Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: David Marcus on 18 Nov 2006 20:21 Franziska Neugebauer wrote: > David Marcus wrote: > > Franziska Neugebauer wrote: > [...] > >> > >> Wrong. You have been discussing matrices. At least William Hughes > >> did. Are you both writing at cross purposes? > > > > WM is always writing at cross purposes to everyone! > > I don't take him too seriously as long as it is fun. I think we'll have more fun if we talk to him on his terms rather than try to get him to be precise and formal. He doesn't know how to do the latter. So, he just reverts to repeating himself. -- David Marcus
From: David Marcus on 18 Nov 2006 20:24 Franziska Neugebauer wrote: > David Marcus wrote: > > At least I defined what I meant. And, I never said "equilateral"! > > I prefered "monangle" or simply "angle". That's a good name for it. -- David Marcus
From: Randy Poe on 18 Nov 2006 20:32 mueckenh(a)rz.fh-augsburg.de wrote: > Randy Poe schrieb: > > > Lester Zick wrote: > > > On 16 Nov 2006 11:32:41 -0800, mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > > > > >David Marcus schrieb: > > > > > > > >> > > Indeed. If people *object* to an axiom, that is philosophy. > > > >> > > > > >> > But if people choose a set of axioms, that is what? > > > >> > > > > >> > > Everyone is welcome to choose their own axioms. > > > >> > > > > >> > That's mathematics? > > > >> > > > >> Of course. > > > > > > > >And if people not decide to use an axiom, that is what? > > > > > > Mathematics obviously. David says so. > > > > Any reasonable person would say so. It's mathematics on > > a system which is missing that axiom. It's still perfectly good > > mathematics. > > > > Haven't you seen people discuss ZF vs. ZFC for instance? Both are > > mathematics. ZFC has the axiom of choice, people working in ZF > > choose not to use that axiom. > > Haven't we seen people discuss the axiom of infinity? Yes. And it seems to be one you reject. Now the mathematics that you are arguing with includes the axiom of infinity. You are perfectly welcome to develop your own, though most of us can't figure out how you could do anything rigorously involving such things as the set of natural numbers without it. Nevertheless, you are free to develop a mathematics without that axiom, developed rigorously and logically from the axioms you do accept. The problem is that "rigorously and logically" does not describe your method of drawing "conclusions". - Randy
From: David Marcus on 18 Nov 2006 20:32 Franziska Neugebauer wrote: > David Marcus wrote: > > Franziska Neugebauer wrote: > >> mueckenh(a)rz.fh-augsburg.de wrote: > >> > Franziska Neugebauer schrieb: > >> >> mueckenh(a)rz.fh-augsburg.de wrote: > >> >> > What *counts* is this: It is asserted that the number of natural > >> >> > numbers is omega > >> >> > >> >> "It" is asserted that the *set* of natural numbers is (does exist) > >> >> and is named omega. The *cardinality* of omega is aleph_0. > >> > > >> > The cardinality of omega is omega. > >> > >> The cardinality of omega is |omega| not omega. > > > > Kunen's "Set Theory" defines |A| to be the least ordinal that can be > > bijected with A. So, with this definition, |omega| = omega. > > You are absolutely right. But we debate the wrong thing instead of True. It is best to keep the ordinals, cardinals, and natural numbers separate, especially for people like WM. > ,----[ WM in <1163782643.189768.296460(a)e3g2000cwe.googlegroups.com> ] > | What counts is this: It is asserted that the number of natural > | numbers is omega (the elements of the first column = number of lines). > | By adding 1 element to the first column we see that, if this assertion > | is correct, the number increases to omega + 1. But are there enough > | lines? No. That means: The finite natural numbers are not sufficient > | to stand in bijection with all omega elements of the column, but only > | with the finite ones. Therefore the unavoidable conclusion is: There > | are not omega finite numbers. > `---- It must have taken WM a lot of work to become this confused. If he stopped calling everything a number, it might help. As it is, he is hopelessly muddling ordinals and cardinals. -- David Marcus
From: David Marcus on 18 Nov 2006 20:35
Virgil wrote: > In article <1163874845.525985.180420(a)m73g2000cwd.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > I agree. But on the other hand, the cardinality of the sequence (d nn), > > i. e. omega, cannot be > > greater than the cardinality of every single line of that "list" or > > "matrix". > > WM seems to be conflating the set of lines with individual lines. That would be consistent since WM doesn't believe in sets. -- David Marcus |