An uncountable countable set
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From: imaginatorium on 16 Sep 2006 11:55 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)? If it is more limited, can you give an example of a function that does not have a "formula"? If it is not more limited, then your statement appears to refer to any set which can be bijected with the naturals - is that what you mean? Brian Chandler http://imaginatorium.org
From: Tony Orlow on 16 Sep 2006 14:03 imaginatorium(a)despammed.com wrote: > 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 > What is the average value of the reals in [0,1]?
From: Tony Orlow on 16 Sep 2006 14:05 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. > >> 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)... Snicker all you want. > > <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.) At least you're open minded. > >> Americans are tired of being attacked too. Have we asked for it? > > This seems not to be related to sci.math. Everything is related to math. > > Brian Chandler > http://imaginatorium.org >
From: Tony Orlow on 16 Sep 2006 14:06 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!
From: Tony Orlow on 16 Sep 2006 14:10
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. > > If it is not more limited, then your statement appears to refer to any > set which can be bijected with the naturals - is that what you mean? > With a monotonically increasing or decreasing function, yes, the inverse of that function giving the number of elements within any given value range. > > Brian Chandler > http://imaginatorium.org > |