An uncountable countable set
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From: Virgil on 16 Sep 2006 15:10 In article <450c3e19(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > imaginatorium(a)despammed.com wrote: > > Tony Orlow wrote: > >> Mike Kelly wrote: > >>> Tony Orlow wrote: > >>>> Mike Kelly wrote: > > > >>>>> Perhaps to demonstrate your firm grasp of these matters you could > >>>>> define "probability" and then explain how one determines the > >>>>> "probability" that "a natural" has some property P? > >>>> Does the set of naturals with property P have a mapping function from > >>>> the naturals? :) > >>> Explain what it means for a property P to have a mapping function from > >>> the naturals xD > >> There is a formula such that maps each unique natural to a unique > >> element of the set, such that no element is omitted. > > > > Can you define what you mean by a "formula"? Just any string of symbols > > that another mathematician might understand? Is a mapping function > > somehow more limited than a set theory function (if you know enough > > about that to answer, which I fear is unlikely)? > > No, an invertible mapping function is precisely a bijection, although > for IFR to work, it needs to be order-isomorphic, that is, either > monotonically increasing or decreasing. > > > > > If it is more limited, can you give an example of a function that does > > not have a "formula"? > > Certain mappings from the naturals, that place elements of the set out > of their natural quantitative order, do not work with IFR. Thus TO's theories do not work for sets merely as sets, but only for totally ordered sets. Since Cardinality is independent of ordering, but TO-ality is totally dependent on total ordering, TO-ality is not a measure of the size of the set itself, but more a measure of which order relation has been imposed upon the set. The set of naturals can be reordered to be order isomorphic to the ordering of the rationals, so that, according to TO-ality, the set has at least two sizes, one for each ordering. Cantor's cardinalities have the great advantage of being determined only by the set itself, independently of any relations, order or otherwise, which might be defined on the set.
From: imaginatorium on 16 Sep 2006 15:26 Tony Orlow wrote: > imaginatorium(a)despammed.com wrote: > > Tony Orlow wrote: > >> MoeBlee wrote: > >>> Tony Orlow wrote: <...> > >> So, that having been said, when there are an infinite number of > >> rationals for every half-open unit interval, and only one natural in > >> every such interval, how does it make sense that there are not > >> infinitely many more rationals than reals? Are the extra naturals that > >> make up the difference squashed down towards the infinite end of the > >> line, where there's no rationals left? Like I said, it's poppycock. > > > > > > Ah, the "infinite end of the line". What's that like, then? Sort of the > > end at the end that isn't there? > > In case you didn't note the sarcastic tone, that's me making fun of your > logic, not some part of my theory. Sheesh! If you point to some part of my (or mathematics in general's) logic, and show that it is laughable, you do indeed make a point. Unfortunately (for you), what you point at as "poppycock" is _precisely_ the bit that is so munged up inside your head that frankly I see no hope of it ever getting straightened out. There are no "extra" anythings that would get squashed up. There is an unending supply of naturals, and an unending supply of rationals, so neither can in any sense ever be less in supply than the other. Can you really not understand the joke about the man granted three wishes by a fairy (or somesuch) - first he asks for an unemptiable bottle of Guinness. However much you drink, there is always more Guinness in the bottle. Then for a second wish he asks for another one. This is a joke, Tony, a joke that can be understood by many people with no pretensions whatsoever to understand set theory, let alone to believe themselves capable of replacing it. People see it as a joke because it is obviously absurd to think of two unemptiable bottles of Guinness as somehow providing more Guinness than one. Good luck with your book <g> Brian Chandler http://imaginatorium.org
