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

 Messages: [ Previous | Next ]
 apoorv Posts: 53 Registered: 4/11/13
Re: Ordinals describable by a finite string of symbols
Posted: Jul 23, 2013 12:07 PM

On Tuesday, July 23, 2013 8:41:54 PM UTC+5:30, apoorv wrote:
> On Tuesday, July 23, 2013 4:22:46 AM UTC+5:30, fom wrote:
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> > On 7/22/2013 3:33 PM, apoorv wrote:
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> >
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> > > 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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> > >>
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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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> > >> If I am reading your notation correctly,
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> > >>
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> > >> g(x)= Goedel number of 'x'
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> > >> Actually, your notation confuses me (due
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> > >> to relative lack of recent experience).
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> > >> To stipulate something along the lines
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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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> > >> would seem to be
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> > >> g(1)= g(f(1,w))
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> > >> g(2)= g(f(2,w))
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> > >> g(3)= g(f(3,w))
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> > >> which would seem to violate the idea that
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> > >>
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> > >> the Goedel numbering corresponds with a
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> > >>
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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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> > >> to be thought as accidental.
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> > Did I miss a reply somewhere?
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> I would like to clarify this with reference to the formula sub(x,x,y) ,which is central to the fixed point lemma.
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> Suppose we have some coding for formulas with one free variable u .
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> We let
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> 1 equal Code of formula u= 1^2,
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> 3 equal code of formula u=2^2
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> 5equal code of formula u =3^2
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> Etc
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> So we generalise and say,
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> 2x-1 is the code the formula u=x^2
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> Or x is the code of formula 4u=(2x-1)^2 , where the brackets are for ease of writing.
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> Now sub (x,x,y) says
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> y is the code of formula obtained if we substitute x for the free variable in the formula whose code is x.
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> Or
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> y is the code of the formula 4x =(2x-1)^2
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> Now the important thing is that
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> sub(x,x,y) is a formula in which x is not free
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> And therefore, if k is the code of sub(x,x,y), then
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> Sub(K,k,y) is not the result of substituting k for the free variable x in sub (x,x,y).
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> Alternately ,if we regard sub(x,x,y) as a convenient shorthand , but not a generalisation of the
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> Many formulae
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> sub(1,1,y),
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> sub(2,,2,y)Etc
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> Then sub(x,x,y) really i will have a code which is not a number but a function of x.
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> I may appear to be thoroughly confused but my own thinking has not helped me to
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> Dispel the confusion.
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> -Apoorv

I think I can put the issue more sharply.
The code of any formula is a number.
Now sub(x,x,y) says
Y is the code of [the formula obtained when we replace the free variable in the formula whose
Code is x by x]
So sub(x,x,y) is equivalent to " y is some number"
So sub(x,x,y) does not have x as a free variable.
On the other hand ,if we accept that x is indeed free in sub(x,x,y), then it is like
Saying that the code of a formula in which x is a free variable is not a number but some function of x or some formula with x free.
-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