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Re: Definition of Mathematics
Posted:
Feb 29, 2012 1:09 PM
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Your point is irrelevant. If G is not provable in T, then it means that G is independent of all axioms in T. You can set it to true or false and T remains consistent if it is consistent to begin with.
If I say A is not provable in ZFC, then it means that A is independent of ZFC. This understanding of provability is generally accepted in the mathematical community.
So, what is the problem with making G an axiom in T to make it an extension? T remains consistent with the addition of G.
When T has no axioms, T is guaranteed to be consistent. When you add the first axiom to T, T is still guaranteed to be consistent. The question is, how do you add a second axiom to T without violating consistency? If you can solve this one, then the rest is just a matter of induction unless your theory is not countably axiomatizable. This is provable because the verification of consistency is done recursively on every addition of a new axiom to T.
Does that give you enough faith in T being consistent when T is finitely axiomatizable?
I know, I know, Godel is always right! T can never be complete and consistent. Besides, if uncountability exists, then T being complete can never be finitely axiomatizable! How do you check infinitely many axioms? You can't even write them down! ;-)
Well, you cannot have an uncountable number of threads anyway. That kind of mathematics does not exist today.
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