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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 ]
 namducnguyen Posts: 2,777 Registered: 12/13/04
Re: The Invalidity of Godel's Incompleteness Work.
Posted: Oct 20, 2013 2:55 AM

On 04/10/2013 9:47 PM, 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.

More to the point, on Incompleteness, the 2 assumptions

C1 = T is consistent.
C2 = T is adequate enough to describe the concept of the natural
numbers.

are distinct as well as necessary. But since it's impossible to
logically guarantee that if T satisfies the _informal_ notion of C2,
T would also satisfy the _formal_ notion of C1, Incompleteness is an
invalid theorem.

Note: by "formal" notion here, we'd mean a notion where the concept
of rules of inference is required, but also where the concept of the
natural numbers (or related notions such as recursion, induction,
language structure, etc...) is _NOT_ required.

>
> 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.
>

--
-----------------------------------------------------
There is no remainder in the mathematics of infinity.

NYOGEN SENZAKI

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