An uncountable countable set
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From: Tony Orlow on 16 Sep 2006 09:15 Virgil wrote: > In article <450b4773(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Mike Kelly wrote: >>> Tony Orlow wrote: > >>>> 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. If you are talking >> about a property that defines a subset of the naturals, it's bound to be >> defined by some formula on the elements of N, no? How else do you intend >> to define this set? >> >>> Pretend it does. Then explain how one determines the probability that a >>> natural has property P. Go wild. For bonus credit explain what happens >>> when property P doesn't have a mapping function from the naturals =D! >> Suppose it has the property that it is divisible by 3. > > That is a special case, and general rules cannot be demonstrated only by > special cases, even when the special case is dealt with honestly. > > > >> So, we can map >> the set of naturals satisfying that property using f(x)=3*x. The inverse >> functions is....anyone? That's right, g(x)=x/3. Huh! So, over the entire >> real line we could expect 1/3 of all naturals to satisfy this property > > > Non sequitur. What is true for finite sets does not automaticaly carry > over to infinite ones. > >> Where P does not have a mapping, this technique cannot be applied. Do >> you have a specific example, for bonus credit? > > Let P be the property that for given natural n, the decimal expansion of > the square root of n has "more" even digits that odd digits. > > >>> Here are a few questions for you to practice your new theory on : >>> >> I'll assume you mean the finite naturals, of size "aleph_0". If you mean >> the hypernaturals through Big'un, replace as desired. >> > >>> 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. > >>> 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. <snip)
From: Tony Orlow on 16 Sep 2006 09:18 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.
From: imaginatorium on 16 Sep 2006 10:55 Tony Orlow wrote: > Virgil wrote: > > Tony Orlow <tony(a)lightlink.com> wrote: > >> Mike Kelly wrote: > >>> Tony Orlow wrote: > >>> 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. > <snip) _How_ would you draw a ball from a vase containing an infinite set of balls. In a vase containing a finite set, you could divide them into two equal subsets, choose one, and repeat, making a sequence of binary choices. Given a finite set of numbers, you can calculate the average value of the number chosen at random. But this doesn't work with an infinite set of numbers, since there is no "right-hand" end, and thus no mid-point. Just something else that's beyond you, I suppose. Brian Chandler http://imaginatorium.org
From: imaginatorium on 16 Sep 2006 11:46 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? > then a set which is denser in every part of that range has more elements > than one which covers the same complete range with a lesser density. > That's the proper generalization from finite to infinite, and the > standard mistake to consider the real line to have whatever length "Standard mistake"? Meaning a mistake written in maths books? Care to point to a maths book (title, page number etc) including this "mistake"? (You wouldn't of course be referring to your own half-baked confusion, would you?) > happens to be convenient, rather than a fixed infinite length for > purposes of comparison. A "fixed infinite length". Very funny (the first time, but has worn a bit thin by now)... <snip> > ... The suggestion I am making is straightforward, the only > explanation I can see for the refusal to consider it on the part of > "educated" mathematicians is an emotional response on their part because > they have invested so much of themselves in a clearly flawed system, and > are tired of being attacked for it. Very funny. Well, a bit. You have noticed that while a number of people have tried to help you articulate your own ideas, such as they are, no-one "educated" in mathematics has been persuaded by your prattling on about "mistakes", "flawed systems" and so on in mathematics. This could of course be because you are so clever you are simply ahead of the entire world of mathematics. We'll have to wait for the book, I suppose. (The other possibility is, I suppose, entirely unthinkable, at least by you.) > Americans are tired of being attacked too. Have we asked for it? This seems not to be related to sci.math. Brian Chandler http://imaginatorium.org
From: imaginatorium on 16 Sep 2006 11:49
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 |