Cantor Confusion
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From: mueckenh on 6 Dec 2006 06:11 Bob Kolker schrieb: > Eckard Blumschein wrote: > > > Obviously, you meant not either but neither? > > > > I wonder why the mathematicians believe to require one-point > > compactification. I consider the rationals as genuine numbers, being as > > close as you like to the fictions infinity and real numbers. The exact > > numerical representation of pi requires the fiction of actual infinity. > > > Rational numbers are non-genuine. Nowhere in the physical world outside > of our nervouse systems do they exist. The nervous system is a part of the physical world. The integers belong to the nervous system. They would not exist without the nervous system. And they cannot surpass its limits. That is the same case as with the human soul. It is clear that the mind is heavily influenced and personality can be changed by drugs and poison and by removing parts of the brain. But some people nevertheless believe in the immortal soul. I think that is just as strange as the belief in one's own capability of imaging actual infinity. And, remarkably, the creator of this theory believed in both, the soul/God and actual infinity, trying to prove the latter by the existence of the former. Regards, WM
From: mueckenh on 6 Dec 2006 06:19 Bob Kolker schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > There is no reason to give up continuity for applied mathematics. Ony > > those who want to learn the real truth may bother. > > And what is the "real truth"? For instance the fact that the cardinal number of the set of all even natural numbers cannot be larger than every even natural number, and, as a consequence, the complete set of even natural numbers cannot exist. Regards, WM
From: Franziska Neugebauer on 6 Dec 2006 06:23 mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: [...] >> What limits thought to only the accessible part of the universe? > > The fact that any brain consists of not more than this limited part of > the universe. > > And the fact that the power set of your neurons and ideas How do you know that there is an inaccessible, unlimited part of the universe beyond this accissible one? If you don't: Do you think it is meaningful at all to distinguish the accessible from the inaccessible? > and other contents of our brain is a finite set. No single "content of our brain" is a set Z-set theoretically. "Contents of our brain" is the subject of neuro sciences not of mathematics. Hence you are off topic. > And the power set of this set is a finite set too. And so on, in > infinity ... (potential infinity , of course) You still have not yet understood the concept of inifinite sets. F. N. -- xyz
From: mueckenh on 6 Dec 2006 06:31 William Hughes schrieb: > > > > And there is no natural number x, which is not element of a line. > > Let us write this out, taking care to note > that line n contains all the elements of any line > less than n. > > We have > > For every natural number n there exists a line L(n), such that > every natural number m <= n is an element of L(n) > > You cannot simply exhange the quantifiers to get > > There exists a line L, such that for every natural number n, > every natural number m<=n, is contained in L. I can simply do that for any finite linear (= totally orderd) set. So I can do it for every line. > > You need to give some other argument to show that L exists. > This you have not done. > > It is in fact easy to show that L cannot exist. > > If X is such that for every natural number n, n is > an element of X, then X is a potentially infinite set. > > No line L is a potentially infinite set. > > Therefore, there does not exist a line L such that > > for every natural number n, every natural number > m <= n is an element of L. Therefore, there is no set of all finite lines, i.e., there is no set containing all natural numbers as elements. > > > The fact that there each natural > number is a member *some* line, does not > mean there is one line which contains all > natural numbers. > > The point remains. > > A set with a largest element can have elements > all of which are natural numbers. > > A potentially infinite set without a largest element > can have elements all of which are > natural numbers. Correct, although this set can never be considered complete. > > There cannot be a bijection between a set with a largest > element and a potentially infinite set without a largest > element. > > Therefore there cannot be a bijection between the > diagonal and a line. > > Do you intend to keep claiming that a bijection can exist > between the diagonal and a line?. I claim that every element of he diagonal must be an element of a line. If the "infinite set of finite numbers" existed, then we would have a bijection between the diagonal and a line. (A diagoal cannot exist without being in a line.) This was my starting point: 1 1 2 12 3 123 .... n 123...n .... w 123... w+1 123...w But this bijection obviously fails. Therefore, there is no "infinite set of finite numbers". Regards, WM
From: mueckenh on 6 Dec 2006 06:34
stephen(a)nomail.com schrieb: > If you think sets grow, then you do not understand set theory. I know that in modern set theory sets do not grow. But I heavily doubt the relevance of modern set theory. Regards, WM |