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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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 namducnguyen Posts: 2,777 Registered: 12/13/04
Re: The Invalidity of Godel's Incompleteness Work.
Posted: Oct 18, 2013 5:16 PM

On 18/10/2013 5:34 AM, Peter Percival wrote:
> 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.

What definition of "invalidity" were you referring to _here_ ? Mine?
>
> Rosser's strengthening of Gödel's incompleteness theorem omits "omega".

So in his theorem, Rosser did _not_ include the assumption something
_similar to my_ "adequate enough to describe the concept of the natural
numbers" for his underlying T?

--
-----------------------------------------------------
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