Re: Cardinality of Real Numbers


Martin Shobe 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.

As with the diagonal proof, the proof sketched there does not
require the contradiction form. There is no reason to assume,
as the first line states, "some sequence x1, x2, x3, ... has
all of R as its range."

Replace that with: "Let x1, x2, x3,... be a sequence of reals".

The proof shows that there exists c not in the sequence. Thus
the original sequence, which had no restrictions put on it,
is incomplete. Therefore there does not exist a sequence
which includes all of the reals.

- Randy

.

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: abundance of irrationals!)
    ... All I know is that what we know about infinite ... > the sets I call finite have larges members. ... The set of all finite naturals is not infinite, ... Sets defined by mapping functions from the naturals to the reals which have ...
    (sci.math)
  • Re: abundance of irrationals!)
    ... >> less than sqrthas no largest member. ... > The set of all finite naturals is not infinite, ... >> I WILL claim that your incomplete definition of cardinality ... > Sets defined by mapping functions from the naturals to the reals ...
    (sci.math)
  • Re: abundance of irrationals!)
    ... the sets I call finite have larges members. ... I WILL claim that your incomplete definition of cardinality ... >> the naturals ... > Yes a function from the naturals to the reals, ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... If a quantitative set is mapped in ascending order from the naturals, with each increment in the domain, the range increases by some amount. ... Like it's the number of unit intervals, and the number of reals in the unit interval. ... You are using a form of infinite induction, making a claim for an infinite set based on all finite initial segments of it. ... don't have a definition for an arbitrary set of its "standard ordering" ...
    (sci.math)