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Gödel's first incompleteness theorem states that any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and comp
by tdmackey 16y ago
Gödel's first incompleteness theorem states that any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory (Kleene 1967, p. 250) per http://en.wikipedia.org/wiki/Gödels_incompleteness_theorems http://en.wikipedia.org/wiki/Gödels_incompleteness_theorems
Just because you title something an "Unanswerable question" doesn't mean it actually is, and certainly doesn't mean the first incompleteness theorem is relevant.
- jaekwon 16y agoIs "The above question has no solution", which is choice C, not a Godel sentence?
- deleted 16y ago[deleted]
- jaekwon 16y agoCan you prove or disprove this statement? "Godel's first incompleteness theorem is a Godel sentence" Anyways, it is my belief that Godel's Incompleteness Theorem is false by definition of Truth. That is, what is true is provable and vice versa. Once you depart from this definition, you get statements that are paradoxical, and by my definition of Truth, the Godel Incompleteness Theorem is certainly paradoxical (thus False).
- deleted 16y ago[deleted]
- tdmackey 16y agoSo be it, but it doesn't imply that Godel's incompleteness theorem applies. Metaphorically they may somehow be similar given someone's viewpoint, but it doesn't make it so.
- ElliotH 16y agoBut he proved his theorems. Thus it is provable and by your own definition is true regardless of what the theorem actually says.
- deleted 16y ago[deleted]
- Chirono 16y agoThe 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.
- tdmackey 16y agoNo. Godel's incompleteness theorems say nothing about non-arithmetical or non-mathematical statements nor do they apply in contexts where no formal system exists. Some sort of paradox, sure, but Godel doesn't apply. Truth is not a mathematical concept, and determining the "truth" or "falsehood" of a sentence has nothing to do with Godel's incompleteness theorems.
- JoeAltmaier 16y agoBoolean logic is an "arithmetic system"
- jheriko 16y agoer.. truth is precisely a mathematical concept, it underlies the concepts of proof and theorem pretty heavily, not to mention formal logic.