Cantor Confusion
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From: Han de Bruijn on 13 Oct 2006 04:52 Virgil wrote: > In article <ddeb9$452e55fe$82a1e228$16456(a)news1.tudelft.nl>, > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >>Set Theory is simply not very useful. The main problem being that finite >>sets in your axiom system are STATIC. They can not grow. > > One can quite easily construct set valued functions which do grow. WE > just do not call them sets. > >>Which is quite contrary to common sense. > > Does "Mueckenh" insist that everything must grow? > > THAT is quite contrary to common sense. I don't know about Mueckenheim's opinion in these matters, but an old Greek philosopher - Heraclitos - has said: "panta rei kai ouden menei" (everything flows and nothing remains the same). And I agree with that. Han de Bruijn
From: Han de Bruijn on 13 Oct 2006 05:00 David Marcus wrote: > Han de Bruijn wrote: > >>Dik T. Winter wrote: >> >>>In article <1160646886.830639.308620(a)c28g2000cwb.googlegroups.com> >>>mueckenh(a)rz.fh-augsburg.de writes: >>>... >>> > If every digit position is well defined, then 0.111... is covered "up >>> > to every position" by the list numbers, which are simply the natural >>> > indizes. I claim that covering "up to every" implies covering "every". >>> >>>Yes, you claim. Without proof. You state it is true for each finite >>>sequence, so it is also true for the infinite sequence. That conclusion >>>is simply wrong. >> >>That conclusion is simply right. And yours is wrong. Completed infinity >>does not exist. So _each_ finite sequence "means" the infinite sequence. > > Are you saying that "completed infinity" does not exist in standard > mathematics? If so, please define "completed infinity". If not, then > please define "exist". Completed infinity does "exist" in standard mathematics. It's embodied by Cantor's Set Theory (: cardinals, ordinals, aleph_0 and some such). Define "exist". Existence in physics is given by nature itself. And I am a physicist by education. The mantra is: A little bit of Physics would be NO idleness in Mathematics. (A bit cryptic - so it seems - but it will do.) Han de Bruijn
From: Han de Bruijn on 13 Oct 2006 05:06 David Marcus wrote: > Han de Bruijn wrote: > >>Dik T. Winter wrote: >> >>>In article <1160647755.398538.36170(a)b28g2000cwb.googlegroups.com> >>>mueckenh(a)rz.fh-augsburg.de writes: >>>... >>> > It is not >>> > contradictory to say that in a finite set of numbers there need not be >>> > a largest. >>> >>>It contradicts the definition of "finite set". But I know that you are >>>not interested in definitions. >> >>Set Theory is simply not very useful. The main problem being that finite >>sets in your axiom system are STATIC. They can not grow. Which is quite >>contrary to common sense. (I wouldn't imagine the situation that a table >>in a database would have to be redefined, every time when a new row has >>to be inserted, updated or deleted ...) > > Is your claim only that set theory is not useful or is contrary to > common sense? Or, are you claiming something more, e.g., that set theory > is mathematically inconsistent? I said that set theory is not *very* useful. I have developed (limited) set theoretic applications myself, so I don't say it is useless. Yes, a great deal of set theory is contrary to common sense. Especially the infinitary part of it (: say cardinals, ordinals, aleph_0). I'm not interested in the question whether set theory is mathematically inconsistent. What bothers me is whether it is _physically_ inconsistent and I think - worse: I know - that it is. Han de Bruijn
From: Han de Bruijn on 13 Oct 2006 05:09 Alan Morgan wrote: > In article <990aa$452e542e$82a1e228$16180(a)news1.tudelft.nl>, > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >>Dik T. Winter wrote: >> >>>In article <1160646886.830639.308620(a)c28g2000cwb.googlegroups.com> >>>mueckenh(a)rz.fh-augsburg.de writes: >>>... >>> > If every digit position is well defined, then 0.111... is covered "up >>> > to every position" by the list numbers, which are simply the natural >>> > indizes. I claim that covering "up to every" implies covering "every". >>> >>>Yes, you claim. Without proof. You state it is true for each finite >>>sequence, so it is also true for the infinite sequence. That conclusion >>>is simply wrong. >> >>That conclusion is simply right. And yours is wrong. Completed infinity >>does not exist. > > sqrt(-1) doesn't exist either. Frankly, I have a much harder time > believing in "imaginary" numbers than I do believing in infinite > sets. Imaginary numbers have an image as vectors in the plane (Euclidian 2-D space). They can be conceived as ordered pairs of real numbers: (a,b) . Is that so hard to "believe"? Han de Bruijn
From: Han de Bruijn on 13 Oct 2006 05:10
Virgil wrote: > In article <1160675643.344464.88130(a)e3g2000cwe.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > >>With the diagonal proof you cannot show anything for infinite sets. > > Maybe "Mueckenh" can't but may others can. Define "can". Han de Bruijn |