Uirgil
Posts:
183
Registered:
4/18/12


Re: Cantor's first proof in DETAILS
Posted:
Nov 14, 2012 2:18 AM


In article <6a63fbfdf7e7458faf65fae2c805c951@d17g2000vbv.googlegroups.com>, Zuhair <zaljohar@gmail.com> wrote:
> On Nov 14, 12:45 am, "LudovicoVan" <ju...@diegidio.name> wrote: > > "Zuhair" <zaljo...@gmail.com> wrote in message > > > > news:3929e6b62932401dba0a0a440bb18277@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. > > > > LV > > Now I think I'm beginning to somewhat perhaps understand your > argument. I think (I'm not sure though) that what you want to say is > that when we are having arguments with "LIMITS" then we must design > the whole argument such that the Limit comes from the sequence, and if > this design was not made then the argument is inherently deficient as > far as the truth of inferences derived from it is concerned. So what > you are trying to say is that Cantor's argument began with incomplete > arsenal so it ended up with misleading inferences. You are making an > argument at TRUTH level of the matter, and yet it is concerned with > formal technicality as well, which is an argument beyond the strict > formal technicality. > > Anyhow if I'm correct, this form of reasoning for it to stand the > quest, then there must be a clear line of justification for it. For > instance the argument about whether the reals are countable actually > means 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 that > countability of the reals (which means bijectivity of reals to N) can > only be reached about by circumventing N and using another countable > infinite set N* as the domain for any sequence in an argument using > limits is really strange somehow.
It is worse, mathematically speaking, than merely strange, it is nonsense. > > 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 is > not 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 show > the missing real.
?"Cease"? > > [3]However we also have the corollary that even when we use N* as the > domain of those functions, still we can by a single common well > defined way define another sequence with exactly the same range of > those functions but from domain N, and we can apply Cantor's argument > and 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 order > concept that must use N* instead. (That's your reply). > > But again: why? what is the higher order part of the argument that you > see it demanding circumventing the heart of the subject (which is N > really) to some N*. > > Is it the definition of Limit. > > But limit is defined in this argument as the least upper bound, and I > don't see in the definition of L that I wrote (which is the standard > by the way) anything that has to do with necessarily picking it up > from 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 say > pick 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 argument > and you clearly can make L be the 1_th digit of (x_n) [Remember a_0 > is x_0, so x_{1} lies "before" a_0]. > > Or you'll say that {1,0,2,3,...} is also a kind of high order > countable set? > > Your argument is simply shunning one of the most important two sets in > this argument, that is N, and using some replacement, without any > clear justification. > > Zuhair
Right! The fact that one can make the proof seem false by changing it does not make the original proof false.

