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Topic: Definition of Mathematics
Replies: 62   Last Post: Mar 6, 2012 1:22 PM

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 David Yen Posts: 630 Registered: 3/1/11
Re: Definition of Mathematics
Posted: Feb 29, 2012 1:09 PM

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.

Date Subject Author
2/27/12 Zaljohar@gmail.com
2/27/12 Jim Burns
2/28/12 Zaljohar@gmail.com
3/3/12 Charlie-Boo
2/27/12 LudovicoVan
2/28/12 Shmuel (Seymour J.) Metz
2/28/12 Tony Bones Orlow
2/29/12 MoeBlee
2/29/12 LudovicoVan
2/29/12 microm2011@hotmail.com
2/29/12 LudovicoVan
2/29/12 MoeBlee
2/29/12 LudovicoVan
3/3/12 Charlie-Boo
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2/27/12 namducnguyen
2/28/12 Frederick Williams
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2/27/12 Tony Bones Orlow
2/28/12 Frederick Williams
2/28/12 Tony Bones Orlow
2/28/12 Ki Song
2/28/12 Tony Bones Orlow
2/28/12 Ki Song
2/29/12 Frederick Williams
2/29/12 Frederick Williams
2/29/12 MoeBlee
3/3/12 Charlie-Boo
3/4/12 Tony Bones Orlow
2/27/12 ruffin
2/28/12 Zaljohar@gmail.com
2/28/12 Zaljohar@gmail.com
2/28/12 Zaljohar@gmail.com
2/28/12 G. A. Edgar
2/28/12 G. A. Edgar
3/3/12 Charlie-Boo
3/3/12 namducnguyen
2/28/12 Frederick Williams
2/28/12 Zaljohar@gmail.com
2/28/12 David Yen
2/28/12 David Yen
2/28/12 ruffin
2/29/12 David Yen
3/1/12 ruffin
3/1/12 David Yen
3/1/12 David Yen
3/4/12 David Yen
3/4/12 David Yen
2/29/12 Jim Burns
3/2/12 Zaljohar@gmail.com
3/4/12 Jim Burns
3/4/12 Tony Bones Orlow
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