Re: Zenkin's paper on Cantor

From: Eray Ozkural exa (erayo_at_bilkent.edu.tr)
Date: 10/07/04 

Date: 7 Oct 2004 01:51:36 -0700

daryl@atc-nycorp.com (Daryl McCullough) wrote in message news:<ck26jh030rb@drn.newsguy.com>...
> Eray Ozkural exa says...
> >
> >Ralph Hartley <hartley@aic.nrl.navy.mil> wrote
>
> >> If you give me a list of real numbers, presented in that way, I can give
> >> you a number not on your list.
> >
> >A timely observation which takes us to the heart of the matter. I will
> >argue that I cannot "give you a list".
>
> You can give me a *procedure* which, given a number n
> returns the nth real number (between 0 and 1, for simplicity),
> which in turn is a procedure which, given a number m, returns
> the mth bit of the binary expansion of that real.

Yes, of course, Daryl, that is quite obvious. I can for instance use a
regular subdivision procedure which, given a number n, returns the
number "0.binary(n)".

I can also guarantee that the nth number has at least n digits after
the (binary) point (I think this is easily done). But the above method
doesn't hold then, you have to use something like "0.<n 0's>1".

Now, giving the list in this fashion would be potential infinity,
since it is compressed in a finite nonhalting program. My post is in
error, the second proof also can be said to be valid from a
constructivist perspective, if the reals chosen are deliberately taken
to be computable in exactly this way.

I think this way, the essence of the proof is actually demonstrated
better; note the similarity of my representation above to the fact
that the measure of the countable reals in (0,1) is 0. (but of course
different representations are possible)

I am pleased with this interpretation of the proof. The later steps of
the proof can be merged in a nonhalting procedure which uses the
generator above, the reasoning part is still infinitary, but I think
it is inductively true (I can try to explain this part if it doesn't
make sense to you). Now, everything is explained bottom-up, when we
are dealing only with a *computable* list of computable reals. (The
actual infinity remains in the continuum, the proof has nothing to do
with it.)

But I do not think that Cantor's original proof is in fact
constructive in the sense of *constructivism*. (I'm aware that a
realist might call it constructive!)

Regards, 

--
Eray Ozkural

Relevant Pages

  • Re: Zenkins paper on Cantor
    ... Now, giving the list in this fashion would be potential infinity, ... note the similarity of my representation above to the fact ... that the measure of the countable reals in is 0. ... constructive in the sense of *constructivism*. ...
    (sci.math)
  • Re: Infinity and indefinite extensibility
    ... infinity is the upper bound. ... It's value is not defined in the real numbers but establishes a boundary for the set of reals. ... The limits of a finite computer only determine the number of beats one gets before one runs out of representation. ...
    (comp.databases.theory)
  • Re: Computable functions/reasls: followup.
    ... The computable-function definition above still applies, ... Russian-style constructivism is BISH + MP, ... which were reals that you couldn't tell whether or not were rational. ... special about the rationals; they could be replaced by the integers, ...
    (sci.logic)
  • Re: how to list all of the real numbers
    ... those with the undecideable continuum hypothesis, yet none of them, ... continuum's cardinality is thus simply equivalent to a lesser cardinal ... does not hold where the reals are a set. ... Infinity in numbers does actually exist. ...
    (sci.math)
  • Re: how to list all of the real numbers
    ... does not hold where the reals are a set. ... could be used as a proof for the existence of limit ordinals. ... concept of "adjacent points" is provably inconsistent in geometry. ... and the infinity of the codomain. ...
    (sci.math)