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Topic: The Invalidity of Godel's Incompleteness Work.
Replies: 87   Last Post: Oct 25, 2013 2:44 PM

 Messages: [ Previous | Next ]
 Peter Percival Posts: 2,623 Registered: 10/25/10
Re: The Invalidity of Godel's Incompleteness Work.
Posted: Oct 5, 2013 9:05 AM

Nam Nguyen wrote:
> Two major theorems Godel's Incompleteness are:
>
> - Incompleteness: Any formal system T that is consistent _and_ adequate
> enough to describe the concept of the natural numbers, would have G(T)
> as a statement that is true but not provable in T.
>
> - Completeness: Any consistent formal system has to have a model.
>
> On the Incompleteness, since the requirement that T be _informally_
> adequate enough to describe the concept of the natural numbers is _not_
> a syntactical notion [as that of a T's consistency], it's logically
> invalid to assume that T always be syntactically consistent, simply
> because we _informally assume_ T adequately describe the concept of
> the natural numbers. QED.

Gödel's first published proof of his incompleteness theorem was entirely
syntactical. It described a formula, which I'll call G, such that
neither G nor ~G is a theorem of a theory he called P, just so long as P
is omega consistent. Do you think that proof is invalid? If so, where
in that proof does the first error occur?

> On the Completeness, since it's still entirely logically possible that
> it's impossible to know the truth value the formula cGC, or ~cGC,
> (defined below) in the natural numbers, it's still entirely logically
> possible that it's impossible to have a model for PA + cGC, or PA +
> ~cGC. Hence it's logically invalid to assert in meta level that a
> consistent formal system must necessarily have a model.

That a consistent set of first order formulae has a model in proved in
texts on logic, including Shoenfield's that you are fond of mentioning.
Where in Shoenfield's text does the first error occur?

> Note: cGC <-> "There are infinitely many counter examples of Goldbach
> Conjecture".
>
> Any constructive response would be welcomed.

Really?

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Date Subject Author
10/4/13 namducnguyen
10/5/13 Peter Percival
10/6/13 LudovicoVan
10/6/13 LudovicoVan
10/9/13 fom
10/18/13 Peter Percival
10/18/13 namducnguyen
10/19/13 Peter Percival
10/19/13 fom
10/19/13 Peter Percival
10/19/13 fom
10/19/13 namducnguyen
10/19/13 fom
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 fom
10/19/13 fom
10/20/13 namducnguyen
10/20/13 fom
10/20/13 namducnguyen
10/20/13 fom
10/20/13 fom
10/20/13 namducnguyen
10/20/13 fom
10/20/13 namducnguyen
10/20/13 fom
10/20/13 fom
10/20/13 namducnguyen
10/20/13 fom
10/20/13 namducnguyen
10/20/13 fom
10/19/13 namducnguyen
10/19/13 fom
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 fom
10/19/13 fom
10/19/13 namducnguyen
10/19/13 fom
10/19/13 fom
10/19/13 Peter Percival
10/19/13 namducnguyen
10/19/13 Peter Percival
10/20/13 namducnguyen
10/20/13 fom
10/20/13 namducnguyen
10/20/13 namducnguyen
10/20/13 fom
10/20/13 namducnguyen
10/20/13 fom
10/20/13 namducnguyen
10/20/13 fom
10/20/13 namducnguyen
10/20/13 fom
10/24/13 namducnguyen
10/24/13 fom
10/24/13 namducnguyen
10/24/13 Peter Percival
10/24/13 namducnguyen
10/24/13 Peter Percival
10/24/13 fom
10/24/13 fom
10/20/13 fom
10/25/13 Rock Brentwood