On 8 Apr., 20:58, Dan <dan.ms.ch...@gmail.com> wrote: > On Apr 8, 8:38 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > > > > > > > On 8 Apr., 19:13, Dan <dan.ms.ch...@gmail.com> wrote: > > > > > I use it as it is in order to contradict it. > > > > Should you contradict something, you can't use the contradicted thing > > > as an argument to support something else . > > > The contradiction has the same character as the famous proof that > > sqrt(2) is not rational. It is assumed to be rational > > > Here I assume that actual infinity exists. The result is > > uncountability, i.e., the existence of a set that has uncountably many > > different elements (elements that can be distinguiushed by infinite > > sequences of digits or bits). But the Binary Tree shows that it is > > impossible to distinguish uncountable many reals. > > > > Every time you draw a line , you make "finished infinity" . > > > That is not the case, since there is no finished infinity. But it is > > in vain to discuss this by means of analogies and generalities. Try to > > distinguish more than countably many infinite paths in the Binary > > Tree. You will fail. > > > > The same goes for you and Set Theory .Can you imagine a world where > > > there never was any 'Set Theory' for you to rant about?How would you > > > get attention? > > > I have done a lot of different things before. But I must confess that > > set theory is the most interesting one, in particular the > > psychological aspect that so many intelligent and bright people could > > fall victims to it. > > > Regards, WM > > Analogy is how mathematics got started in the first place . And I see > no particular way in which this analogy is wrong . > For a more formal description , consider measure theory (that > discipline where you roughly "use a ruler" to measure the total length > of different subsets of the Real Line .)http://en.wikipedia.org/wiki/Lebesgue_measurehttp://megamindacademy.com/wp-content/uploads/2011/08/Line-Segments.jpg > > The segment [0,1] has total measure 1 . There are no gaps . > Consider, by contrast , a singular point on the line , A. > The measure of A is the distance from A to A , namely, 0 . > > The measure of a set containing only two points , A and B , is still 0 > + 0 = 0 > As you continue to add points in a sequential , countable manner , the > measure of your 'point set' is still 0 . > > The measure of a countable set is 0 .http://www.proofwiki.org/wiki/Countable_Sets_Have_Measure_Zero
No. There is no countable set because countability is nonsense. Every element you count belongs to a small minority (in fact of measure zero) of a majority of infinitely many uncounted element. (Yes I know the sum of the series (1/2^n).)
> The measure of the whole line segment [0,1] is 1 . > That's why you can never fill the continuum with points in a stepwise > manner . > The continuum is 'larger' than any countable set , hence > uncountable .
Even if this were correct, then there exists the contradiction that the continuum is constructed in the Binary Tree by countably many steps, none of which adds more than one path to the tree. > > >But the Binary Tree shows that it is > >impossible to distinguish uncountable many reals. > > What do you even mean by that?
The system of all real numbers of the unit interval expressed as paths, sequences of bits:
0. / \ 0 1 / \ / \ 0 10 1 ...
The number of nodes is countable. Therefore, when during construction, more than one path per node is constructed, then there is no reason to believe that the Cantor-list cannot accomplish the same trick.
Further I can construct a Cantor-list, the anti-diagonal d = d_1, d_2, d_3, ... of which is in the list:
For all n: the list contains infinitely many digit sequences d_1, d_2, ..., d_n. Note: for ALL n. And more is not possible.
> And why should distinguishably in the > empirical sense even be a criteria for what's real?
Numbers are means to distinguish things. Undistinguishable numbers are nonsense. In no case they have any reality. The real numbers obey trichotomy. For every pair a, b we have a < b or a = b or a > b. If numbers have do deinition, then this cannot be decided. > > I'm sure you've had ample opportunity to discuss your "binary tree" > and in what ways it is deficient with respect to the real continuum. > So, instead of taking on a futile endeavor , I'd like to cite an > anecdote : > > "A student of philosophy, eager to display his powers of argument, > approached Diogenes, introduced himself and said, "If it pleases you, > sir, let me prove to you that there is no such thing as motion."
That had already been proved by Zenon, a while before Diogenes was born. And that there are no such things as things also has been proved or at least claimed
And you come to me and will argue that there is no such thing as the continuum whithout the silly notions of countability and finished infinity.