Scott
Posts:
66
Registered:
2/2/07
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Re: New essay on Goedel's Incompleteness Theorem
Posted:
Sep 30, 2009 9:26 PM
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On Sep 30, 5:23 pm, stevendaryl3...@yahoo.com (Daryl McCullough)
wrote:
> Scott H says...
>
> >1. Every statement has a Goedel number.
> >2. If a statement is 'about' a number, then its Goedel number is also
> >'about' that number.
>
> That doesn't make any sense. What is the number 4 about?
4 isn't the Goedel number of a statement that is about anything.
> >3. Some statements are about Goedel numbers.
> >4. Therefore, there are Goedel numbers that are about Goedel numbers.
>
> I don't think it makes any sense to say that a Godel number is "about"
> anything. In any case, I don't see where you are getting an infinite
> sequence of G, G', G'', etc.
I say that a Goedel number is about something when its statement is
about something. The infinite sequence arises when the substitution or
'arithmoquine' function is applied repeatedly.
> G could be said to be "about" its Godel number, since it says
> that its Godel number is not the Godel number of a provable sentence.
> But then G' is the Godel number of G, so G' is "about" whatever G is
> about. So there is no G'', G''', etc.
I'll be more accurate: G' is about G'', the Goedel number of G', which
G is about. As I told Aatu, I'll leave it open whether G, G', G'', ...
are all basically the same statement.
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