
Re: some amateurish opinions on CH
Posted:
Apr 8, 2013 2:58 PM


On Apr 8, 8:38 pm, WM <mueck...@rz.fhaugsburg.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_measure http://megamindacademy.com/wpcontent/uploads/2011/08/LineSegments.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 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 .
>But the Binary Tree shows that it is >impossible to distinguish uncountable many reals.
What do you even mean by that?And why should distinguishably in the empirical sense even be a criteria for what's real?
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." Whereupon Diogenes immediately got up and left. "

