Cantor Confusion
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From: Lester Zick on 16 Nov 2006 12:45 On Thu, 16 Nov 2006 02:10:17 -0700, Virgil <virgil(a)comcast.net> wrote: >In article <MPG.1fc5d0ae1682037a98990a(a)news.rcn.com>, > David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > >> Virgil wrote: >> > In article <1163426009.651510.237050(a)h48g2000cwc.googlegroups.com>, >> > mueckenh(a)rz.fh-augsburg.de wrote: >> > > It is difficult to answer this question, because the expression "set" >> > > is occupied in modern mathematics by collections of elements which are >> > > actually there (you don't know what that means, imagine just a set as >> > > you know it). Such infinite sets do not exist. >> > >> > While infinite collections in any physical sense are not possible, why >> > are imaginary infinities, such as sets of numbers must be, unimaginable? >> >> For that matter, we can always switch from Platonism to formalism and >> declare the question of whether sets really exist to be a philosophical >> question. > >I suspect that WM will be as bad at philosophy as he is at mathematics, >but at least it will no longer be a mathematical problem. So mathematical problems are those which are declared not to be mathematical problems? ~v~~
From: Randy Poe on 16 Nov 2006 13:36 Lester Zick wrote: > On Thu, 16 Nov 2006 01:35:12 -0500, David Marcus > <DavidMarcus(a)alumdotmit.edu> wrote: > > >Virgil wrote: > >> In article <1163426009.651510.237050(a)h48g2000cwc.googlegroups.com>, > >> mueckenh(a)rz.fh-augsburg.de wrote: > >> > It is difficult to answer this question, because the expression "set" > >> > is occupied in modern mathematics by collections of elements which are > >> > actually there (you don't know what that means, imagine just a set as > >> > you know it). Such infinite sets do not exist. > >> > >> While infinite collections in any physical sense are not possible, why > >> are imaginary infinities, such as sets of numbers must be, unimaginable? > > Why are square circles unimaginable? They aren't. We can define "the set of squares S". We can define "the set of circles C". Most of us (except you, apparently) can easily imagine defining the set SC = C intersect S. It happens under the rules of Euclidean geometry that the set SC is empty. But that doesn't stop most of us (except you, apparently) from imagining a geometry where it is not empty. Furthermore, the fact that SC is empty in Euclidean geometry would be a theorem. There's nothing inherently wrong with defining the set. Most of us (except you, apparently) can see that forming such a definition tells us nothing about whether the set is empty or not. It's just a shorthand, the name of the membership test for SC. Most of us (except you, apparently) can then easily imagine a proof that SC is empty which relies on argument from axioms, not such handwaving as "the definition is false". - Randy
From: Lester Zick on 16 Nov 2006 13:44 On Thu, 16 Nov 2006 01:48:52 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >Lester Zick wrote: >> On Sat, 11 Nov 2006 13:50:38 -0500, David Marcus >> <DavidMarcus(a)alumdotmit.edu> wrote: >> >stephen(a)nomail.com wrote: >> >> I think a lot of this "opposition" would go away if the word >> >> "transfinite" instead of "infinite" had been used to describe >> >> a set that can be put into a one-to-one correspondence with >> >> a proper subset of itself. The word "infinite" sends people >> >> down strange philosophical paths, as does the word "infinity" >> >> despite the fact that it is not really even used in set theory. >> >> Noone would argue about "transfinity". >> > >> >You could be right. Although, it seems unfair of the cranks to dictate >> >what words mathematicians can appropriate. It is hard to make up good >> >names. We have enough names like "second category" as it is. >> >> Once more we have this sloppy word usage on the part of those who self >> righteously proclaim their mathematical rectitude. What exactly does >> "crank" mean besides "crank(x)=disagree(u)"? Modern mathematikers >> routinely appropriate words and make up private definitions for them >> as if they were the sole arbiters of truth in mathematical terms. Of >> course it's hard to make up good names especially when mathematikers >> insist on private definitions cast in parochial terms of modern math. > >"Crank" means someone who makes up pejorative names for people "Crank" means someone who makes up private definitions for "crank" and insists their usage is definitive and correct. > whose >language they do not understand Hell I don't understand a lot of languages. So do you. Doesn't make them private or non private. > and thinks that definitions which are >explained in many books are "private". "Many books" being your definition of "true"? ~v~~
From: mueckenh on 16 Nov 2006 14:21 William Hughes schrieb: > No. The length of the diagonal is the supremum of the lengths > of the lines (this is easy to show). So if there is no > longest line the diagonal must be longer than any line. The length of the diagonal cannot surpass the length of any line. (This is easy to show for every matrix.) So, there is a line which has the same length as the diagonal, i.e., infinite length. > > In order to see that, add one element to every column and > > to every line. Now the order type of any column is omega + 1, the > > length of the matrix has order type omega + 1, the order type of any > > line is n+1 < omega, and the width of the matrix has order type omega. > > The diagonal is a bijection between columns and lines. It des no exist. > > This shows that the diagonal in the original matrix did not exist > > either, > > No. > > Recall, adding one element to every line does not > change the number of columns. Therefore the set of columns is less than the set of lines. There cannot exist a bijection. The original matrix has no diagonal Regards, WM
From: mueckenh on 16 Nov 2006 14:23
Lester Zick schrieb: > Pi is certainly accessible through circular arcs and their diameters. > Are you suggesting circles and their diameters don't exist? It looks > to me like you're confusing the accessibility of calculations for > fractional expansions with the accessibility of the number itself. If > one ignores geometry and makes arithmetic the sole mathematical > paradigm of course your conclusion follows. That's the price to be > paid for exclusive reliance on infinite set analytical techniques as > exhaustive criteria for the properties of numbers. Sqrt(2) does exist as the diagonal of the square. But I call that an idea in order to distinguish it from numbers which can be written in lists and can be subject to a diagonal proof. I call only those entities numbers which can be put in trichotomy with each other. If you use the first 10^100 digits of pi as a natural number and exchange the last digit by 5, then you cannot determine which one is larger. Regards, WM |