```Date: Nov 14, 2012 2:12 AM
Author: Zaljohar@gmail.com
Subject: Re: Cantor's first proof in DETAILS

On Nov 14, 12:45 am, "LudovicoVan" <ju...@diegidio.name> wrote:> "Zuhair" <zaljo...@gmail.com> wrote in message>> news:3929e6b6-2932-401d-ba0a-0a440bb18277@y6g2000vbb.googlegroups.com...> On Nov 13, 11:16 pm, Uirgil <uir...@uirgil.ur> wrote:>> <snip>>> >> Your alleged argument against the Cantor proof does not work against> >> either Cantor's proof, nor Zuhair's proof, nor my proof for that matter,> >> since your N* is irrelevant for all of them.>> > I showed in the Corollary that even if he use N* as the domain of> > (x_n), still we can prove there is a missing real from the range of> > (x_n). So Cantor's argument or my rephrasing of it both can easily be> > shown to be applicable to N* (any set having a bijection with N) as> > well as N.>> You are simply missing the point there: we don't need N* to disprove Cantor,> we need N* to go beyond it and the standard notion of countability.  In> fact, that there is a bijection between N* and N is a bogus argument too, as> the matter is rather about different order types.>> -LVNow I think I'm beginning to somewhat perhaps understand yourargument. I think (I'm not sure though) that what you want to say isthat when we are having arguments with "LIMITS" then we must designthe whole argument such that the Limit comes from the sequence, and ifthis design was not made then the argument is inherently deficient asfar as the truth of inferences derived from it is concerned. So whatyou are trying to say is that Cantor's argument began with incompletearsenal so it ended up with misleading inferences. You are making anargument at TRUTH level of the matter, and yet it is concerned withformal technicality as well, which is an argument beyond the strictformal technicality.Anyhow if I'm correct, this form of reasoning for it to stand thequest, then there must be a clear line of justification for it. Forinstance the argument about whether the reals are countable actuallymeans literally whether there is a bijection between the reals and N,so N is at the heart of the subject. Now to go and say thatcountability of the reals (which means bijectivity of reals to N) canonly be reached about by circumventing N and using another countableinfinite set N* as the domain for any sequence in an argument usinglimits is really strange somehow.What you are having is the following:[1]When we use N as the domain of injections (x_n), (a_n) and (b_n),then Cantors argument PROVES and SHOWS that there is a real that isnot in the range of those functions.[2]When we use N* as the domain of injections (x_n), (a_n) and (b_n),then Cantor's argument will seize from working in the same way to showthe missing real.[3]However we also have the corollary that even when we use N* as thedomain of those functions, still we can by a single common welldefined way define another sequence with exactly the same range ofthose functions but from domain N, and we can apply Cantor's argumentand SHOW a missing real in the rang of those functions!Now you call [1] deficient, [2] apt to reality standards [3] bogus.Why? because we used N in an argument that involves a higher orderconcept that must use N* instead. (That's your reply).But again: why? what is the higher order part of the argument that yousee it demanding circumventing the heart of the subject (which is Nreally) to some N*.Is it the definition of Limit.But limit is defined in this argument as the least upper bound, and Idon't see in the definition of L that I wrote (which is the standardby the way) anything that has to do with necessarily picking it upfrom some Omega_th end point? that has no meaning at all, so why?Should I adopt this rational of yours then I'd ask you: why not saypick L from the -1_th starting point. i.e. choose your domain to be{-1,0,1,2,3,...} since this clearly also preclude Cantor's argumentand you clearly can make L be the -1_th digit of (x_n) [Remember a_0is x_0, so x_{-1} lies "before" a_0].Or you'll say that {-1,0,2,3,...} is also a kind of high ordercountable set?Your argument is simply shunning one of the most important two sets inthis argument, that is N, and using some replacement, without anyclear justification.Zuhair
```