An uncountable countable set
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From: Tony Orlow on 17 Sep 2006 11:14 Aatu Koskensilta wrote: > Tony Orlow wrote: >> Well, I understand that, but it doesn't seem to me that one CAN >> measure an infinite set with any accuracy without involving some >> notion of measure into the properties of the elements which define the >> set. > > One obviously can, in so far as cardinality can be said to be a > "measure" of an infinite set. Whether comparison of sets in terms of > cardinality meets the criteria associated with some notion of > "measurement" can of course be questioned, but has pretty much nothing > to do with modern set theory. Can cardinality be said to be "accurate", in the sense that it detects all changes in set size? Not if a proper superset is not larger. It's a gross measure, a classification scheme, but not an accurate measure of any sort. > >> Cardinality gives a certain gross measure of complexity, but to >> consider the alephs to be any exact numbers of any sort is >> unjustified, in my opinion. > > You can consider them inexact numbers if you want. This has no effect on > the mathematical content of the set theoretical theory of cardinality. But it does have an effect on how cardinality is treated. If set theory is a generalization of all mathematics, then none of the general conclusions it draws should be at odds with any of the specific conclusions drawn throughout mathematics. That is, if it applies to all mathematics, then it applies to a superset of any field of mathematics, and should not contradict any such subset. > >> Yes, there are some concepts such as limiting density and Lebesgue >> measure, which come to different conclusions regarding the same sets. >> That is, they detect differences between some sets where cardinality >> does not. Doesn't this mean that bijection alone misses real >> distinctions in element count, or size, which can only be detected >> using more sophisticated methods? > > Cardinality is one property of sets, and obviously there are others - > which might in some context be more appropriate measures of "size" for > some mathematical purposes. But as said, when we consider infinite sets > notions such as "size" are ambiguous, and we might well get different > answers for different mathematical explications of different informal > ideas of what "size" means, even when those ideas are equivalent when > applied to finite sets. In case of general abstract set theory in which > we're dealing with sets that do not come with associated structure > cardinality is the only notion that can be universally applied. Can you give an example of transfinite set theory determining the cardinality of an infinite set with NO reference to the stricture of the set? Even the most basic limit ordinal is based on an inductive structure based on the successor function. > If > you're not interested in that, you're of course free to study structured > sets and devise measures that in some way better reflect some informal > notions of size associated with those sets and structures. If your > pursuits in this directions are in some interesting way connected to > something mathematicians find interesting, it is possible that they > would take interest in your new notions of size, but if you just go on > about how cardinality misses something without connecting your criticism > to actual mathematics, or produce interesting new mathematics - possibly > in some alternative framework - you'll just be ignored by everyone but > people like Virgil, who are perhaps even more eccentric than you. (You > might have noted that I haven't had anything substantial to say about > your ideas, and have only offered general reflections, for exactly that > reason.) Yes, you and I have not talked a lot in the past. I don't think you are familiar with my IFR and N=S^L approaches for quantitative and symbolic sets. I've put forth two new number systems which may be significant. Virgil will discount all this, until someone he respects gives it some credit. Perhaps we can discuss those things in the future. I am only now getting a new web site begun, and will include papers on these things. In the meantime, thanks for your reflections. > >> If general set theory comes to a conclusion that contradicts the >> conclusions of a theory that takes into account more details of the >> situation, then hasn't the generalization failed in the specific case? > > No. It just means there are many different notions of "size" that might > apply to sets of some kind. A contradiction would occur only if the > *same* mathematical notion of "size" led to different answers in the > same situation. Now, you might wish argue that we should call > cardinality something else than a "measure of the size of a set". But, > as you surely realise, there is no hope of changing entrenched technical > or informal terminology, nor is there really any reason to do so, simply > because the only measure of size in abstract set theory that applies > generally is cardinality. > Sure, it's an uphill battle to disagree with tradition. I didn't ask for a free ride. :) The question is this. If cardinality DOES apply generally, then why does it contradict so many specific cases? If I have a rule that all species of mammals produce milk and urine, then that should not have any exceptions, or it's not a general rule. If I have a rule