Virgil
Posts:
4,661
Registered:
1/6/11
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Re: Mathematics in brief
Posted:
Dec 10, 2012 3:22 PM
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In article
<26c455c2-c3d6-4ac1-a0a2-823fab37df1c@r4g2000vbi.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.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 ending that I do not publish).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).
--
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