Cantor Confusion
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From: William Hughes on 14 Nov 2006 08:57 Franziska Neugebauer wrote: > William Hughes wrote: > > > If we add one element to each of the columns then > > the supremum of the lengths of the initial segments of the columns > > changes to omega +1. If we add one elelment to each of the lines > > the supremum of the lengths of the lines does not change, it > > remains omega. > > Could you explain in a formalized way what "add one element to each of > the columns" means? > True formalism would take a lot of effort. However, WM and I would seem to be talking about the same thing here. The original columns are {1,1,1,...} {2,2,2,...} {3,3,3,...} .... Clearly adding one element to each of the columns can only have an effect if we add it "at the end". So, if this makes any sense this is what we will do. The element to add for each column has not been specified, but we will assume that it is given by the obvious pattern. So the new columns are {1,1,1,....1} {2,2,2,...,2} {3,3,3,...,3} ... - William Hughes
From: Franziska Neugebauer on 14 Nov 2006 09:18 William Hughes wrote: > Franziska Neugebauer wrote: >> William Hughes wrote: [...] >> Could you explain in a formalized way what "add one element to each >> of the columns" means? >> > > True formalism would take a lot of effort. However, WM and > I would seem to be talking about the same thing here. > > The original columns are > > {1,1,1,...} > {2,2,2,...} > {3,3,3,...} > ... > > Clearly adding one element to each of the columns can only have an > effect if we add it "at the end". So, if this makes any sense this is > what we will do. The element to add for each column has > not been specified, but we will assume that it is given by the > obvious pattern. So the new columns are > > {1,1,1,....1} > {2,2,2,...,2} > {3,3,3,...,3} OK F. N. -- xyz
From: Sebastian Holzmann on 14 Nov 2006 10:56 mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: > I think that the rise of non-euclidean geometries was a big > disadvantage of mathematics, because from then on every guy felt > entitled to create his own system of axioms. Too bad the earth is not flat anymore...
From: William Hughes on 14 Nov 2006 11:09 mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > ...The set > > {{1, 2}, {3, 4}} > > has two elements. It's cardinality is 2. And each of the elements is > > a set containing two elements. > > Yes, here you are talking about two unordered pairs of numbers. You use > a convenient way to denote that. Nothing else stands behind the {, } > symbols. In particular N = 1,2,3,... = {1,2,3,...}. That does not > prevent to build a set {{1,2,3,...}, 1} with two elements. > And noting that omega = {1,2.3 ...} we get that the set {omega,1} has two elements. William Hughes
From: Michael Stemper on 14 Nov 2006 13:02
In article <1163506063.849027.169180(a)m7g2000cwm.googlegroups.com>, mueckenh write: >Dik T. Winter schrieb: >> In article <1163433510.518520.69770(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: >> > Dik T. Winter schrieb: >> > > Not in set theory. AC is an axiom that may or may not hold, depending >> > > on the branch you are following. >> > >> > Zermelo considered it as "has to be taken". >> >> Perhaps he did. In modern mathematics there is no such imperative. >> Euclid considered that the parallel postulate "has to be taken". >> This does not mean that in modern mathematics it has to be taken. >I don't. The parallel axiom is necessary, correct, true in any >Euclidean plane. That's only because if you're working in a problem space where the parallel postulate is false, you're not in a Euclidean plane -- by definition. -- Michael F. Stemper #include <Standard_Disclaimer> Twenty-four hours in a day; twenty-four beers in a case. Coincidence? |