" as that is pretty hard to decide.
Suppose I have a statement S that I can't prove. Is it false, or am I just not clever enough to prove that it is true?
The normal definition of S being false is that the negation of S is true.
Part of what the incompleteness theorem says is that in any system of logic that doesn't contradict itself, there will be statements that are neither provably true nor provably false. Thus you can take these statements to be true OR false as an axiom and it won't lead to contradictions.
- jaekwon 16y agoGodel's "proof" states that G is undecidable, and since that is what G states, G must be True, and this G is a True statement that cannot be proven (or disproven). I say that the conclusion that G is True does not follow. Calling G True is no better than calling G False. Nobody can prove GIT (Godel's Incompleteness Theorem). I tried to disprove it, but I can't do that either. Godel's Incompleteness Theorem itself is a Godel Sentence. You can add GIT as an axiom in my system, then it would become True. I'm saying that you don't need to do that to have a complete system. You can either have a complete and consistent system, OR you can have GIT.
- jaekwon 16y agoWell, my definition does not imply that what is False is easy to decide, and that's OK. If N != NP, then the statement "X is factorable" would be hard to decide, even if it's False (or True for that matter). Godel, in the proof of GIT, chose inconsistency (by concluding that G is True, even though it is also False). So, by your own conclusions, I choose GIT to be false, and there are no contradictions.