Re: Cardinality of Real Numbers

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
.

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: Uncountable sets in CZF?
    ... a trivial surjection from R onto any subset of N there is a bijection. ... I don't base my arguments (that the reals and naturals are equivalent) ... from some proper subset of the naturals to the reals, ...
    (sci.math)
  • Re: Uncountable sets in CZF?
    ... It doesn't follow that there's a bijection. ... > naturals to the reals. ... What that means is that one of the reasons that people call the reals ... That implies it is not a mathematical fact and to promote the other ...
    (sci.math)
  • Re: Zenkins paper on Cantor (reply of Dr. Zenkin)
    ... What is "a function which is a bijection"? ... exists another function (algorithm), f^-1, with the property for all b ... well-defined (exists for all naturals n, m) and S is a bijection - ... of reals, S". ...
    (comp.theory)
  • Re: Zenkins paper on Cantor (reply of Dr. Zenkin)
    ... What is "a function which is a bijection"? ... exists another function (algorithm), f^-1, with the property for all b ... well-defined (exists for all naturals n, m) and S is a bijection - ... of reals, S". ...
    (sci.math)