An uncountable countable set
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From: Tony Orlow on 17 Sep 2006 10:18 Han.deBruijn(a)DTO.TUDelft.NL wrote: > Mike Kelly wrote: > >> astounded that you are claiming that employing a mathematical argument >> that is not your own invention is plagiarism. Perhaps you are simply >> unaware of the meaning and connotations of the word. Plagiarism is >> dishonest and in many cases criminal. A fairly hefty accusation. > > I've been looking for a good English equivalent of the Dutch word > "meeloperij" and found "plagiarism" as my best match. I think that you > are right and that it's actuallty a mismatch. I apologize for this > fact but I don't know what shorthand expression to substitute instead. I like "regurgitation", but you could try "aping", "mimicking", "parroting" or "reciting the creed" if you prefer. Hope that helps! English is a rich language. ;) > > What I meant to express is that you are about to be parrotting Very good!!! (except only one 't') > mainstream arguments, without adding to it much thoughts of > yourself. And that is quite senseless because we have gone > through all this already. > > Han de Bruijn >
From: Tony Orlow on 17 Sep 2006 10:27 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. In general, it doesn't make sense to talk about probability without a uniform probability distribution over a finite set. However, since probability is really a percentage, any subset which is a finite fraction of the whole can certainly have a probability associated with it: that fraction. 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. > >> Finally, please stop with the scare quotes. They make you look like a >> "tool". > > Sorry. I don't know what "scare quotes" are > and I don't know what I'm doing wrong here. You're scaring Mike. Now he has to change his pants again. Bad! > > Han de Bruijn > :) Tony
From: Tony Orlow on 17 Sep 2006 10:32 Virgil wrote: > In article <450c86f8(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <450c7444(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>> Virgil wrote: >>>>> In article <450c6210(a)news2.lightlink.com>, >>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>>> Let's use 1/aleph_0 for r and see what happens. >>>>> >>>>> Since probabilities are necessarily real numbers and 1/aleph_0, whatever >>>>> it may be, is not a real number, it is also not a probability. >>>> Oh Papa San Grasshopper Grampa Boyeeee! Man! That's so cute. What is a >>>> probability? A real between 0 and 1? >>> Inclusive. >> Indeedly. And within what interval lies the multiplicative inverse of a >> number at least equal to 1? Is it [0,1]? > > What "number" does TO refer to? A real number, dumbass, not "1+chicken". > Since Aleph_0 is not a real number, neither would it have a real > number reciprocal. The set of non-zero real numbers is a group, so that > only real numbers can have real number multiplicative inverses. It's not a real number at all, in the sense of not being a number at all. It's not a count of anything and it doesn't have any valid arithmetic. It can't be expressed except by invoking it's holy name. If it's not greater than any finite number, then it's not a number. >>>> Does 1/aleph_0 lie within the real interval [0,1]? >>> AS "1/aleph_0" is not a real number at all, and real intervals contain >>> nothing other than real numbers, "1/aleph_0" does not lie within ANY >>> real interval. >> Can we not say that the multiplicative inverse of any real number >> greater than one is less than 1, and at least equal to 0? > > It doesn't matter what TO tries to say about it, it still will not make > either aleph_0 into a real number, nor any alleged reciprocal of aleph_0 > into a real number. Because aleph_0 ISN'T a number. It's a phantom with a name and cult that worships it. "No Largest Finite!!! (GONGGG!!!) Huyah huyah huyah Ommmmmmmm......ega!" Chant it, Baby. You'll find it one day. You're potentially infinite.
From: Aatu Koskensilta on 17 Sep 2006 10:36 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. > 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. > 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. 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.) > 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. -- 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 10:38
Virgil wrote: > In article <450c8974(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <450c6449$1(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>> Virgil wrote: >>>>> In article <450c3d37(a)news2.lightlink.com>, > >>>>> TO keeps harping on the "length" of the real line as a part of his >>>>> mythology. > >>>> Does it exist? It's the number of naturals on the line. >>> Between what two points does TO find that measure of length? >> Between which two naturals does Virgil find a difference of aleph_0-1? > > The difference (distance?) between any two naturals is a natural, which > Aleph_0 - 1 is not. > > The distance between any two "points" on the real line is a real number, > the (absolute) difference between the real numbers for these points. And there is no such difference which is infinite on the entire real line. How then do you fit an infinite number of unit intervals in that space? >>>> Is it fixed, or does it stretch and shrink at whim? >>> The "length" of the real line does not exist at all. >> Then neither does the count of naturals, > > What TO means by "the count of naturals" only he knows, but the > cardinality of the set of naturals exists. > Right, because cardinality is only a number in the finite case, and otherwise is an amorphous equivalence class. > > >>>>> Length measurements on a line are distances on that line between two >>>>> points on that line. >>>> Measurements of line SEGMENTS, yes. >>> What other sorts of "length" measurements does TO claim he can make on a >>> line? >> I claim that, even if no length can be determined due to endlessness of >> a given line, that if that line exists, it is always as long as itself. >> Can you disagree? >> >>>>> Which points on the real line does TO use to measure the length of that >>>>> line? >>>> It is not measurable, but like all objects and properties, is equal to >>>> itself. >>> If the line is not measurable then it does not have any length at all, >>> just as TO has no sense at all. >> If you think the line is not measurable, then quit trying to pretend you >> are providing any kind of measure for sets which traverse the line. > > There is a standard measure for the distance between any two numbers on > the real line which is given by the absolute difference between those > real numbers. But that in no way gives any measure for the line in its > entirety, because it has no end point numbers from which to take an > absolute difference. > > If TO's mind is too perplexed to see that, he needs a shrink. > Yeah, a shrink will help me with infinity. Good suggestion. Either you have a complete real line or you do not. If you are comparing infinite sets over this entire possible interval, then that interval exists as a range for the sets under consideration. As we can compare finite sets over finite ranges given the mapping functions which define them based on standard sets such as the naturals, we can do the same for infinite sets, over the range R. If not, why not? > > >> My logic is clear and simple. > > And wrong! Define "wrong". I thought for you it was all about internal consistency, and there was no objective measure of truth. Are you waffling on that point? |