Date: Nov 12, 2012 4:50 AM
Author: Zaljohar@gmail.com
Subject: Re: Cantor's first proof in DETAILS
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