Cantor Confusion
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From: Virgil on 10 Nov 2006 19:08 In article <1163182382.226922.297900(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > In mathematics counting and comparing numbers comes before most other > things. Not because I think so, but because mathematics developed this > way. Geometry didn't. > > > > > becauses you > > > do not know that limits are inside set theory. > > > > Where are they? That there are "limit ordinals" does not mean there are > > "limits" in set theory. > > Why are they called so? Why are rings called rings? Many mathematical meanings of words have little or no connection to their non-mathematical meanings. > > > > > Since Cantor quite a bit has changed. Limits are *not* part of set > > theory, they belong to topology and other things build on set theory. > > Experience has shown that practically all notions used in contemporary > mathematics can be defined, and their mathematical properties derived, > in this axiomatic system. (Hrbacek and Jech, p. 3) But this does not > include limits? Not without topology.
From: Dik T. Winter on 10 Nov 2006 19:11 In article <1163181580.683374.243410(a)e3g2000cwe.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > No. When we start with finite triangles, also each column has a finite > > length. So when we have "completed" the triangle, the height of the > > triangle it the supremum of the set of the number of lines, and is > > also the supremum of the lengths of the finite columns. The width > > is the supremum of the lengths of the lines. > > The hight (column) has actually many digits (maximum). This is *not* the maximum based on the finite triangles. But I have no idea what you *mean* here with actually many. Each column has an n-th digit for each natural n, and no other digits. > All numbers are > there, yielding a set of cardinality omega., The width (lines) has not > actually many digits n(supremum). All numbers are there, but no one has > infinitely many digits. The diagonal must have both, but cannot. The width has the same number of digits as the columns, but there is not a single line that has them all. The first line has the first digit (amongst otheres), the second line has the second digit. And so on and so forth. For each n there is a line that has a n-th digit. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: William Hughes on 10 Nov 2006 19:27 Franziska Neugebauer wrote: > William Hughes wrote: > > > Franziska Neugebauer wrote: > >> William Hughes wrote: > >> > >> > The length of the set of natural numbers is the supremum of the > >> > lengths of the initial segments. > >> > >> Since when do sets have a length? > > > > Substitute size or cardinality if you do not like the term > > length in this context. > > It is clear to *me*. Please do me a favor and ask *WM* whether > substituting "length" by "cardinality" changes - in his eyes - the > subject of your current discussion. I am quite willing to go to a much more formal set of terms and definitions, but I try to use the language preferred by whomoever I am discussing something with so if WM insists on using informal language that is what I will use.. Yes, my switch from "length of a line or a diagonal" to "length of the natural numbers" was an attempt to move back to slightly more formal language. Whether this will work I do not know. Note. It has become clear that at the base we have two statements. a: Any infinite set of numbers must contain an infinite number b: It is possible to have an infinite set of numbers that does not contain an infinite number. that cannot both be true. WM claims that a is true and that furthermore b leads to a contradiction. However, all of the contradictions he has been showing depend on assuming that a is true. I would like to discuss statements a and b directly, however, I do not think that WM will cooperate. Note that WM has the standard crank habit or not replying to a message unless he thinks he has a counterargument. - William Hughes
From: David Marcus on 10 Nov 2006 20:43 William Hughes wrote: > a: Any infinite set of numbers must contain an infinite number > > b: It is possible to have an infinite set of numbers that does > not contain an infinite number. WM, please tell us if you agree or disagree with the statements a and b above. -- David Marcus
From: Dik T. Winter on 10 Nov 2006 20:55
In article <4rjeevFrhvopU2(a)mid.individual.net> Bob Kolker writes: > Dik T. Winter wrote: > > What surprises me is that opposition against it in this way comes from > > especially the German posters in this group. They all want to stress > > the difference between actual infinity and potential infinity, that > > are indeed not mathematical terms, and most other posted do not know > > the difference. Is it still the old religious issue? I think so, > > based on the postings of at least one of the German posters. > > If Cantor's last name had been Shickelgrueber or von Richthoffen, none > of this would be written. Perhaps. Though I do not think so. But what I am wondering about is the preoccupation with "actual infinity" and "potential infinity" with some of the posters. I think you are objecting to my "German posters". I can shrink this to "German physicists". In this newsgroup there are actually three posters that refer to these terms. You may try to find out who those three are. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |