Scott
Posts:
66
Registered:
2/2/07
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Re: New essay on Goedel's Incompleteness Theorem
Posted:
Sep 30, 2009 5:19 PM
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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
'about' that number.
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.
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