Cantor Confusion
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From: Eckard Blumschein on 5 Dec 2006 06:35 On 12/5/2006 2:50 AM, Dik T. Winter wrote: > In article <1165263838.656385.305770(a)l12g2000cwl.googlegroups.com> "MoeBlee" <jazzmobe(a)hotmail.com> writes: > > Eckard Blumschein wrote: > > > Correct. There are people who extend the reals to include oo. > > > > Would you give an example of a text that does this? > > > > What we sometimes do is add two points (called 'oo' and '-oo') to the > > real number system so that we have a different, extended system (which > > is not a complete ordered field). But that does not meant that we > > consider oo and -oo to be real numbers. > > In the one-point compactification of the real line or the complex plane > a single point at infinity is added. But also in that case it is either > a real nor a complex number. And again, the result is not a field, and > also not odered. 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.
From: Bob Kolker on 5 Dec 2006 07:26 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. For the same reason, integers are non-genuine. The only genuine things in the cosmos are physical. Bob Kolker
From: mueckenh on 5 Dec 2006 07:27 William Hughes schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > William Hughes schrieb: > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > William Hughes schrieb: > > > > > > > > > > > > > We extend this to potentially infinite sets: > > > > > > > > > > A function from the set potentially infinite set A to the > > > > > potentially infinite set B is a potentially infinite set of > > > > > ordered pairs (a,b) such that a is an element of A and b is > > > > > an element of B. > > > > > > > > > > We can now define bijections on potentially infinite sets > > > > > and extend the bijection equivalence relation to include > > > > > potentially infinite sets. Thus we can define > > > > > equivalence classes under bijection of potentially infinite sets. > > > > > Thus we can define "cardinal numbers" of potentially > > > > > infinite sets. > > > > > > > > > There is only one "cardinal number". In order to apply any of Cantor's > > > > proofs of higher cardinal numbers, a set of aleph_0 must be complete. > > > > But it cannot be complete in potential infinity. > > > > > > You now agree that a potentially infinite set can have > > > a cardinal number and that this cardinal is not > > > a natural number. > > > > > I wrote: a "cardinal number". oo is not a cardinal number in the sense > > of set theory. > > > > > We have: there exists a bijection between sets or potentially infinite > > > sets > > > A and B iff the cardinal number of A is the same as > > > the cardinal number of B. > > > > > > Now apply this. > > > > > > The natural numbers form a potentially > > > infinite set. The diagonal contains the potentially infinite set > > > of natural numbers. > > > > There is nothing to contain! You are too much caught in the terms of > > set theory. You cannot have the complete set because then it would be > > complete, i.e., actually existing, i.e., actually infinite. > > > If you want to avoid the word contain, reword > "The diagonal contains the potentially infinite set > of natural numbers." as "if x is an element of the potentially > infinite set of natural numbers, then x is an element of > the potentially infinite set of elements of the diagonal". And as well: If x is an element of the potentially infinite set of natural numbers, then x is an element of a line (and all natural numbers y < x are also elements of that very line). And there is no natural number x, which is no element of a line. Regards, WM
From: mueckenh on 5 Dec 2006 07:34 Franziska Neugebauer schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > Franziska Neugebauer schrieb: > >> mueckenh(a)rz.fh-augsburg.de wrote: > >> > Dik T. Winter schrieb: > >> [...] > >> > Everybody knows what the number of the EC states is. > >> > >> Everybody except you knows that EC states are not part of any set > >> theory. > > > > Oh, there are two of us. You forgot Cantor.Beiträge zur Begründung > > der transfiniten Mengenlehre. (That *is* a set theory). "Unter einer > > "Menge" verstehen wir jede Zusammenfassung M von bestimmten > > wohlunterschiedenen Objekten in unsrer Anschauung oder unseres Denkens > > (welche die "Elemente" von M genannt werden) zu einem Ganzen." > > Or do you insist on living creatures? > > Since "Cantor" is still present in the subject I have to ask you whether > you want to discuss anachronisms or if you want to learn how > contemporary set theory works. Contemporary set theory *is* an anachronsm. Compare, for instance, P. Lorenzen: Die endlichen Weltmodelle der gegenwärtigen Naturwissenschaft zeigen deutlich, wie diese Herrschaft eines Gedankens einer aktualen Unendlichkeit mit der klassischen (neuzeitlichen) Physik zu Ende gegangen ist. Befremdlich wirkt dem gegenüber die Einbeziehung des Aktual-Unendlichen in die Mathematik, die explizit erst gegen Ende des vorigen Jahrhunderts mit G. Cantor begann. > > > Further I am in accordance with he sentence: "Sets are not objects of > > the real world: they are created by our mind, not by our hands." Of > > course I understand by EC states the mind-created set of EC states. > > I will not discuss this anachronisms but modern concepts instead: After you will have learned how it works, you wil see that it is but an anachronism. > > I am especially interested in the growing number and the growing set. The number of your contributions has increased by 1 with your post I just answer. The same holds for the set of your contributions. Regards, WM
From: mueckenh on 5 Dec 2006 07:36
Franziska Neugebauer schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > Franziska Neugebauer schrieb: > >> mueckenh(a)rz.fh-augsburg.de wrote: > >> > So you can construct the set of all real numbers (of the interval > >> > [0, 1] in binary representation) by: > >> > > >> > 0.0 > >> > 0.1 > >> > 0.01 > >> > 0.11 > >> > ... > >> > > >> > This set is countable. > >> > >> 1/3 is missing. > > > > Of course. It is not a potentially infinte sequence. > > 1/3 is not a sequence at all. It is a rational number. > Some correspondents try to think themselves. I encourage you to join them. Regards, WM |