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 '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.

