Virgil
Posts:
8,833
Registered:
1/6/11


Re: Mathematics in brief
Posted:
Dec 10, 2012 3:22 PM


In article <26c455c2c3d64ac1a0a2823fab37df1c@r4g2000vbi.googlegroups.com>, WM <mueckenh@rz.fhaugsburg.de> wrote:
> On 10 Dez., 10:37, Zuhair <zaljo...@gmail.com> wrote: > > > > This proof is given by constructing the whole Binary Tree by countably > > > many actually infinite paths (i.e. finite paths with infinite endings > > > like 010101... or 000... or 111... or 001001001... or any desired > > > ending that I do not publish). My proof rests upon your inability to > > > find out what paths are missing. > > > > But those are not all the reals, there can be reals that do not end > > with a tail that is a repeated segment, > > what you are speaking about is > > actually the rationals which are known to be countable. > > Anyhow I don't know the details of how did you construct your binary > > tree, can it for example represent irrational numbers like the square > > root of 2 (which doesn't end by a repeated segment tail) > > or the transcendental reals. > > Of course I can use also tails that consist of the bits of sqrt(2) or > of pi or I can mix arbitrary tails. The point is, that you cannot > define a real number or an infinite tail by following its nodes. It is > simply impossible. You can only give a finite definition of infinite > tails (like I did above  who could follow all bits of pi?). But there > are only countably many finite definitions. > > > > > I can see that the number of paths in any FINITE binary > > > > tree is less than the number of its nodes? but can that feature > > > > survive at infinite level? and what is the proof? I do see that the > > > > number of nodes in your tree is countable. But would it follow that > > > > the number of paths must be so at infinite level? > > > > > Find a path that I have not used! > > > > That is easy, you claim that there is a bijection between N and the > > set of all infinite paths in your tree, correct! Then simply apply the > > diagonal argument of Cantor and you will get a linear graph > > that is not among the paths of your tree, and this will correspond to > > a real, since it reflects a binary sequence, so your tree is NOT of > > all paths that reals corresponds to. In other words your tree if > > countable then it is incomplete. > > No. The idea that real numbers could be defined by infinite sequences > of nodes without a finite definition defining them, is simply wrong. > > > > > > How I see matters is that if I assume that there is a bijection > > > > between N and the set of all infinite paths in your tree, then I can > > > > easily construct a diagonal using Cantor's argument, and this diagonal > > > > would provably be a path that is not in that Tree. So either your tree > > > > must have uncountably many paths (with countably many nodes) or your > > > > tree has countably many paths but is incomplete, i.e. there are > > > > infinite binary paths that are not paths of it. > > > > > Or the idea of countable and uncountable sets is humbug. Why do you > > > refuse to take into account this possibility? Because you cannot > > > believe that many thousands of mathematicians have behaved like fools > > > in the last hundred years? No reason to be ashamed. I have been among > > > them myself for a long time. > > > > To be honest my answer is YES. I find it hard to believe that many > > thousands of mathematicians behaved as you stated for a whole century > > to time, not only that, among those thousands are people who are > > considered geniuses of all times like Harvey Friedman for example, > > Frege, Tarski, Godel, Hilbert, Von Neumann, Quine, and many many > > others, who are authorities by known standards. To go say that they > > acted as fools etc... is insulting really. How can you account for > > such a claim. To be honest I don't see your claim to be reasonable. > > Anyhow everything in this strange world is possible, but for one to > > hold of such claim, then he must demonstrate a solid I mean really > > really solid argument to verify his stance, since it is way against > > what one should expect really. > > Answer these questions honestly to yourself. You need not publish the > answers. > Have you ever realized that 0.111... is *not* an infinite sequence but > only a finite expression allowing you to determine every digit of an > infinite sequence?
Just as we never actually use numbers, real or otherwise, but only representations of them (names/numerals for them). 

