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, reexpress 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

