Cantor Confusion
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From: Virgil on 10 Nov 2006 18:53 In article <1163180802.424978.193240(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > > > > > > Exactly this is done by Cantor's definition given above: "It is allowed > > > to understand the new number omega as limit to which the (natural) > > > numbers n grow". This is a definition, at least for those > > > mathematicians who know what "limit, number, grow" means. > > > > I seriously doubt that Cantor considered it to be a definition. If he > > did, he wouldn't have said, "It is allowed to understand". Regardless, > > it is not a definition by modern standards. > > Who judges what modern standards are, in your opinion? Not those who have none. > > > Nor, is it the defintion of > > omega in modern mathematics. And, since the set of mathematicians who > > know the mathematical meaning of the words "limit", "number", "grow" is > > probably empty, your final sentence seems vacuous. > > > You carried out a survey? Or is that your guess? I know some > mathematicians who know what a mathematician should understand by > "limit", "number", "grow". Like whom? > > > Your changing "finished entity" to "finished infinity" is a bit of a > > stretch. Regardless, Hrbacek and Jech are clearly making a philosophical > > comment, not discussing mathematics. > > They were discussing mathematics. Philosophy belongs to mathematics. There may be some benighted mathematicians who claim so, but there are as many philosophers, and others, who will claim that mathematics belongs to philosophy. > It > is not uncommon to meet a PhD in mathematics, who, in Germany, has the > title "doctor of philosophy". Many, if not most, doctorates, other than those in medicine, awarded in the USA, and many other countries, are nominally "of philososphy", regardless of the actual subject of study. So a Ph.D. is hardly an indication of one who has studied philososphy in any formal way.
From: Virgil on 10 Nov 2006 18:57 In article <1163181215.887845.121170(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > > Here we can set up a bijection, expressing the lines by the natural > > > indexes of the elements: > > > > > > 1 1 > > > 2 1,2 > > > 3 1,2,3 > > > ... ... > > > n 1,2,3,...,n > > > ... ... > > > omega 1,2,3,... > > > > > > which shows that omega or aleph_0 does not exist. Then how is it that it exists in your table?
From: Virgil on 10 Nov 2006 19:01 In article <1163181580.683374.243410(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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). 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. Live with that delusion and die in self imposed ignorance.
From: Virgil on 10 Nov 2006 19:04 In article <1163181843.615394.171730(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > > WM's logic (if we can call it that) > > Please do not! > You have no balls at noon. Wm has no balls at all. > You have Tristram Shandy complete his diary. No, he only completes one day at a time. > You have a diagonal longer than every line. We do if the diagonal has no end and every line does have an aend. > You can make two balls of one. Right. > You have a countable model of an uncountable theory. Not us! > > I would be dismayed if you found any parallel between my thinking and > your "logic". Well, I'm not sure we even would call it thinking. > > Regards, WM
From: David Marcus on 10 Nov 2006 19:05
Franziska Neugebauer 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. > > 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. He has his own defintion for the word "definition". By "infinite triangle", I meant a function with domain {(n,m)| n,m in N and m <= n} and range N. -- David Marcus |