
Re: Cantor's first proof in DETAILS
Posted:
Nov 12, 2012 4:50 AM


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 online 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

