An uncountable countable set
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From: Han de Bruijn on 19 Sep 2006 05:00 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". Han de Bruijn
From: Mike Kelly on 19 Sep 2006 05:02 Han de Bruijn wrote: > Mike Kelly wrote: > > > Han de Bruijn wrote: > > > >>Mike Kelly wrote [ OK, let's keep the quotes intact this time ]: > >> > >>>Han.deBruijn(a)DTO.TUDelft.NL wrote: > >>> > >>>>Mike Kelly wrote: <snip> > >>>>Once you stop > >>>>thinking this way, everything falls in its place and you will see that > >>>>it is quite meaningful to discuss the probability of "a natural" being > >>>>divisible by 3. > >>> > >>>It is meaningful to say that a natural drawn uniformly at random from a > >>>set of consecutive naturals 1 thru 3n has a 1/3 probabaility of being > >>>divisible by 3. Nobody disputes this. But talking about the probability > >>>of "a natural" being divisible by 3 implies a uniform distribution over > >>>the naturals. Such a thing does not exist. > >> > >>The core of the matter is that THE naturals can only exist as a set of > >>consecutive naturals 1 thru n where n is large and undefined. > > > > Why? > > Because completed infinities do not exist. Why? "Exist" in what sense? > >>Any such set is equipped with a uniform distribution. And hence "THE" naturals. > > > > Such a set doesn't contain all naturals, so in what sense is it "THE" > > naturals? > > 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? > >>This does not say that meaningful answers (i.e. independent of n) can > >>always be obtained. But: mainstream mathematics HAS found a way out of > >>al this. It's called the theory of "natural densities" or some such .. > >>Why not substitute? But this belongs to another thread: > >> > >>http://groups.google.nl/group/sci.math/msg/225dca8f63d0d6ae?hl=en& > > > > Not changed much, have you? > > Give me one good reason why I should. You are making a fool of yourself by clinging to comprehensively refuted arguments. You are being rude by refusing to follow etiquette. You are being dishonest by misrepresenting what people say to support your position. Of course, I wasn't telling you that you *should* change because clearly you can't. I was just making an observation. -- mike.
From: Mike Kelly on 19 Sep 2006 05:09 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. > >>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. -- mike.
From: Han de Bruijn on 19 Sep 2006 05:11 Virgil wrote: > In article <70eb8$450e4cce$82a1e228$14803(a)news1.tudelft.nl>, > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >>Virgil wrote: >> >>>In article <cf41b$450a799d$82a1e228$10666(a)news1.tudelft.nl>, >>> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >>> >>>>But, fortunately, reality is more simple than this. Every equality is >>>>an equivalence relation. And every equivalence relation is an equality. >>>>So the bijection between tally marks or collected pebbles with counted >>>>objects means, indeed, that tally marks and moving pebbles ARE numbers. >>> >>>Not in mathematics. >> >>Not in _your_ mathematics. > > Where, in mathematics or anywhere else, is a pebble indistinguishable > from a nick in a stick? Of one can distinguish between them, then they > are not the same thing. "Indistinguishable" is not the same as "indistinguished". Two things are equal if we DON'T WANT to distinguish them. Not if they are indistinguishable. Any two things are distinguishable and equality would not even exist if it were dependent on that. Han de Bruijn
From: Mike Kelly on 19 Sep 2006 05:16
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. -- mike. |