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fom
Posts:
1,969
Registered:
12/4/12


Re: Prime Interger Topology
Posted:
Oct 21, 2013 7:52 AM


On 10/21/2013 3:24 AM, William Elliot wrote: > On Sun, 20 Oct 2013, fom wrote: >> On 10/20/2013 9:58 PM, William Elliot wrote: >>> >>> Why relations? Aren't functiosn used to manage the undue complexity of >>> relations? >> >> This is the pair coding function from Goedel's >> lemma in Kaye's "Models of Peano Arithemtic" >> >> < x, y > = [ ( x + y )( x + y + 1) / 2 ] + y >> >> Given any "definite" model of my axioms, I >> could collapse the relations into sets of >> numbers arithmetically. >> >> Those sets probably could not be represented recursively. > > (0,0) = 0 > (x+1, 0) = (x + 1)(x + 2)/2 = x(x + 1)/2 + x + 1 = (x,0) + x + 1 > > (x, y+1) = (x + y + 1)(x + y + 2)/2 + y + 1 > = (x + y)(x + y + 1)/2 + x + y + 1 + y + 1 > = (x,y) + x + y + 2 >
Oh! The pairing function could.
I meant the codings on the relations in the theory.
Given a definite model, those relations would be given definite values. Applying the coding to the relations would map each relation back into the natural numbers.
It is the set of such relations to which I refer.



