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Unformatted text preview: G ödel‘s Fi rst I ncompleteness Theorem – “ Any adequate axiomatizable theory is incomplete. In particular the sentence ‘This sentence is not provable’ is true but not provable in the theory.” G iven: G∈T G – Provable ∈ - a member of the set of T – All statements that can be proved using axioms Thus for any sentence S, 1.) S <-----> <S> <S> - S is in the set of provable statements S - S is provable <-----> - if and only if 2.) Since we can computably generate the set of axioms, we can also computably generate the set of proofs. Hence, provable means computable. And computable implies it’s adequately definable. Therefore, provable is definable. 3.) Now Suppose, S is the statement, “This set is unprovable” (Note: a clearly defined statement) 4.) In this case the S is true when we have the statement below being true. S <-----> ~<S> ~ - Not <S> - S is in the set of provable statements S –the statement S, is provable 5.) In other words, In the above statement S is true if and only if S is not in the set of provable statements is true, when S is in fact not in the set of provable statements. ((S <-----> ~<S>) <-----> ~<S>) T T T F F F T F T Therefore when S is true, S is not provable. This shows that there are some clearly definable statements, that in fact when they are true. They cannot be proved. * Source for the whole proof is the website: http://www.math.hawaii.edu/ ...
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- Fall '10