An uncountable countable set
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From: Aatu Koskensilta on 17 Sep 2006 10:41 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. -- Aatu Koskensilta (aatu.koskensilta(a)xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Tony Orlow on 17 Sep 2006 10:45 Mike Kelly wrote: > 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 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. > >> 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. > Mike, you haven't responded to my use of IFR to derive the very conclusion that Han is espousing. Why is that? I proved to you that 1/3 of N are multiples of 3. What more do you need? TOny
From: Tony Orlow on 17 Sep 2006 10:47 Aatu Koskensilta wrote: > Tony Orlow wrote: >> Do you think he would have disagreed with you, Aatu? > > I think there's much I disagree about with Virgil. In particular his > conception of what mathematics is about seems extremely wrongheaded. > Well, I agree, but I'm not into ganging up on people. Still, given your thoughtful posts, I am curious to know in which ways you think he is wrongheaded in his conception, if not simply by being over-technical when discussing general concepts. It seems to me that he seems overly invested in certain ways of think, though I have no doubt that many would say the same of me. :) Thanks, Tony
From: stephen on 17 Sep 2006 11:06 Mike Kelly <mk4284(a)bris.ac.uk> wrote: > Han.deBruijn(a)DTO.TUDelft.NL wrote: >> Mike Kelly wrote: >> >> 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. You gotta love Han's claim: The probability of picking a number divisble by 3 from the set of all natural numbers, which does not exist, is 1/3. <snip> > 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. But that is just fine according to Han's logic. No uniform distribution exists over the set of all naturals because the set of all naturals does not exist. Therefore the probability of choosing a number divisible by 3 from the set of all naturals is 1/3, according to the non-existent uniform distribution on the non-existent set. Stephen
From: Mike Kelly on 17 Sep 2006 11:10
Tony Orlow wrote: > Mike Kelly wrote: > > 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. > > 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? > > > > > 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. > > Just because someone disagrees with the transfinite portions of set > theory doesn't mean they reject all of set theory. Clearly those of us > who object do so on the basis of the conclusions drawn in infinite case, > which derive from the axiom of infinity and/or the axiom of choice. So, which do you reject? The axiom of infinity or the axiom of choice? >As > far as probability goes, it certainly depends on the concept of sets, > since probability more or less measures a subset of events with respect > to the entire set of possible events. However, the same question remains > as with the rest of transfinitology - is the cardinality generalization, > based solely on raw bijection, really the most appropriate > generalization from the finite to the infinite for sets? Irrelevant. >Do we need to > know the last element and exact range to derive a probability for > something as simple as "n is a multiple of 3"? No, but we need to know that it is possible to define a uniform distribution on the set. > > 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. > > I think if Wolfgang and Han were offered a more sensible treatment of > the infinite case, they might find it more palatable. Uh, like yours you mean? Snicker. Han rejects completely the existence of a "completed infinity". This is pretty close to the opposite of what you're trying to do. > > If you refuse the idea of infinite sets, what does it mean to you to > > say a function has domain and range R? > > > > What does it mean to you, if not that one can use that range as a > constant for infinite sets? Why can't we say that, over the entire range > of R, the naturals have twice the density of the evens, and so are twice > as large a set? You're free to do so. The evens do have have a density of 1/2 in the naturals (not in R, I'm not sure you meant to say R. I think they both have 0 density in R...). If you want to think of this as meaning the set is twice the "size" then you're free to do so. >Why should set theory contradict so basic an understanding? It doesn't. You really don't get it, do you? I've told you half a dozen times : cardinality doesn't claim to be the only or the "correct" generalisation of size to infinite sets. -- mike. |