Re: Cardinality of Real Numbers

In article <8pegh1t8h5c6k0q80hhstul6l18uakbq36@xxxxxxx>,
Martin Shobe <mshobe@xxxxxxxxxxxxx> wrote:

> On Thu, 01 Sep 2005 23:39:16 -0600, Virgil
> <ITSnetNOTcom#virgil@xxxxxxxxxxx> wrote:
>
> >In article <u7lfh19ci265649u3nurks2u3b1281csc6@xxxxxxx>,
> > Martin Shobe <mshobe@xxxxxxxxxxxxx> wrote:
> >
> >> >What do you mean: "Cantor's first requires the well-ordering be
> >> >order-equivalent to N?" Do you mean that to say that Cantor's first
> >> >applies to a bijection from N to R only, or what?
> >>
> >> Yes. Cantor's first assumes the existance of a bijection between the
> >> natural numbers and the reals. From this, a contradiction is reached
> >> by showing that there must be a real mapped to a natural number that
> >> is also mapped to a number larger than any natural number.
> >
> >As I read it, Cantor's first starts with an arbitrary injection from the
> >naturals to the reals, and shows that there is some real not in the
> >image of that injection. Thus no such injection can be a surjection. No
> >contradiction required.
> >
> >Many mathematicians, and I believe Cantor was one of them, did not much
> >like proofs by contradiction, and go to considerable lengths to avoid
> >them where possible. In this case no great lengths were required.
>
> This is where I got Cantor's first proof from
>
> http://en.wikipedia.org/wiki/Cantor%27s_first_uncountability_proof
>
> In this article, the proof is a proof by contradiction. As I don't
> have access to the originals, I can't tell you if that was actually
> Cantor's proof.
>
> Martin

Wikipedia sometimes gets things wrong. Its original remarks on Euclid's
proof of the infinitude of primes also mistakenly claimed that the proof
was by contradiction, but has since been corrected. Euclid's proof was
also a direct proof.

See. for example,
http://64.233.167.104/search?q=cache:7F2OFHlvlKIJ:www.fh-augsburg.de/~mue
ckenh/Infinity/MA2-040405.doc++%22Cantor%27s+first+proof%22&hl=en
for a rough version in English, which shows it as a direct proof.
.

Relevant Pages

  • Re: Real Discontinuity in Cantor Diagonal
    ... arrive at a contradiction. ... Assume that there is an infinite list of all reals ... We don't use "diagonalization" in this case. ... This can happen in the Naturals: each Natural that is not in the list ...
    (sci.logic)
  • Re: Cardinality of Real Numbers
    ... Cantor's first assumes the existance of a bijection between the ... >> natural numbers and the reals. ... From this, a contradiction is reached ... >naturals to the reals, and shows that there is some real not in the ...
    (sci.math)
  • Re: Cardinality of Real Numbers
    ... >>> natural numbers and the reals. ... From this, a contradiction is reached ... >>naturals to the reals, and shows that there is some real not in the ... The proof shows that there exists c not in the sequence. ...
    (sci.math)
  • Re: Cantor and the binary tree
    ... >> all the reals in this interval, and suppose further that we can ... >> This is a contradiction; therefore our hypothesis (that I can have ... > Indeed it shows that there are more numbers than digits, ... > more reals this way than naturals, it would seem, but it doesn't ...
    (sci.math)
  • Re: Cardinality of Real Numbers
    ... Cantor's first assumes the existance of a bijection between the ... > natural numbers and the reals. ... Cantor's first starts with an arbitrary injection from the ... naturals to the reals, and shows that there is some real not in the ...
    (sci.math)