iOn 9 Jan, 23:41, Virgil <vir...@ligriv.com> wrote: > In article > <69862de0-c105-424e-88f3-26ac3f9a7...@r14g2000vbe.googlegroups.com>, > > WM <mueck...@rz.fh-augsburg.de> wrote: > > On 9 Jan., 18:57, Zuhair <zaljo...@gmail.com> wrote: > > > On Jan 9, 8:15 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > > > > > On 9 Jan., 13:24, Zuhair <zaljo...@gmail.com> wrote: > > > > > > Actually Cantor's diagonal argument > > > > > proves that the number of those tuples MUST be uncountable. > > > > > In fact, it does! But we know that the number of those tuples is > > > > countable! > > > > No we don't know that! > > > Perhaps you don't. But everybody can learn it. > > No one need do so when not trying to pass one of WM's courses. > > > > > > > > > > > > > > there is NO seal on how many "Omega" sized > > > tuples of FINITE initial segments of reals do we have! > > > Omega-sized sequences of finite initial segments are not different > > from omega-sized sequences of digits. Both do not belong to the set of > > finite definitions. Do you really believe that by using finite initial > > segments instead of simple digits you make infinite sequences finite? > > > > there is no > > > argument (even at intuitive level) in favor of having countably many > > > such Omega sized tuples. > > > But there is a simple argument that omega sized tuples are infinite > > and are not suitable as finite definitions. > > > > Actually the ONLY arguments that we have is > > > in favor of having uncountably many such Omega sized tuples, which are > > > Cantor's arguments. The set of All finite initial segments of reals is > > > COUNTABLE! Yes! BUT the set of all Omega sized tuples of finite > > > initial segments of reals is NOT countable!!! > > > That is not of interest. A Cantor-list does not use omega-sized tuples > > but only finite initial segments. > > > > And also I want to stress that all of > > > what I'm saying is abiding by the axiom of Extensionality of course, > > > no doubt, and accordingly I do NOT hold that in a formal consistent > > > theory we can distinguish UNdistinguishable elements or objects, since > > > this is clearly nonsense. However the concept of Uncountability > > > doesn't lead to that, > > > Try to read the literature of great mathematicians, for instance > > matheology § 187. > > Even the best may have lapses, and WM quote mines those lapses with much > greater effectiveness than he manages to do mathematics. > > > > > Or try to think with your own head > > Since WM only manages to think with someone else's head, if at all, he > should reck his own rede. > > > Construct a list of all rationals of the unit interval. Remove all > > periodic representations and keep only the finite ones > > Since any sensible method of listing of rationals will not be done in > terms of their decimal or other base expansions, why do it the hard way? > > , but all. > > > Then the list contains all finite initial segments which a possible > > anti-diagonal can have. So the anti-diagonal can at no finite position > > deviate from every entry of the list. > > There will be no finite digit position in the list up to which position > the list itself does not contain duplicates, however any nonterminating > decimal will not be in any list such as WM has described, so his list is > incomplete. > > > That means it can never deviate > > from every entry. > > On the contrary, if every listed decimal terminates then every > non-terminating decimal is unlisted. > > > According to Cantor, anti-diagonal deviates from > > every entry of the list at a finite position. > > In AT LEAST one finite position! > > An antidiagonal might even differ from some entries at EVERY digit > position. > > > But even according to > > Cantor it deviates never(at no finite position) from all entries. > > WM's Quantifier dyslexia strikes again. > > In order for an infinite sequence of digits to differ from all the > infinite sequences of digits in a list, it only needs to differ suitably > from each entry at one digit position, and that position can be > different for different entries in the list. > > And any such infinite list of infinite sequences of digits, from a set > of two or more digits, is provably incomplete since Cantor showed that > there is always a sequence missing from that list. > -- I beleive answers is at a very basic level, what distuingish a natural from a real and so on. I do not know much about math but it seem to me that this must be its foundation.