that all mammals give live birth, well, that's generally true, but for a couple of exceptions. But if I have a rule that all mammals have four legs, we start to wonder how a good a rule it is, and if I say all mammals are nocturnal, we reject the rule. It's a matter of how many specific instances there are that contradict the rule So again, the question is, in how many specific situations, where we have more information, do the general rules fail? I think, in any case where the infinite set is better structured and defined than raw set theory allows, the conclusions of abstract set theory fail, and are supplanted by better rules. This leads me to believe that trying to measure infinite sets without any additional structure is simply impossible. Perhaps this seems crankish, but I am hopeful that in some sense you can agree that there is an issue there. :) Tony
From: Mike Kelly on 17 Sep 2006 11:16 Tony Orlow wrote: > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > Mike Kelly wrote: > > > >> Given that any second-year student of probability theory knows that > >> there are no uniform distributions over countable sample spaces, [ ... ] > > > > This "given" is most disturbing. Mainstream mathematics is so certain > > about its own right that no sensible debate is possible. > > There IS no LUB on the finites, omega notwithstanding. Omega's a > phantom. That's why you can't get any average value or any uniform > probability distribution. Vaguely correct, minus reflexive whining about Omega. >In general, it doesn't make sense to talk > about probability without a uniform probability distribution over a > finite set. That's absurd. I don't think you meant to say what you said here. Of course there are non-uniform probability distributions and probability distributions on infinite sets. > However, since probability is really a percentage, That is, a real number between 0 and 1. > any > subset which is a finite fraction of the whole can certainly have a > probability associated with it: that fraction. Only if a uniform distribution can be defined on the whole. > This discussion could not have occurred, say, regarding the primes, > because over the infinite range of R, n has 0% chance of being prime, > rather than a 1/3 chance. Still, as every natural has an equal chance, > in theory, of being selected from the vase o' balls, every natural has a > chance which is not strictly 0. And the same goes for a random natural > being prime. > > The agreement that I think Han and I came to in "Calculus XOR > Probability" was that such probabilities are infinitesimal. Probabalities are never infinitesimal. They are real numbers between 0 and 1. -- mike.
From: Mike Kelly on 17 Sep 2006 11:23 Tony Orlow wrote: > 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. Yes I have. I'm not sure how to provide a direct link to my post but it's number 2180 on Google Groups when the thread is sorted by date. While you're reviewing things you've missed, you haven't responded to a number of my posts in the last couple of days.You never responded to post 2176, about the ...1111 thing, for example. >Why is that? I proved to you that 1/3 > of N are multiples of 3. What more do you need? You proved that the density of the multiples of 3 in N is 1/3. It hardly needed proving, I would not dispute it if you just claimed it without proof. What I dispute is your complete non sequitur "OBVIOUSLY, therefore, the probabaility of selecting a multiple of 3 when you select a random natural is 1/3." This only makes any sense at all if one can select a (uniformly) random natural. One can't. You didn't address that point in any way in your post. -- mike.
From: Tony Orlow on 17 Sep 2006 11:31 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, thank you for withdrawing your assessment of my crankhood. That was the first thing you said that I rejected (though I think you said it once or twice before, but we hadn't conversed much so I didn't take it personally). I am well aware my position is "provably false", but I am also aware that that depends on the axioms assumed and the rules regarding logical inference. Hopefully, a recognition of that fact convinced you I am not just being a stupid yankster. I don't deny being called eccentric. That seems to be the general opinion of the world around me, and I'm used to it. After all, I waste my time on math questions. What could be more eccentric? ;) So, I don't reject that either. You may not have any inclination to pay attention to my ideas, and you may wonder why Virgil wastes his time, but Virgil is doing a service. He's defending his territory, which is under attack. It's not just me doing the attacking, but Virgil and I have done a lot of sparring, and we both enjoy it. I've gained a lot from it anyway. I don't know about Virgil, so maybe he IS wasting his time. Or, maybe, he's investing it in calling me to the carpet, because he knows this fight has to happen. In any case, I respect Virgil, though he doesn't ever seem to grow, and wish him well. So, while I may seem confused or unintelligible, well, I think we have some interesting conversations ahead of us. We'll see. Have a nice day! Tony
From: Tony Orlow on 17 Sep 2006 11:33
stephen(a)nomail.com wrote: > 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 Han doesn't think the set of all finite naturals is a completed infinity. You forgot that crucial point. Oooops. |