|
|
Re: Cantor's Proofs
Posted:
Oct 16, 2011 12:14 PM
|
|
On Oct 16, 4:08 pm, Dick <DBatche...@aol.com> wrote:
.............................................................
>
> > The definition of uncountable is unlistable.
> > So to a constructivist the computable numbers
> > are uncountable. (They are subcountable but
> > this is a different thing).
>
> > - William Hughes- Hide quoted text -
>
> The topic of Cantor's Second proof always raises a great deal of
> interest. Unfortunately, it also raises a great deal of heat. I think this is largely caused by the limited vocabulary available to those taking the "classical" view.
> Coming from recursive function theory I can say "The set of computable
> reals is not recursively-enumerable but is Goedelizable." Tjis, I
> think is clear. Things are much more difficult for the classisist. The
> only words available are "enumerable" and "uncountable."
**** And this is all is needed to understand the really easy diagonal
proof of Cantor's Theorem. You can use Goedelizable, recursively en.
and the lot just I can use a guitar and a piano to give my argument;
anyway, we don't need goedizable and recursion and guitar and a piano
and stuff to fully understand Cantor's proof.****
He ants to
**** Who what? ****
> say that the computable reals cannot be listed but are enumerable (Goedelizable) He may say 'The computable reals are noy listable but are enumerable." However, "listable" is a synonym for "enumerable" so he ends up saying "The computable reals are bot enumerable but are enumerable." Whish looks like nonsense.
**** No, it doesn't. It though looks like a very boring tautology. You
wrote "The reals are bot [sic] A but are A", with A = enumerable. I
guess you meant one of the above was "not A". ****
>
> Though the statement is paradoxical I think there is some wisdom in it.
**** Perhaps but it is highly doubtable that it will yield some
fruitful idea within mathematics. A statement which contains both a
claim and its negation is logically a contradiction and it doesn't
show nicely within mathematics. ****
In physics, an electron must sometimes be regarded as a wave,
> sometimes as a particle. The physisist has no difficulty with the statement "An electron is not a wave but is a wave."
**** I think they'd do and I don't think this is what they say. I
think that what they say is more or less like "an electron is
something whose exact nature we can't still define but sometimes we
succeed to measure/probe/check/"see" it as a wave and sometimes as a
particle, and the electron shows characteristics of both these
features."
In general I don't think it is a very good idea to deduce stuff in
mathematics from stuff we do in physics. We very well may be pretty
wrong, though there's hardly any doubt that physics is an excellent
must to inspire work within mathematics. ****
A productive set
> (such as the set of constructable reals) has the same ambiguity.
> Sometimes it is countable, but look at it in a different way and it is
> noy.
> I think a productive set is rather like a petilant rwo-year old.
> Unless you are very careful in the phrasing, the answer to any
> question will be an angry "NO!!!"
> Many of the properties of a productive set are counter intuitive. It
> is no wonder that the use of a productive set in Cantor's Second proof
> has raised uneasiness in generations of students and continues to
> raise contriversy. Would it not be better to teach Cantor's First
> proof, which is free from "productive sets" and has the additional
> virtue that it emphasises the "completeness" of the Reals, a
> distinguishing feature shared with no other field?
**** There are other fields (lots of, in fact) that are complete: the
reals, the complex, the p-adic fields, for any prime p...
Tonio ****
>
> Dick-
|
|
