To me the implication is that no matter how much we learn, we will still be wrong. Not because we dont know everything, but because what we do know is fundamentally uncertain. We are not unsure only about mathematics. Physics for example will always exhibit paradoxes like those of quantum theory. People unaccountably will always do things which are bad for them. And my socks will continue to disappear in the dryer.
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Not all natural numbers represent a formula. It is crucial that the formal arithmetic be capable of proving a minimum set of facts. This is a uniform procedure. In other words, suppose that there is a deduction rule D1, by which one can move from the formulas S1,S2 to a new formula S. Because each deduction rule is concrete, it is possible to effectively determine for any natural numbers n and m whether they are related by the relation. Suppose the theory has deduction rules: D1, D2, D3, ….
Let R1, R2, R3, … be their corresponding relations, as described above. Every provable statement is either an axiom itself, or it can be deduced from the axioms by a finite number of applications of the deduction rules. A proof of a formula S is itself a string of mathematical statements related by particular relations each is either an axiom or related to former statements by deduction rules , where the last statement is S.
This is because the relation between these two numbers can be simply "checked". Formally this can be proven by induction, where all these possible relations whose number is infinite are constructed one by one.
The detailed construction of the formula Proof makes essential use of the assumption that the theory is effective; it would not be possible to construct this formula without such an assumption. Note that for every specific number n and formula F y , q n, G F is a straightforward though complicated arithmetical relation between two numbers n and G F , building on the relation PF defined earlier. This formula has a free variable x. We have here the self-referential feature that is crucial to the proof: A formula of the formal theory that somehow relates to its own provability within that formal theory.
Very informally, P G P says: "I am not provable". We therefore conclude that P G P is not provable. Consider any number n. But we have just proved that P G P is not provable. We have sketched a proof showing that: For any formal, recursively enumerable i. This makes no appeal to whether P G P is "true", only to whether it is provable.
Truth is a model-theoretic , or semantic, concept, and is not equivalent to provability except in special cases. Therefore, within this model, P.
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Proof sketch for Gödel's first incompleteness theorem
Incompleteness: The Proof and Paradox of Kurt Gödel