An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: Tony Orlow on 17 Sep 2006 20:06 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, such that all individual probabilities sums to 1? I think that was the salient import of the XOR thread. > >>>> 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. > >>>> 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.
From: Virgil on 18 Sep 2006 00:52 In article <450de164(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > >> Mike Kelly wrote: > >>> 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. And as TO keeps saying he cannot count the naturals... > > > > > 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. There can be lots of distributions on the naturals, just not a uniform one. > > Do they exist for "uncountably" (aka actually) infinite sets? Is there > an average value of the reals in [0,1]? There is an expected value, definable in terms of integrals, which for the uniform distribution works out to be equal to \integral[x=0..x, x] / \integral[x=0..1, 1] = 1/2 > > That you are unable to grasp it points to a problem with you, not with the > > treatment. > > That I am able to grasp it enough for you to leave questions unanswered > bodes ill with you. That areas of mathematics contradict the > "foundations" thereof is unacceptable. That any areas of mathematics contradict those foundations is mere delusion. > I believe it was Virgil (could be wrong) who used the term, today or > yesterday, "over the range of R". Maybe it was Stephen. In any case, > it's obvious what it means. it was TO that used it, and it is not at all obvious what it means. > > But so, you agree that, given certain considerations, the set of even > naturals can be said to be half the size of the set of naturals? And in certain other conditions twice the size, or any other multiplier one wants. It is only in cardinality that the size is fixed. >If so, > then aren't the conclusions of cardinality not generally true, if > equivalent cardinality is taken to be "the same size, in every respect"? Straw man fallacy!!! Cardinality does not say "the same size in every respect", it merely says the same size in one respect, bijectability. Only TO conflates the meaning that way. > > > > > The evens have a density of 1/2 in the naturals. Nobody disputes this. Absent any definition of "density", no one concedes it either. > > The problem is that the general foundation contradicts many of the > specifics. I see no contradictions that are not the directly attributable to TO's misrepresentations. > > > >>>> 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. > >>> > >> If it claims that the vase is empty, when sequences tell us the vase is > >> not, then a set is not a generality of a sequence. > > > > Nonsensical sentence as far as I can tell. What would it mean for a set > > to be "a generality of a sequence"? > > It would mean that set theory was the entire foundation upon which is > built the theory of sequences. Apparently, it is not. All sequences are definable as functions whose domains are either the set of naturals or some bounded subset of the naturals. In that sense, the entire theory of sequences can be derived from set theory. What part of sequences does TO suggest is not so derivable? > > > > >> If set theory claims > >> to subsumes all of math, then it cannot contradict any other part of math. > > > > Set theory doesn't claim to subsume all of math. People use it in > > (almost) every area of math because it works extremely well. > > Does it cover the entire field of sequence theory or not? Can > conclusions drawn from set theory be properly applied to such specific > sequences as in the ball-and-vase problem? If not, then why is the > set-theoretic answer considered correct? As a matter of common sense, if everything that is put into a vase is taken out before the contents are analyzed, there is nothing left in the vase. TO seems to think otherwise. > > > > > Meanwhile, you continue to argue against a strawman. Cardinality > > doesn't claim to be the best or only generalisation of size to infinite > > sets - only the most general. > > If it were generally true, it wouldn't contradict any specific > conditions, would it? And it doesn't. > And you continue wasting your time believing that you can produce size > for infinite sets without introducing measure to the elements. And TO believes he can measure the sizes of sets by insisting that no set may exist without a "natural" order defined on it on which its "size" depends. So that the same set given a different order is a different size.
From: Virgil on 18 Sep 2006 00:58 In article <450de2e8(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > >> Mike Kelly wrote: > >>> 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, such that all individual > probabilities sums to 1? I think that was the salient import of the XOR > thread. Look up measure theory and Lebesgues integration. > > > > >>>> 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 Because that is the way the definitions work. Look them up for yourself.
From: Han de Bruijn on 18 Sep 2006 03:21 stephen(a)nomail.com wrote: > Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > >>mueckenh(a)rz.fh-augsburg.de wrote: > >>>Representation is number. There is no difference. Numerals have no >>>"soul". > >>Whew! I've never heard someone expressing this fact so lucidly! > > So 3/2 is a different number than 6/4? The representations > differ, and if there is no difference between representation > and number, then different representations must imply different > numbers. For that matter 6 birds is a different representation > of 6 than 6 cars. And 6 birds is a different representation > of 6 than 6 different birds? How many 6's are there? Try to understand what equality means. Han de Bruijn
From: Han de Bruijn on 18 Sep 2006 03:25
Tony Orlow wrote: > Hi Han - How are you? Welcome to the thread. > Have a nice day! Same to you! Han de Bruijn |