An uncountable countable set
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From: Mike Kelly on 19 Sep 2006 06:38 Han de Bruijn wrote: > Mike Kelly wrote: > > > Han de Bruijn wrote: > > > >>Mike Kelly wrote: > >> > >>>Han de Bruijn wrote: > >>> > >>>>All naturals do not exist. What is "all"? > >>> > >>>Huh? So some naturals don't exist? What does that mean? How can > >>>something that doesn't exist be a natural number? > >> > >>"All naturals" is undefined, void of meaning. Got it? > > > > Not really, no. If something is true for "All naturals" it is true for > > any object which is a natural number. The set of all naturals is the > > set which has as an element every object which is a natural number and > > no element which is not a natural number. > > > > I suppose you're just going to respond "but there is no such set!" - > > presupposing your conclusion; that there are no "completed infinities". > > Vigorous assertion is so convincing. > > So there is no lack of understanding on your part. Good! So you admit you don't actually have an argument beyond assertion? At least mathematicians recognise that the "existence of completed infinities" is not provable one way or the other. -- mike.
From: Han de Bruijn on 19 Sep 2006 06:38 Mike Kelly wrote: > Han de Bruijn wrote: > >>Mike Kelly wrote: >> >>>Han de Bruijn wrote: >>> >>>>Mike Kelly wrote: >>>> >>>>>Han.deBruijn(a)DTO.TUDelft.NL wrote: >>>>> >>>>>>Mike Kelly wrote: >>>>>> >>>>>>>Infinite natural numbers. Tish and tosh. Good luck explaining that idea >>>>>>>to schoolkids. >>>>>> >>>>>>Look who is talking. Good luck explaining alpha_0 to schoolkids. >>>>> >>>>>Sure, the theory of infinite cardinals is beyond (most)schoolkids. But >>>>>this is a bad analogy, because school kids don't need to know about >>>>>cardinals but they do need to know how to work with natural numbers. My >>>>>point, if you really missed it, was that Tony's ideas of "infinite >>>>>natural numbers" don't match up to our "naive" or "intuitive" idea of >>>>>what numbers should be - how we were taught to do arithmetic in school. >>>>>I for one don't understand what the hell an "infinite natural number" >>>>>is. And yet supposedly the advantage of his ideas are that they're more >>>>>intuitive than a standard formal treatment. >>>> >>>>My point is that the pot is telling the kettle that it's black (: de pot >>>>verwijt de ketel dat ie zwart is). Your aleph_0 is in no way better than >>>>Tony's "infinite natural number". >>> >>>Your analogy is terrible, as usual. >>> >>>My point was that Tony's "infinite natural numbers" are not compliant >>>with everyday arithmetic. Aleph_0 is part of a formalisation that leads >>>to an arithmetic that works exactly as we expect it to. >> >>"... that works exactly as we expect it to". Ha, ha. Don't be silly! > > So, what part of the arithmetic on natural numbers defined rigorously > as sets doesn't match up to the "naive" arithmetic we were taught at > school? I thought you meant the arithmetic with transfinite numbers. No? Han de Bruijn
From: Han de Bruijn on 19 Sep 2006 06:42 Mike Kelly wrote: > Han de Bruijn wrote: > >>Mike Kelly wrote: >> >>>Han de Bruijn wrote: >>> >>>>Mike Kelly wrote: >>>> >>>>>Han de Bruijn wrote: >>>>> >>>>>>All naturals do not exist. What is "all"? >>>>> >>>>>Huh? So some naturals don't exist? What does that mean? How can >>>>>something that doesn't exist be a natural number? >>>> >>>>"All naturals" is undefined, void of meaning. Got it? >>> >>>Not really, no. If something is true for "All naturals" it is true for >>>any object which is a natural number. The set of all naturals is the >>>set which has as an element every object which is a natural number and >>>no element which is not a natural number. >>> >>>I suppose you're just going to respond "but there is no such set!" - >>>presupposing your conclusion; that there are no "completed infinities". >>>Vigorous assertion is so convincing. >> >>So there is no lack of understanding on your part. Good! > > So you admit you don't actually have an argument beyond assertion? At > least mathematicians recognise that the "existence of completed > infinities" is not provable one way or the other. No argument beyond assertion AND (empirical) scientific evidence: http://huizen.dto.tudelft.nl/deBruijn/grondig/natural.htm#oo Han de Bruijn
