
Re: Cantor's first proof in DETAILS
Posted:
Nov 14, 2012 5:49 AM


On Nov 14, 10:18 am, Uirgil <uir...@uirgil.ur> wrote: > In article > <6a63fbfdf7e7458faf65fae2c805c...@d17g2000vbv.googlegroups.com>, > > > > > > > > > > Zuhair <zaljo...@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"? > > Yes, Cease, i.e. stop, of course I'm speaking about stopping in the sense of running the exact particulars of the argument per se, that's why I said "...in the same way" for example when you use some N* which has an omega_th position as the domain then for example Result 7 cannot be proven in exactly the same straightforwards way as it is proved with N, to prove it you need to define it indirectly in terms of bijections from N* to N ...., which is a long way. But ultimately you will also succeed in finding a missing real as I pointed out. That is merely a temporary conundrum with the argument that has no significance to the reality of the matter, and has no philosophical value whatsoever.
Zuhair > > > > > > > > > > [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.

