An uncountable countable set
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From: Han de Bruijn on 18 Sep 2006 06:36 Mike Kelly wrote [ OK, let's keep the quotes intact this time ]: > Han.deBruijn(a)DTO.TUDelft.NL wrote: > >>Mike Kelly wrote: >> >>>You claimed that you have a very much better understanding of >>>probability than me. Since you know nothing of my knowledge of >>>probability other than that I disagree that it is meaningful to discuss >>>the probability of "a natural" being divisible by 3, [ ... snip ... ] >> >>What more evidence do we need, huh? > > Given that this is a *theorem* of probability theory I am mystified as > why this is evidence that I don't understand probability. Do you have > some alternative probability theory? The problem is the layer below Probability theory: Set theory. You say it correctly here: >>The good news is that you are doing wrong only _one_ thing: infinitary >>reasoning. You think that completed infinities do exist. > > If you don't accept the existence of a set of natural numbers then you > don't accept the set theory that probability theory is based upon and > you haven't suggested an alternative. Indeed, it seems somewhat odd to > complain about the conclusion of a theorem discussing an object you > don't accept even exists. The infinitary part of set theory that underpins probability theory is IMHO the only problem. >>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. Any such set is equipped with a uniform distribution. And hence "THE" naturals. 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& Han de Bruijn
From: Han de Bruijn on 18 Sep 2006 06:53 Mike Kelly wrote: I want to snip all this but Mike doesn't like it: > Han.deBruijn(a)DTO.TUDelft.NL wrote: > >>Mike Kelly wrote: >> >>>[ ... snip ... ] It's not clear to me that providing finite examples then >>>saying "obviously this holds for infinite cases too" without any >>>justification whatsoever should be at all convincing to anyone. >> >>It may be not clear to any mathematician, but it is clear to any >>scientist. The reason is that infinities do not really exist. >>They only exist as an attempt to make the "very large" rigorous >>in some sense. The moment you forget this, you get into trouble. > > But we are discussing whether there exists a uniform distribution over > the naturals. If you don't think this claim means anything at all then > why do you dispute it? If you reject the existence of the set of > natural numbers then you reject the set theory probability is based on. > So why bother to argue against individual theorems? You don't accept > *any* of probability theory. > > It seem your argument is based on the idea that infinites do not exist > in physical reality. But mathematics is abstract, so this seems an > absurd objection. End of virtual snipping. > If you refuse the idea of infinite sets, what does it mean to you to > say a function has domain and range R? I don't really refuse the idea of infinite sets. (How else could I do my calculus stuff ?) I only refuse the idea that the infinite can be something which is essentially different from the _finite_. In short: infinity is just finity in disguise. That's why I have no problem with limits. But I _have_ a problem with aleph_0. No, I have no problem with R, because I can add uncertainity to its members, and then the reals become just floating point numbers in the PC on my desk. Reals are very handsome idealizations of floats. But aleph_0 is not an idealization and it's not even handsome ... Han de Bruijn
From: Han de Bruijn on 18 Sep 2006 07:00 Tony Orlow wrote: > Han, if I prove inductively, say, that 2^x>2*x for all x>2, do you find > it objectionable to say that this also applies to any infinite value, if > such a thing existed, given that any infinite value would be greater > than any finite value, and therefore greater than 2? Give me one reason, Tony, why I would find such a theorem interesting in the first place. I'd prefer the ultimate terseness in mathematics, especially if it comes to infinities. > I think if Wolfgang and Han were offered a more sensible treatment of > the infinite case, they might find it more palatable. Affirmative. Han de Bruijn
From: Han de Bruijn on 18 Sep 2006 07:05 Tony Orlow wrote: > The agreement that I think Han and I came to in "Calculus XOR > Probability" was that such probabilities are infinitesimal. Affirmative. Sometimes people _do_ agree, even in 'sci.math'. Han de Bruijn
From: Han de Bruijn on 18 Sep 2006 07:08
Aatu Koskensilta wrote: > Tony Orlow wrote: > >> I haven't found myself rejecting one thing that Aatu has said so far. > > That's nice then. I find your posts much more entertaining than those of > Virgil, who doesn't seem to know when to stop - not that it's any > business of mine to tell people how to waste their time on USENET. And > on further reflection I think I was too hasty in calling you a crank - > you're probably merely eccentric. That said, I'm afraid I still have no > inclination to study your contributions, and they appear quite confused > or unintelligible to me in any case. Well, huh, how about the mere _volume_ of his postings? Terseness, Tony! Han de Bruijn |