An uncountable countable set
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From: Han.deBruijn on 18 Sep 2006 09:12 Mike Kelly wrote: > Tony Orlow wrote: > > it's not true that you can put ten balls in a vase and take one out, > > over and over, and ever get an empty vase. > > Because you can't physically perform an infinite sequence of operations > in finite time. No. That's not the real reason. There are many mathematical models of physical phenomena for which "an infinite sequence of operations in finite time" is required, like e.g. in Zeno's paradox. The real reason is that the infinity in the "balls in a vase" paradox is of a vastly different nature: it cannot be understood as a limiting case of something finite. It's a jump. And nature doesn't jump. Han de Bruijn
From: Han.deBruijn on 18 Sep 2006 09:14 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. Han de Bruijn
From: Mike Kelly on 18 Sep 2006 09:27 Tony Orlow wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > >> Mike Kelly wrote: > >>> 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. > >> To what? > > Ahem! No answer? To a request for clarification of a curt dismissal? Hmmm... Irrelevant to whether there can be a uniform probability distribution on the naturals. You know, the topic at hand? It's pretty rich for you to get uppity about one missed question on what should have been an obvious point. You don't respond to salient points /all the time/. Heck, in the last few days you've completely ignored several of my posts then complained that I didn't respond to your posts! While we're on the subject of unanswered questions : Which axioms or logic of ZFC set theory do you reject? Do you understand now that the sum of an infinite series need not exist? Do you admit that your attempt at a proof that the naturals can be bijected with their powerset was totally bogus? Do you understand that Cardinality doesn't claim to be the only or best analogy to "size" for finite 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, but we need to know that it is possible to define a uniform > >>> distribution on the set. > >> Which requires an average, which requires a range. > > > > What does "requires an average" mean? > > It means it requires a count and a sum, and the notion of division. *sigh*. So what does "requires a count and a sum, and the notion of division" mean? And what does any of this have to do with probability? Are we talking about probabiltiy theory here or your own vague ideas about what you think probability theory should be? > > Loosely speaking, to define a uniform distribution to select an element > > from a set one assigns a contsant probability to each element such that > > they all sum to 1. No such contstant exists for countable sets. > > They cannot exist for "countably infinite" sets, since those have no > upper bound (omega notwithstanding). Without an upper bound, there's no > mean, and no distribution. > > Do they exist for "uncountably" (aka actually) infinite sets? Yes, continuous probability distributions. >Is there an average value of the reals in [0,1]? Yes, 1/2. >No, that also would require the conception of a value less than any finite, an >infinitesimal probability for each real, which would sum to 1. No, it requires that the integral is 1. A probability measure is required to be 1. In the discrete case this means the summation of all the elementary probabilities must be 1. For continuous probability distributions, the integral must be 1. >So, you probably reject that notion > as well. Why? > However, the average value of the reals in [0,1] is quite > obviously 1/2. Sure. >So, you have a bit of a problem there. Why? >Yes, there is an > average of the reals in that interval. Set theory contradicts this area > of mathematics, which means it isn't the foundation for all math. Of course it doesn't. That you know so little about probability theory that you don't know the difference between a discrete and a continuous probability distribution is not an indication that it is full of contradictions. > >>>>> 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. > >> That is true, for the reason I just mentioned. The standard streatment > >> is nonsensical. > > > > You really do have a hugely inflated opinion of yourself. > > That's a survival technique in Kroneckerland. > > > The standard treatment makes a huge amount of sense. > > Define "huge" and "sense". For example, all the mathematics built upon it would seem to indicate that it makes sense. > > That yo
From: Mike Kelly on 18 Sep 2006 09:32 Tony Orlow wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > >> Mike Kelly wrote: > >>> 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. > >> Okay. I misspoke. But what about uniform probability distributions on > >> infinite sets in general? > > > > They don't exist on countably infinite sets. One can have continuous > > uniform distributions on real intervals [a,b]. > > That answers my question from another post, but then, what is the > probability of each real being chosen, 0. >such that all individual probabilities sums to 1? They don't. They integrate to 1. The PDF is 1/(b-a) between a and b and 0 everywhere else. >I think that was the salient import of the XOR thread. If that was the salient point of the XOR thread then the rebuttal is really rather short : discrete and continuous distributions are different. These differences lie in dissimilarity and the fact that they are not the same. > >>>> However, since probability is really a percentage, > >>> That is, a real number between 0 and 1. > >> Yes. > >> > >>>> 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. > >> Why? A probability IS a fraction. A random number n has x chance of > >> being in any subset of N which is x portion of N. > > > > Only if there is a uniform distribution on N. There isn't. > > Why does that matter, if the definition of the subset gives it a > limiting density within the set? The is what a probability really is, a > relatively finite portion. Because for a discrete distribution, the elementary probabilities have to sum to 1. It's the definition of a probability measure. > >>>> 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. > >>> > >> Everything between 0 and 1 is a real number. An infinitesimal is a real > >> less than any finite real, the recioprocal of any infinite real, which > >> is greater than any finite real. The reciproacl of anything greater than > >> 1 lies in [0,1]. > > > > *sigh*. Probabilities are *standard* real numbers between 0 and 1. > > > > *Standard* probabilities are *standard* real numbers from 0 to 1. Right! -- mike.
From: Mike Kelly on 18 Sep 2006 09:40
Han de Bruijn wrote: > imaginatorium(a)despammed.com wrote: > > > _How_ would you draw a ball from a vase containing an infinite set of > > balls. > > Yaaawn! This has been discussed, at length, as well: > > http://huizen.dto.tudelft.nl/deBruijn/grondig/natural.htm#bv > > Han de Bruijn Why provide a link that is completely irrelevant to the question asked? -- mike. |