From: Aatu Koskensilta on 16 Sep 2006 15:27 David R Tribble wrote: > The problem is that using too simple a language can lead to > further confusion, or at least maintain the misunderstandings. > I've been chastised for talking about "set size" instead of using the > more specific term "cardinality", for instance. When discussing technical subjects there is no hope of avoiding technicalities and technical concepts and terminology. But if someone wonders about a feature of, say, a proof of Cantor's theorem it is seldom useful, and often counterproductive, to bring in the heavy machinery of first order logic and this or that formal set theory. > So there may be > times when a more technically precise meaning is called for in > order to cut through the confusion of multiple meanings. Of course, clarity is to be always sought, and in making clear exactly what is meant by such ambiguous words as "size" in some context is often necessary. -- 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 16 Sep 2006 16:44 Virgil wrote: > In article <450bf8df(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: > >>>>> What is the probability that a number is greater than n? >>>> (aleph_0-n)/n <1 >>> (aleph_0 - n)/n will still be infinite, so TO is wrong to call it a >>> probability at all, much less a probability less than 1. >>> >> Oooops, you're right. ANswering too many interrogatories at once. That >> should be aleph_0-n/aleph_0. Sorry 'bout that. > > Still wrong, as aleph_0-n/aleph_0 = aleph_0-(n/aleph_0) = aleph_0, > if it means anything at all. > I suspect that TO meant to write "(aleph_0 - n) / aleph_0", which at > least makes a little sense. Yes, this time I left out the parentheses. Oops. Does it make " a little sense" now? >>>>> What is the probability that a number is greater than another number >>>>> (also randomly chosen)? >>>> 1/2 >>> Given that any natural is "randomly chosen", which is an impossibility >>> in the first place, the probability that another natural, equally >>> randomly chosen, will be greater than the first is as close to 1 as >>> makes no matter. >> Take your vase and shake it a countably infinite number of times, and >> draw a ball. That should be random enough for you. > > To chose something from a set "at random" has a technical meaning in > probability theory, meaning that each member of the set must have > exactly the same probability of be selected as any other member. Yes. For the finite naturals that would be 1/aleph_0, there being aleph_0 of them. > > For a set with a countably infinite number of elements, such as the set > of naturals, that is not possible. Because the set does not have size aleph_0? > > But one can come close. Given any r strictly between 0 and 1, by > assigning the probability of (1-r)*r^n to each n in N = {0,1,2,3,...}, > one gets a legal probability distribution on N ( the sum of all such > probabilities equals 1) with the probability of consecutive naturals > being as nearly equal as one chooses, short of perfect equality, by > choosing r close to, but slightly less than, 1. > > Of course, if r were allowed to actually equal 1, all of those > probabilities become zero and no longer add up to anything except zero. That's an interesting concept. Let me cut and paste into notes...... Let's use 1/aleph_0 for r and see what happens. That gives 1-1/aleph_0*1/aleph_0^n, or ((aleph_0-1)/aleph_0)/aleph_0^n, or (aleph_0-1)/aleph_0^(n+1), or 1/aleph_0^n-1/aleph_0^(n+1). That seems too small.
From: Tony Orlow on 16 Sep 2006 16:45
Virgil wrote: > In article <450bf9ae(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <450b4cb2$1(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>> Virgil wrote: >>>>> In article <49edf$450aacfc$82a1e228$14539(a)news2.tudelft.nl>, >>>>> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >>>>> >>>>>> Mike Kelly wrote: >>>>>> >>>>>>> Han de Bruijn wrote: >>>>>>> >>>>>>> Plagiarism? I don't get it. Who is plagiarising what? >>>>>> "Your" would-be arguments against mine are not really yours. They are >>>>>> just a _plagiary_ of well-known "arguments" employed by the mainstream >>>>>> mathematics community. >>>>>> >>>>>> Han de Bruijn >>>>> What is common knowledge can be used by anyone without plagiarizing. >>>>> >>>>> Otherwise only its original author could use "2 + 2 = 4". >>>> Yes, and Virgil would be collecting the royalties from every first-grade >>>> class. >>> Does TO credit me with discovering/inventing/creating "2 + 2 = 4" ? >>> That fact has been around for millennia. >>> How old does TO take me for? >> I couldn't guess, but I heard you fart dust devils and used to date >> Methuselah's sister. > > You must have heard such things from someone else whose pretentions I > have also been puncturing. Yes, I think it was the dirt by the road, who claims you refer to it as a "young'un" or "greenhorn" or something. |