Cantor Confusion
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From: William Hughes on 11 Nov 2006 09:50 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > > > > Lines with finitely many indexes cannot exhaust the column with its > > > infinitely many > > > > An infinite number can. > > A God can do everything he wants. Infinite numbers are of the same > power? > No. God can do anything. Infinite numbers cannot. However, Infinite numbers can do some things. In particular an infinite number of finite lines can "exhaust": an infinite number of indexes. > Can you apply your reasoning to the following matrix? > > 1 > 21 > 321 > 4321 > ... > Reversing the order of the lines does not change thing. There are an infinite number of lines. Each line is finite. There are are omega lines and omega columns. There is no last line. There is no omega th line. There is no inifinite line. > What ordinal numbers have the following sequences? > None. These are not ordinals. An ordinal number must consist of an initial segment of ordinals. However, in a slightly sloppy way they can be said to represent ordinals (note that this representation is no longer uniqe). Note that you can represent any countable ordinal (not just omega) by using just finite integers, you do not need (athough you can use if you want) infinite ordinals. > 3,4,5,...,1,2 A infinite set consisting of finite integers. This represents omega+1 (For n>=3 you have put n in the spot one would usually have n-2, 1 is in the spot where you would usually have omega, and 2 is in the spot where you would usually have omega+1) > > 3,4,5,...,omega,1,2 > An infinite set consisting of the finite integers and the non-natural number omega. This represents omega+2 (For n>=3 you have put n in the spot one would usually have n-2, 1 is in the spot where you would usually have omega+1, and 2 is in the spot where you would usually have omega+2) What matters is not which ordinals you use to occupy places, but where you put the "..." For example. The both sequences 5,4,3,2,1,6,7,8 ... omega, omega*omega, omega+5*omega^3 + 7, 1,2.3, ... represent the ordinal omega. - William Hughes
From: stephen on 11 Nov 2006 12:42 Dik T. Winter <Dik.Winter(a)cwi.nl> wrote: > 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. 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". Stephen
From: David Marcus on 11 Nov 2006 12:49 mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > And > > so, apparently, he thought that if a set was actually infinite, it should > > have an actually infinite element, or somesuch. > > It should have an integer number counting its elements. Please explain what you mean by this. In particular, what does "counting its elements" mean? > Of course this > number had to appear in the sequence of numbers if it was a number. But > where? Obviously after all natural numbers. But then, what is the > number of 1,2,3,...,omega ? > It cannot be omega + 1, because, according to his equation above, > 2,3,4, ....,1 has already the number omega + 1. > Therefore omega cannot appear in the sequence. > > So we have the irrevocable dilemma: > 1) omega is the number of countable many numbers like 1,2,3,... or > 7,8,9,... and as such a single term in the sequence following the > counted terms like 1,2,3,...,omega or 7,8,9,...,omega > 2) omega + 1 = 2,3,4,...,1 and therefore omega cannot not be a term in > the sequence but omega is the first part of it, omega = 2,3,4,... > > This dilemma is what I have been trying to explain for years now. Oh. -- David Marcus
From: David Marcus on 11 Nov 2006 12:50 imaginatorium(a)despammed.com wrote: > David Marcus wrote: > > Franziska Neugebauer wrote: > > > David Marcus wrote: > > > > > > > [...] you are working with an infinite triangle [...] > > > > > > ,----[ http://en.wikipedia.org/wiki/Triangle ] > > > | A triangle is one of the basic shapes of geometry: a polygon with > > > | three vertices [...] > > > `---- > > > > > > Are there really three vertices in WM's "triangle"? > > > > No. But, I don't think an "infinite triangle" needs to be a triangle. > > However, I'm open to suggestions for what to call it. > > You could call it a "two-sided triangle". This might turn out to be > useful in a quiz some day. I like that! -- David Marcus
From: David Marcus on 11 Nov 2006 12:54
Franziska Neugebauer wrote: > David Marcus wrote: > > Franziska Neugebauer wrote: > >> I would prefer discussions either with precisely defined (formalized) > >> "infinite triangles" or better without all these words borrowed from > >> Cantor/geometry/physics. > > > > In that case, if you discuss anything with WM, you will be > > disappointed. > > Depends on your expectations. You will find a plethora of absurd > constraints for thinking. Very true. > > He has his own defintion for the word "definition". > > He does not at all enact the notion of definition. > > > By "infinite triangle", I meant a function with domain {(n,m)| n,m in > > N and m <= n} and range N. > > OK. > > So you are writing about abstract entities like functions and domains > whereas WM does not. In WM's view a /notation/ like > > 1 > 1 2 > 1 2 3 > ... > > *is* the object under consideration whereas for you it is merely an > illustration or reference the abstract entity. Are you sure? If you ask him if the the above object has less than five lines, I think he will say no. > What WM calls "diagonal" > means the geometrical object contained in that notation either on > screen or written on paper. Mathematically "diagonal" simply means the > sequence of f(i,i) i e N which by definition has cardinality |N|. > > WM does not belong to the "factinista". He is ignorant about the fact > that nowadays mathematics is a formal science in the first place. It appears to be willful ignorance. -- David Marcus |