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.