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Re: Cantor's first proof in DETAILS
Posted:
Nov 12, 2012 4:50 AM
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On Nov 12, 12:05 pm, Zuhair <zaljo...@gmail.com> wrote:
> Apologies beforehand for this long proof, and for any possible errors,
> typos, mistakes that most possibly would be there with such a long
> draft. I'v written this with the intention to give what I think it to
> be the complete story of Cantor's first proof. So the following is my
> view of this proof, it came from reading on-line proofs other than the
> original one, since I don't have the original article of Cantor.
> References given below.
>
> If a mistake in this proof is noticed, then please feel free to
> outline it.
>
> CANTORS FIRST PROOF OF UNCOUNTABILITY OF REALS
> --------------------------------------------------------------------------- -----
>
> Statement: There is no bijection between the set N of all naturals
> and the set R of all reals.
>
> Proof:
> We prove that for every injection (x_n) from N to R, there
> exist a real J such that J not in the range of (x_n).
>
> Notation: for every x_i, i shall be called the place of x_i in (x_n),
> while x is the value of x_i. Whenever mentioned in this article
> symbols < , > , = and =/= are comparisons of the values of entries of
> sequences mentioned, while the places of those entries shall be
> compared by "lies before" , "lies after" , is the first entry, is the
> last entry, in the same place, etc..
>
> (x_n) is said to have the Intermediate Value property (IVP) iff
> for every two entries x_i,x_j of (x_n) there exist an entry x_k
> of (x_n) such that: x_i < x_k < x_j or x_i > x_k > x_j
A better statement would be:
for every two entries x_i, x_j of (x_n) where x_i < x_j
there exist an entry x_k of (x_n) such that: x_i < x_k < x_j.
Zuhair
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