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! >
> Regards, WM
No we don't know that! there is NO seal on how many "Omega" sized tuples of FINITE initial segments of reals do we have! there is no argument (even at intuitive level) in favor of having countably many such Omega sized tuples. 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!!! Since we have Uncountably many Omega sized tuples of finite initial segments of reals, then we do have UNCOUNTABLY many reals distinguished after them! that's all fairly intuitive, there is nothing wrong with the line of that reasoning at all. 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, so I don't see why you keep asking such SILLY irrelevant questions.