An uncountable countable set
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From: Virgil on 16 Sep 2006 14:37 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. > > > > >>> 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. For a set with a countably infinite number of elements, such as the set of naturals, that is not possible. 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.
From: Virgil on 16 Sep 2006 14:39 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.
From: Virgil on 16 Sep 2006 14:46 In article <450c3c65(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > What is the average value of the reals in [0,1]? By what definition of average? There are a whole bunch of such definitions. Note that many such definitions only apply to finite sets of numbers, and thus won't work. Also, since the set [0,1] is invariant under x --> x^n, its average should be also.
From: Virgil on 16 Sep 2006 14:51 In article <450c3cfe(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > imaginatorium(a)despammed.com wrote: > > Tony Orlow wrote: > >> Randy Poe wrote: > >>> Tony Orlow wrote: > >>>> Randy Poe wrote: > >>>>> Tony Orlow wrote: > > > > <bibble-babble> > > > >> ... If the real line is considered a fixed range of aleph_0, > > > > Remind us what "range" means? Normally a range goes from a left end to > > a right end: in a set with a left end and no right end, where is the > > "range" measured to? > > The range is not measured, but is declared constant over the real line, > whatever that length is. The real line does not have a length,. Length along a line can only be measured between points on that line and the real line does not have end points. > > > > A "fixed infinite length". Very funny (the first time, but has worn a > > bit thin by now)... > > Snicker all you want. As all "lengths" are distances between points, what points does TO suggest one use to measure the "length" of the real line?
From: Virgil on 16 Sep 2006 14:55
In article <450c3d37(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > imaginatorium(a)despammed.com wrote: > > Tony Orlow wrote: > >> MoeBlee wrote: > >>> Tony Orlow wrote: > >>> > >>>> Can a dense set like the rationals, with an infinite number of them > >>>> between any two naturals, really be no greater a set than the naturals, > >>>> which are an infinitesimal portion of the rationals? That's just > >>>> poppycock. > >>> A set is not dense onto itself. A set is dense under an ordering. And > >>> the set of natural numbers is dense under certain orderings. > >>> > >>> MoeBlee > >>> > >> Not in the natural quantitative order on the real line. You cannot say > >> that between any two naturals is another, in quantitative terms. I > >> meant, obviously, dense in the quantitative ordering. But, you knew that. > >> > >> 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? > > > > Brian Chandler > > http://imaginatorium.org > > > > In case you didn't note the sarcastic tone, that's me making fun of your > logic, not some part of my theory. Sheesh! TO keeps harping on the "length" of the real line as a part of his mythology. Length measurements on a line are distances on that line between two points on that line. Which points on the real line does TO use to measure the length of that line? |