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I'm not familiar with logic beyond simple boolean operators and the standard mathematical tools (quantifiers, implication, proof by contradication, etc.)

I've known for a while that Gödel's Theorem(s) state (very loosely!) that given any system of logic, there are true statements within that system that can't be proved within that system.

However, is it always possible to extend a system of logic in which those unprovable true statements from one system can be reformulated in the new system and then proved?

pshmath0
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  • See this one: (the second answers) http://math.stackexchange.com/questions/69353/true-vs-provable?rq=1 – NPHA Jan 11 '16 at 16:03
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    The point should be that you can, but then you would end up with new problems. Let $S$ be a system, and let $p$ be the proposition that we cannot prove. If we obtain a new system denoted $S^+$ where proposition $p$ can be proved, we will have for $S^+$ a proposition $p^+$ that we cannot prove, and so on and so forth, – Kolmin Jan 11 '16 at 16:04
  • @Kolmin I don't mind that because in theory we can keep improving our system as far as we need. I was feeling a little let down when I entertained the possibility that a true statement could never ever be proved true. – pshmath0 May 12 '17 at 12:12

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It is always possible to extend a first-order logical system by adding axioms to make it a complete logic: one in which every sentence is either provable or disprovable. However, if your starting point is a system like first-order arithmetic, then you have no effective (i.e., computable) way of knowing whether an axiom you have added is true in the standard model of the original system. So the resulting "logic" is useful for some theoretical purposes, but fails to satisfy the important property that you can effectively check whether a sequence of statements is a proof.

Rob Arthan
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