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

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 apoorv Posts: 53 Registered: 4/11/13
Re: Ordinals describable by a finite string of symbols
Posted: Jul 23, 2013 11:11 AM
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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:
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> >> On 7/21/2013 1:40 PM, apoorv wrote:
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> >>
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> >>> I needed some clarification on Godel Numbering . I had asked it earlier .
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> >>
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> >>> Maybe I have more luck this time.
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> >>
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> >>>
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> >>> https://groups.google.com/forum/m/#!topic/sci.logic/dFK-EENfh6w
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> >>
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> >>> -Apoorv
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> >>
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> >>>
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> >>
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> >>
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> >> If I am reading your notation correctly,
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> >> g(x)= Goedel number of 'x'
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> >> Actually, your notation confuses me (due
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> >>
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> >> to relative lack of recent experience).
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> >> To stipulate something along the lines
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> >>
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> >> of
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> >> g(1)=godel number of f(1,w)
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> >>
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> >> g(2)=godel number of f(2,w)
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> >>
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> >> g(3)=godel number of f(3,w) etc
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> >>
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> >>
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> >> would seem to be
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> >> g(1)= g(f(1,w))
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> >>
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> >> g(2)= g(f(2,w))
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> >> g(3)= g(f(3,w))
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> >>
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> >>
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> >>
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> >> which would seem to violate the idea that
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> >> the Goedel numbering corresponds with a
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> >> unique naming of symbols.
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> >> Now, if your countable language is indexed
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> >> by the natural numbers and the argument to
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> >> 'g' is the index of the given formula, then
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> >> the numerals on the left have no relation
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> >> to the numerals on the right. In that case,
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> >> the correspondence of your listing would have
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> >>
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> >> to be thought as accidental.
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> >
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>
> 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,
2x-1 is the code the formula u=x^2
Or x is the code of formula 4u=(2x-1)^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 =(2x-1)^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

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

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