In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> 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.
Not in ZF. WM may choose not to deal with infiniteness himself, but has not the power to prohibit others from understanding what is beyond him.
Every > element you count belongs to a small minority (in fact of measure > zero)
There cannot be any such measure without actually infinite sets, so that WM has to rely in the truth of what he denies to argue against it.
> > > 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.
If one had to construct that tree one node at a time, one would never finish, but one can describe what a finished tree would be like without ever having troubled to construct one, just as one can find limits of infinite sequences without ever going through each term one by one. > > > > >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.
While WM is busy constructing the unconstructable, the rest of us have long since found the limit of that construction process and moved on from there.
> > Numbers are means to distinguish things.
WRONG! Numerals are to distinguish numbers, and while there are only countably many numerals, that does not mean that every umber has to have one.
> Undistinguishable numbers are > nonsense.
Undistinguishable numerals may be nonsense, but undistinguishable numbers are not.
> 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.
It is only their names which we may not be able to know. --