From: Mike Kelly on 19 Sep 2006 06:45 Han de Bruijn wrote: > Mike Kelly wrote: > > > Han de Bruijn wrote: > > > >>Mike Kelly wrote: > >> > >>>Han de Bruijn wrote: > >>> > >>>>Mike Kelly wrote: > >>>> > >>>>>Han de Bruijn wrote: > >>>>> > >>>>>>Mike Kelly wrote in response to Tony Orlow: > >>>>>> > >>>>>>>*sigh*. Probabilities are *standard* real numbers between 0 and 1. > >>>>>> > >>>>>>Yes. And infinitesimals are *standard* real numbers in engineering. > >>>>> > >>>>>Engineering is not mathematics. It uses mathematical results. > >>>> > >>>>Sure. And moslems are not praying. Only roman catholics do. > >>> > >>>Bizarre and borderline offensive analogy. Engineering isn't > >>>mathematics, anymore than accounting is mathematics. They both *use* > >>>mathematics. > >> > >>Wrong. They both *create* mathematics as well. I know, because I've been > >>active in both engineering and adminstration. > > > > But not in mathematics, evidently. So how do you tell the application > > of mathematics from mathematics itself, then? > > Good question! I find it impossible to distinguish them, while it seems > that you can do it. Unambiguously ? Please tell me how, because I don't > understand. Mathematics is the derivation of theorems from axioms. The application of mathematics is using the theorems of some mathematical theory to solve problems that the theory can model. > >>>>>>That's why infinitesimal probabilities will become feasible as soon > >>>>>>as mathematics becomes a science which is compliant with engineering. > >>>>> > >>>>>Mathematics is not a science. What exactly would it mean for it to > >>>>>"become a science compliant with engineering"? > >>>> > >>>>Ah! Mathematics is not a science. So mathematics is not serious at all! > >>>>Why didn't you tell me this before? > >>> > >>>How do you get from "mathematics is not a science" to "mathematics is > >>>not serious"? Time to brush up on your English, Han. > >> > >>How can mathematics be serious, if it is not scientific? > > > > Definately a translation issue. Can I expect a lecture from you on the > > proper meaning of the word "science", now? > > Not now. But _you_ started saying that "Mathematics is not a science". > How can you be so sure, if you need an explanation from me? I don't need an explanation from you. You really can't read English, can you? -- mike.
From: Mike Kelly on 19 Sep 2006 06:47
Han de Bruijn wrote: > Mike Kelly wrote: > > > Han de Bruijn wrote: > > > >>Mike Kelly wrote: > >> > >>>Han de Bruijn wrote: > >>> > >>>>Mike Kelly wrote: > >>>> > >>>>>Han.deBruijn(a)DTO.TUDelft.NL wrote: > >>>>> > >>>>>>Mike Kelly wrote: > >>>>>> > >>>>>>>Infinite natural numbers. Tish and tosh. Good luck explaining that idea > >>>>>>>to schoolkids. > >>>>>> > >>>>>>Look who is talking. Good luck explaining alpha_0 to schoolkids. > >>>>> > >>>>>Sure, the theory of infinite cardinals is beyond (most)schoolkids. But > >>>>>this is a bad analogy, because school kids don't need to know about > >>>>>cardinals but they do need to know how to work with natural numbers. My > >>>>>point, if you really missed it, was that Tony's ideas of "infinite > >>>>>natural numbers" don't match up to our "naive" or "intuitive" idea of > >>>>>what numbers should be - how we were taught to do arithmetic in school. > >>>>>I for one don't understand what the hell an "infinite natural number" > >>>>>is. And yet supposedly the advantage of his ideas are that they're more > >>>>>intuitive than a standard formal treatment. > >>>> > >>>>My point is that the pot is telling the kettle that it's black (: de pot > >>>>verwijt de ketel dat ie zwart is). Your aleph_0 is in no way better than > >>>>Tony's "infinite natural number". > >>> > >>>Your analogy is terrible, as usual. > >>> > >>>My point was that Tony's "infinite natural numbers" are not compliant > >>>with everyday arithmetic. Aleph_0 is part of a formalisation that leads > >>>to an arithmetic that works exactly as we expect it to. > >> > >>"... that works exactly as we expect it to". Ha, ha. Don't be silly! > > > > So, what part of the arithmetic on natural numbers defined rigorously > > as sets doesn't match up to the "naive" arithmetic we were taught at > > school? > > I thought you meant the arithmetic with transfinite numbers. No? In what way is the arithmetic of transfinite numbers part of everyday arithmetic??? -- mike. |