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Given a formal logic system W with it's set of axioms, if it is capable of 'handling' basic arithmetic then W can not be used to 'prove' its own consistency. In other words there will always be a statement H in W that is 'true' yet not provable. You could call system W a STATIC logic system . Could there also be a definable and rigorously constructed 'Semi-Variant' Logic system with a set of 'base' axioms that are invariant and a set of 'variable' axioms that can 'change' if presently needed to change for the sake of consistency or completeness. Could the logical foundations of Physics be called a 'Semi-variant' Logic system?

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Are all true mathematical statements provable?

An interesting look for Gödel Theorem.... Check :

Are all true mathematical statements provable?

  • All provable math statements are not necessarily true. A math statement could be proven to be false. If a true math statement X is not provable does this mean any math statement that X implies ,say X1 is also true but not provable?? – user128932 Jul 19 '14 at 4:42
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What do you mean by "W1 does not have H"? I thought that W1 was derived from W by adding some axioms. (Incidentally, I don't know why you're adding "certain relevant and useful axioms" as opposed to just doing the obvious thing and adding H itself as an axiom.) In particular, then, the basic grammar of W1 is the same as that of W, so any statement in W is still a statement in H.

Perhaps what you meant is that in W, H is no longer unprovable. Continuing along, you have a sequence (W,W1,W2,...) where, as you go down the list, you keep adding axioms.

This is not a way around Godel's theorem. Let X be the union of all the sets of axioms from all the Wi. Then (by Godel's theorem!), if X is recursive then your logical system still contains true statements about arithmetic that are not derivable from X.

(Of course if X is not recursive --- for example, if you chose at some point along the way to add every true statement about arithmetic as an axiom --- then Godel does not apply.)

  • If X is the union of all the sets of axioms from all the Wi then if it is recursive it will have a true statement that is not derivable from X , but does this statement hold if X has an infinite number of axioms ? IF X has a finite number of axioms maybe one can add new axioms to $banish$ the true but un-derivable statements . But this new set of axioma, X1 will also have new true-but-unprovable statements. THE point is if X $changing$ to X1 is considered a DYNAMIC logic set with a changing set of axioms Godel Theorem is NOT avoided just $postponed$. – user128932 Jul 13 '14 at 5:24
  • @user128932 : Godel's theorem applies perfectly well whether the number of axioms is finite or infinite. (For example, it applies to Peano arithmetic, which has an infinite number of axioms.) Why would you think otherwise? (I.e. at what step in the proof does it appear to you that the number of axioms is being assumed finite?) – WillO Jul 13 '14 at 15:15
  • So Godels theorem works with a finite or infinite number of axioms but my point is even if there are always true-but-unprovable statements 'floating' around in any formal logic system with a specific $invariant$ list of axioms can this true-but-unprovable statement 'get in the way' of some other proof being attempted? Do true but unprovable statement interfere with someone trying to prove some theorem? – user128932 Jul 15 '14 at 2:20
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    @user128932: Of course the fact that some theorem is not provable from some set of axioms won't (and shouldn't) stop anyone from proving it with a different set of axioms. For example, we know that the Godel sentence for Peano arithmetic is true precisely because we can prove it in a stronger system (with the additional axiom that Peano arithmetic is consistent). See also, e.g. the Paris-Harrington theorem, Goodstein's theorem, etc. – WillO Jul 15 '14 at 12:46
  • So a Godel sentence for Peano arithmetic is true in a 'bigger' or stronger Logic system that has the added axiom that Peano arithmetic is consistent. Could one call the former axiom system and the stronger axiom system (with the Peano arithmetic axiom added) two subsequent instances of a Dynamic Logic system that can 'change' or is variant; as long as any 'change' in axioms or rules of information 'manipulation' do not alter any 'required' logic 'structure'? – user128932 Jul 19 '14 at 4:30

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