An uncountable countable set
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From: Mike Kelly on 17 Sep 2006 08:10 Han.deBruijn(a)DTO.TUDelft.NL wrote: > Mike Kelly wrote: > > > I think Han demonstrated some gross ignorance of basic probability > > theory (along with limits, infinity, the difference between the > > physical sciences and math etc. etc.) and drew a very stupid analogy > > with calculus. > > The left hand of mainstream mathematics (Probability Theory) does not > know what the right hand (Calculus) is doing. Incorrect. You were utterly unable to defend this claim in the Calculus XOR Probability thread and have indicated that you're unwilling to try now. So stop making it. >It's impossible to have a > sensible debate with someone who is as brainwashed as you are, Mike. > And it's everywhere. You are seeing "differences" where there are none. What an empty statement. It's impossible to have a debate with someone who is so is so certain of his own position, Han. It works both ways. That you are incapable of seeing differences even when they are pointed out to you in exruciating detail does not mean they do not exist. > About my supposed "ignorance". Read this: > > http://hdebruijn.soo.dto.tudelft.nl/QED/singular.pdf > > And tell me what the flaws are in the mathematics of this paper. I have > dozens of the kind. I haven't questioned your ability to use the calculus. -- mike.
From: Mike Kelly on 17 Sep 2006 08:22 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.
From: Mike Kelly on 17 Sep 2006 08:28 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. If you refuse the idea of infinite sets, what does it mean to you to say a function has domain and range R? -- mike.
From: Tony Orlow on 17 Sep 2006 09:29 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. 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? 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 more than we need to label the balls we place in the vase, in order to know the vase will never empty. > > 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. > > 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? Why should set theory contradict so basic an understanding?
From: Tony Orlow on 17 Sep 2006 09:56
Aatu Koskensilta wrote: > Tony Orlow wrote: >> Hi Aatu - >> >> I appreciate your desire to accommodate the simply naive and confused >> by addressing issues in terms they can understand regarding >> mathematical questions. I think that's very human and generous of you, >> and good advice to anyone trying to teach. I do think that once we get >> into foundational arguments of the sort going on here, we really can't >> avoid such technicalities, since the validity, if not the soundness, >> of the arguments hinges on such fundamental issues. > > First order logic, rules of inference, and all that form a mathematical > tool that correctly captures, to an extent, the informal notion of > something logically following from a set of premises. This is shown by > the completeness theorem for first order logic with a few conceptual > considerations - famously due to Kreisel in his informal rigour paper. > Now, one might consider the basic idea of logical consequence in first > order context suspect or inadequate in some way, in which case an > alternative notion of logical validity etc. must be provided - most > likely not in the form of any formal system of logic with rules of > inferences and formal axioms, but in informal terms similarly as one > presents the classical or the intuitionistic picture. It is then up to > the individual mathematicians to evaluate the fruitfulness, > plausibility, coherence, applicability and so forth of the presented > idea of logic - in this process formalization might or might not be of > some help, e.g. by enabling a proof that the alternative picture is in > some sense incompatible but still intertranslatable with the classical > picture, or enabling one to establish conclusively that arguments of > this or that kind are not valid under the alternative conception of logic. > > Now, in addition to the question of logic one might simply choose to > reject this or that mathematical principle either as outright false or > simply as unjustified. Of course, without tweaking one's logic it is not > possible to accept a mathematical principle and reject some of its > consequences. Here too formal theories might be of some limited use, > e.g. allowing one to establish that some principle is indeed independent > of some other principle (over some suitable background assumptions). Yes, if one wants to draw different conclusions from the same premises, one has to tweak their logical system, though I have my doubts as to how far that can go. The main questions I see in logic are, as you mention, whether ~x v y really is the same as x->y as first order logic has it, and whether continuous logic (probabilistic) sheds useful light on this area. While that's interesting, I don't see that as the core problem with transfinite set theory. It's a matter of sacrificing core principles of sets, for me. > >> Yes, this is my point. When we speak of the "size" of a set, for >> finite sets it's the count. For infinite sets, some generalization >> becomes necessary. The issue for me is which tenets of finite sets do >> we consider most important to preserve as we generalize. > > The concept of "size" when applied to non-finite sets bifurcates into > many different concepts that just happen to agree on finite sets. The > most general such notion of size is provided by cardinality in the > Cantorian sense, since it doesn't presuppose any numerical ordering or a > notion of density or some such be provided along with the sets compared. 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. Trying to derive measure from a structure where no measure has been introduced is bound to fail. 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. > It also leads to a highly fruitful and beautiful mathematical theory > with applications in almost every area of modern mathematics. This is > not to say that other notions of "size" as applied to sets are ignored; > the idea that there are twice as many naturals as there are odd naturals > can be captured mathematically, although this notion is less general and > applies only in case the sets in question are equipped with additional > structure. > 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? 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? Cheers, Tony |