Re: A question about Caontor's proof of the uncountability of the reals

Dave Seaman wrote:

On Mon, 27 Feb 2006 06:06:23 GMT, Michael Olea wrote:

This, I think, misconstrues my post. I have never had any trouble
understanding Cantor's diagonalization proof: assume the reals are
countable, show that this assumption leads to a contradiction, conclude
that therefore that the assumption was false. This all makes perfect
sense. It is only when the law of the excluded middle comes under fire -
something I've only given any attention recently - that there may be room
for seeds of doubt. Are there conditions under which (P) and (not P) are
not the only possibilities, and therefore establishing that not P is
false need not imply the P is true? So the real question was not about
Cantor's proof of the uncountability of the reals per se, but about the
validity of proof by contradiction in axiomatic systems where Godel's
incompletenes theorems apply.

Are you aware that direct proofs exist?

No, I was unaware of that.

Let f: N -> R be an arbitrary mapping. Show that there exists x in R
that is not in the range of f, and hence f is not a surjection. Since f
is arbitrary, we conclude that no surjection exists.

Cool. Do you know of a link to such a proof, or if it's not too involved do
you mind fleshing out how to show there exist x in R that is not in the
range of f?

This is a direct proof. There is no contradiction in sight, and no
reliance on excluded middle. Does this answer your objection?

It does, but my objection was faulty anyway. I suspected as much (which is
why I asked "what is wrong with this argument"), but I wanted to be clear
just how it was faulty. Jesse F. Hughes' comments cleared it up for me. So
the fact that there are direct proofs does not answer what I was really
asking (even though it removes the objection), but it is good to know, and
I would love to see one of these worked out in detail.

-- Michael


.

Relevant Pages

  • Re: A question about Caontors proof of the uncountability of the reals
    ... understanding Cantor's diagonalization proof: assume the reals are ... countable, show that this assumption leads to a contradiction, conclude ... Cantor's proof of the uncountability of the reals per se, ... Does this answer your objection? ...
    (sci.math)
  • Re: Review of Mueckenheims book.
    ... >> proof by contradiction is wrong ... It is said that Euclid assumed ... Cantor assumed a complete list of reals and showed the existence ... I do not know of such an axiom. ...
    (sci.math)
  • Cantors diagonal proof wrong?
    ... When I was shown Cantor's diagonal proof that the number of reals was not ... infinity, you are making some basic assumptions on what an idea is. ... of the size of an infinite set is a contradiction in itself). ... So you just reverse the digits in the integer to create the real. ...
    (sci.math)
  • Re: infinitely many nns = infinite nns?
    ... proofs, one which shows that there is a FINITE distance (or difference, ... if you must) between any two reals on the line segment (which is why I ... What "infinite sum of finite distances"? ... We are still waiting for you to present the contradiction. ...
    (sci.logic)
  • Re: Anti-diagonalist page
    ... >than the cardinality of the naturals. ... >On the assumption that "the cardinality of the reals is greater than ... there still is that objection. ...
    (sci.logic)
Quantcast