Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Topic: Ordinals describable by a finite string of symbols
Replies: 24   Last Post: Jul 27, 2013 12:38 PM

 Messages: [ Previous | Next ]
 apoorv Posts: 53 Registered: 4/11/13
Re: Ordinals describable by a finite string of symbols
Posted: Jul 24, 2013 2:12 PM

On Saturday, July 20, 2013 11:12:40 PM UTC+5:30, Peter Percival wrote:
> apoorv wrote:
>
>
>

> >
>
> > I know; i was on threads dying out. Godels theorem gives a specific self referential sentence.
>
> > And all I wanted to know is what is its number per the coding used by him. But that is not to be
>
> > Found anywhere ,not in texts, not on the net, and not in this group. So I guess my question
>
> > Is Ilframed, maybe it is some uncomputable number,
>
>
>
> No, it is computable, but you would need to take his "this is not
>
> provable" statement, re-express it using primitive notation (i.e. get
>
> rid of all defined symbols) and then assign numbers to its constituents,
>
> and so on. Getting rid of the defined terms is already a daunting task.
>

If phi(x) is Ay[sub(x,x,y)--> ~provable formula with code y] and
Code of phi(x) is k
Then the self referential sentence sigma is
Phi(k) <--> ~provable phi(k) <-->~provable[~provablephi(k)]
Now for any number alpha, we can prove phi(alpha) then clearly alpha is not k.
On the other hand if k is indeed alpha, then we will never prove phi(alpha),
And therefore never prove that k is alpha.
We could ,for example always prove that 100 is not equal to k, but if k were indeed 100, we
Could never prove that.
This seems to go against the computability of the number k, and hence of the
Code of the self referential sentence.
-Apoorv

Date Subject Author
7/10/13 Aatu Koskensilta
7/10/13 David C. Ullrich
7/10/13 Sandy
7/10/13 fom
7/12/13 apoorv
7/15/13 fom
7/16/13 Shmuel (Seymour J.) Metz
7/19/13 apoorv
7/19/13 fom
7/20/13 apoorv
7/20/13 Peter Percival
7/20/13 apoorv
7/21/13 apoorv
7/21/13 apoorv
7/21/13 fom
7/22/13 apoorv
7/22/13 fom
7/23/13 apoorv
7/23/13 apoorv
7/24/13 apoorv
7/27/13 apoorv
7/27/13 fom
7/10/13 Aatu Koskensilta