apoorv
Posts:
53
Registered:
4/11/13


Re: Ordinals describable by a finite string of symbols
Posted:
Jul 23, 2013 11:11 AM


On Tuesday, July 23, 2013 4:22:46 AM UTC+5:30, fom wrote: > On 7/22/2013 3:33 PM, apoorv wrote: > > > On Monday, July 22, 2013 1:23:53 AM UTC+5:30, fom wrote: > > >> On 7/21/2013 1:40 PM, apoorv wrote: > > >> > > >>> I needed some clarification on Godel Numbering . I had asked it earlier . > > >> > > >>> Maybe I have more luck this time. > > >> > > >>> > > >> > > >>> https://groups.google.com/forum/m/#!topic/sci.logic/dFKEENfh6w > > >> > > >>> Apoorv > > >> > > >>> > > >> > > >> > > >> > > >> If I am reading your notation correctly, > > >> > > >> > > >> > > >> g(x)= Goedel number of 'x' > > >> > > >> > > >> > > >> Actually, your notation confuses me (due > > >> > > >> to relative lack of recent experience). > > >> > > >> > > >> > > >> To stipulate something along the lines > > >> > > >> of > > >> > > >> > > >> > > >> g(1)=godel number of f(1,w) > > >> > > >> g(2)=godel number of f(2,w) > > >> > > >> g(3)=godel number of f(3,w) etc > > >> > > >> > > >> > > >> > > >> > > >> would seem to be > > >> > > >> > > >> > > >> g(1)= g(f(1,w)) > > >> > > >> g(2)= g(f(2,w)) > > >> > > >> g(3)= g(f(3,w)) > > >> > > >> > > >> > > >> which would seem to violate the idea that > > >> > > >> the Goedel numbering corresponds with a > > >> > > >> unique naming of symbols. > > >> > > >> > > >> > > >> Now, if your countable language is indexed > > >> > > >> by the natural numbers and the argument to > > >> > > >> 'g' is the index of the given formula, then > > >> > > >> the numerals on the left have no relation > > >> > > >> to the numerals on the right. In that case, > > >> > > >> the correspondence of your listing would have > > >> > > >> to be thought as accidental. > > > > > > > Did I miss a reply somewhere?
I would like to clarify this with reference to the formula sub(x,x,y) ,which is central to the fixed point lemma. Suppose we have some coding for formulas with one free variable u . We let 1 equal Code of formula u= 1^2, 3 equal code of formula u=2^2 5equal code of formula u =3^2 Etc So we generalise and say, 2x1 is the code the formula u=x^2 Or x is the code of formula 4u=(2x1)^2 , where the brackets are for ease of writing. Now sub (x,x,y) says y is the code of formula obtained if we substitute x for the free variable in the formula whose code is x. Or y is the code of the formula 4x =(2x1)^2 Now the important thing is that sub(x,x,y) is a formula in which x is not free And therefore, if k is the code of sub(x,x,y), then
Sub(K,k,y) is not the result of substituting k for the free variable x in sub (x,x,y). Alternately ,if we regard sub(x,x,y) as a convenient shorthand , but not a generalisation of the Many formulae sub(1,1,y), sub(2,,2,y)Etc Then sub(x,x,y) really i will have a code which is not a number but a function of x. I may appear to be thoroughly confused but my own thinking has not helped me to Dispel the confusion. Apoorv

