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Topic: New essay on Goedel's Incompleteness Theorem
Replies: 123   Last Post: Oct 1, 2009 11:56 PM

 Messages: [ Previous | Next ]
 Scott Posts: 66 Registered: 2/2/07
Re: New essay on Goedel's Incompleteness Theorem
Posted: Sep 30, 2009 5:19 PM

On Sep 30, 4:26 pm, stevendaryl3...@yahoo.com (Daryl McCullough)
wrote:
> Scott H says...
> > If
> > statements can be about Goedel numbers, then Goedel numbers can be
> > about Goedel numbers.

>
> I can't make any sense of that.

I'll try to simplify it:

1. Every statement has a Goedel number.
2. If a statement is 'about' a number, then its Goedel number is also
3. Some statements are about Goedel numbers.
4. Therefore, there are Goedel numbers that are about Goedel numbers.

[...]

> 3. There is a provability predicate Pr for Peano Arithmetic with the
> property that for any formula Phi,
> If Phi is provable from the axioms of Peano Arithmetic
> then
> Pr(#Phi) is true
> Conversely, if Phi is not provable from the axioms of
> Peano Arithmetic, then
> ~Pr(#Phi) is true.

More accurately, if x proves Phi, then the formula Pr(x, Phi) is
provable, and if x doesn't prove Phi, then the formula ~Pr(x, Phi) is
provable.

[...]

> 5. Applying 4 to the formula ~Pr, we have
>
> G <-> ~Pr(#G)
>
> So, what is the G', G'', etc. that you are talking about?

I have called #G, G'. G'' comes in when we transform #G into the
Goedel number of an equivalent statement.

Date Subject Author
9/21/09 Bill97
9/21/09 george
9/21/09 Frederick Williams
9/21/09 Aatu Koskensilta
9/21/09 LauLuna
9/21/09 Scott
9/24/09 Frederick Williams
9/24/09 Scott
9/21/09 Scott
9/22/09 Daryl McCullough
9/21/09 Newberry
9/22/09 Guest
9/22/09 Newberry
9/22/09 Scott
9/22/09 Scott
9/22/09 Newberry
9/22/09 Scott
9/22/09 Newberry
9/22/09 namducnguyen
9/22/09 Scott
9/22/09 namducnguyen
9/23/09 Daryl McCullough
9/24/09 LauLuna
9/24/09 Newberry
9/25/09 Scott
9/22/09 LauLuna
9/22/09 Scott
9/23/09 Scott
9/23/09 Scott
9/23/09 Scott
9/24/09 Scott
9/24/09 Newberry
9/24/09 namducnguyen
9/27/09 Newberry
9/26/09 Scott
9/26/09 David C. Ullrich
9/26/09 Scott
9/26/09 Frederick Williams
9/26/09 Scott
9/27/09 Frederick Williams
9/26/09 namducnguyen
9/26/09 namducnguyen
9/26/09 Marshall
9/26/09 namducnguyen
9/26/09 namducnguyen
9/26/09 Marshall
9/27/09 namducnguyen
9/27/09 Marshall
9/27/09 namducnguyen
9/27/09 namducnguyen
9/27/09 Marshall
9/27/09 Scott
9/27/09 Scott
9/27/09 Scott
9/28/09 Daryl McCullough
9/28/09 David C. Ullrich
9/29/09 Scott
9/29/09 Daryl McCullough
9/29/09 Aatu Koskensilta
9/29/09 Newberry
9/27/09 george
9/27/09 george
9/28/09 Scott
9/28/09 namducnguyen
9/29/09 Frederick Williams
9/29/09 Aatu Koskensilta
9/28/09 Scott
9/28/09 ross.finlayson@gmail.com
9/29/09 Marshall
9/29/09 Aatu Koskensilta
9/29/09 Marshall
9/30/09 Frederick Williams
9/30/09 Marshall
9/30/09 Marshall
10/1/09 Marshall
9/30/09 Marshall
9/30/09 Marshall
9/29/09 Scott
9/30/09 namducnguyen
9/30/09 Scott
9/30/09 Scott
9/30/09 Scott
9/30/09 Scott
9/30/09 Tim Little
10/1/09 Scott
10/1/09 Daryl McCullough
10/1/09 David C. Ullrich
10/1/09 Scott
10/1/09 David C. Ullrich
10/1/09 Marshall
10/1/09 Scott
10/1/09 Tim Little
9/30/09 Scott
9/30/09 Daryl McCullough
9/30/09 Scott
9/29/09 Newberry
9/23/09 Scott
9/23/09 Newberry
9/24/09 Frederick Williams
9/24/09 namducnguyen
9/24/09 Aatu Koskensilta
9/24/09 namducnguyen
9/24/09 namducnguyen
9/24/09 namducnguyen
9/24/09 namducnguyen
9/24/09 ross.finlayson@gmail.com
9/23/09 John Jones
9/24/09 LauLuna
9/24/09 Newberry
9/27/09 george
9/30/09 Scott
9/30/09 Scott
9/30/09 Daryl McCullough
9/30/09 Scott
9/30/09 Daryl McCullough
9/30/09 David Libert
9/30/09 Tim Little
9/30/09 ross.finlayson@gmail.com
9/30/09 Marshall
10/1/09 Marshall
10/1/09 Daryl McCullough
9/30/09 Scott
10/1/09 Daryl McCullough