Cantor Confusion
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From: mueckenh on 12 Dec 2006 05:27 William Hughes schrieb: > Do you now claim > > the natural numbers are not a potentially infinite set? N is a potentially infinite set. > > the natural numbers do not exist? More than enoug do exist. (More than we will ever need could be brought to existence.) > > one can't define functions on the natural number? One can define such functions called sequences, but that does not guarantee that all the pairs do actually exist. The pair (x, f(x)) with x = floor(pi*10^10^100) does not exist, for instance. > > Not that all of the above is completely independent > of whethe the natural numbers actually exist, we just > need that the natural numbers exist. > > I do not claim that a potentially infinite set actually > exists, I merely claim that a potentailly infinite set exists. > That is all I need. But what do you mean and understand by "to exist"? In *every* set theory, starting from Cantor, it means to exist actually. Regards, WM
From: mueckenh on 12 Dec 2006 05:45 William Hughes schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > William Hughes schrieb: > > > > > > > This is the argument you present. > > > > > > > > > > > 1) The set of lines is also the potentially infinite set of natural > > > > numbers. > > > > 2) Every element of the diagonal is in at least one line. > > > > 3) Every initial segment of the diagonal is (in) a line. > > > > > > No, there is one initial segment of the diagonal that is > > > not in a line. > > > > Then name the element of the diagonal please, which supports your > > claim. > > > There is one initial segment of the diagonal which is not defined > by a single element. > > The potentially infinite sequence > > {1,2,3 ...} > > is an initial segment of the diagonal, but it is not in a line, nor > does it have a largest element. If it is not in any line then there must be at least one element of it which is not in any line. If you say it exists, then you must say by what element it differes from any finite initial segment. > > (It is contained in the union of all lines, but the > union of all lines is not a line) That is a void assertion unless you can prove it by showing that element by which the union differes from all the lines. > > > > > > > > > 4) The limit (n -->oo, for the number n of elements) of the diagonal is > > > > oo. > > > > > > Yes and it attains this limit (i.e the limit is a maximum as > > > well as a supremum) > > > > > > > 5) The limit (m -->oo, for the number m of elements of a line) is less > > > > than infinite. > > > > > > No. The limit is oo. (The limit cannot be any finite number) > > > > And it cannot be an infinite number. There is no infinite natural > > number. > > Why do you think it is a natural number? The limit a of a sequence (a_n) is a number which is approximated by the sequence with arbitrary precision. For any eps > 0 we can find a number n_0, such that for any n > n_0 |a_n - a| < eps. For any infinite number, call it omega, and any finite natural number n we have |n - omega| > 1. Therefore no infinite number omega can be the limit of the sequence f(n) = n of natural numbers. > The limit of natural numbers does not have > to be a natural number. This limit is > not a natural number so it can be infinite. This limit is not a natural number so it cannot be. Regards, WM
From: mueckenh on 12 Dec 2006 05:50 Tony Orlow schrieb: > >> Perhaps, with a countable number of bits. But, there are more than any > >> finite number of reals in the unit interval, and this is an infinite > >> number. > > > > Where *are* they? They canot be addressed. They cannot be imagined. > > They cannot be known. And, moreover, the proof proving their existence, > > is false. > > > > Quite a poor existence. > > > > Regards, WM > > > > Well, the proof is simple. Any finite number of subdivisions of any > finite interval will only identify a finite number of real midpoints in > that interval, between any two of which will remain more real midpoints. And they are (can be represened by) rational numbers too. > Therefore, there are more than any finite number of real points in the > interval. Of course, most of them require infinite strings of bits to > specify, but that doesn't mean the point doesn't exist. Tony, have you ever wondered why physics is atomistic but mathematics is (assumed to be) not? Regards, WM
From: mueckenh on 12 Dec 2006 05:59 Michael Stemper schrieb: > In article <1165615427.980137.7670(a)f1g2000cwa.googlegroups.com>, mueckenh writes: > >William Hughes schrieb: > > >> We can now define bijections on potentially infinite sets > > > >Only if we consider them being actually infinite. > > Would you please define "actually infinite" or "actually infinite set"? Actual infinity is the infinity used (or rather claimed to be used) in every kind of set theory. > > > But that would > >exclude them from being potentially infinite. > > Would you please define "potentially infinite" or "potentially infinite set"? A set or sequence like (n) is potentially infinite, if n can surpass any upper bound. For more information on this topic visit: http://www.fh-augsburg.de/~mueckenh/MR/PUundAU.htm Regards, WM
From: mueckenh on 12 Dec 2006 06:03
Dik T. Winter schrieb: > In article <1165872695.605284.98210(a)f1g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > > > So why did you state that I erronously believed that nodes represent > > > > > numbers? > > > > > > > > Because you erroneously did. > > > > > > I did not. Reread what is stated above. > > > > Reread what you wrote on 12 November "What is the node number of 1/3?" > > and "Do you really think the node 1/3 is finitely far from the root in > > the tree?" > > > > Reread what you wrote on 28 November: "But as I did show in another > > article, when you assign bits to nodes, you can show that the nodes > > represent numbers,..." > > I did not state that nodes represent numbers, I stated that it can be > shown that nodes represent numbers. Great. That is set theory from the finest! "It is not the case, but it can be shown or proved to be the case. " Regards, WM |