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

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 Peter Percival Posts: 2,623 Registered: 10/25/10
Re: The Invalidity of Godel's Incompleteness Work.
Posted: Oct 18, 2013 7:34 AM
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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.
>
> 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.
>
> Note: cGC <-> "There are infinitely many counter examples of Goldbach
> Conjecture".
>
> Any constructive response would be welcomed.

For PA, Gödel's incompleteness theorem says that, if PA is omega
consistent, then there is a sentence G, say, in the language of PA such
that neither PA |- G nor PA |- ~G. If that is invalid then there is no
such G, so, in particular one of cGC and ~cGC is a theorem of PA.

Rosser's strengthening of Gödel's incompleteness theorem omits "omega".

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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

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