An uncountable countable set
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From: Tony Orlow on 16 Sep 2006 16:48 Virgil wrote: > 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? Expected value of an element chose at random. Is it impossible to choose a real in [0,1] at random? > > There are a whole bunch of such definitions. And, what are the "whole bunch" of answers to that question? > > Note that many such definitions only apply to finite sets of numbers, > and thus won't work. Says you. > > Also, since the set [0,1] is invariant under x --> x^n, its average > should be also. What does average have to do with exponentiation. (You could use a good opportunity to strut your stuff, so remind me what "invariant" means :)
From: Tony Orlow on 16 Sep 2006 16:51 Virgil wrote: > 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. Well, given the fact that there are n naturals in every real interval of length n, I guess there is no measure of the number of naturals either. > >>> 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? We need not know the endpoints or length of a line to say it is as long as itself. That is given.
From: Tony Orlow on 16 Sep 2006 16:53 Virgil wrote: > 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. Does it exist? It's the number of naturals on the line. Is it fixed, or does it stretch and shrink at whim? > Length measurements on a line are distances on that line between two > points on that line. Measurements of line SEGMENTS, yes. > 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.
From: Virgil on 16 Sep 2006 16:54 In article <450c6210(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > 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? Very little. > > >>>>> 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. But 1/aleph_0, not being a real number, cannot be a probability. > > > > > 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? Because there is no real number which can serve as a probability. There is no real number r such that r + r + r + ... = 1. > > > > > 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. 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.
From: Tony Orlow on 16 Sep 2006 17:00
imaginatorium(a)despammed.com wrote: > 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. Noting is straight. All is curved. > > 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. No extra infinity of rationals between any two naturals, which is equinumerous with the entire set of naturals? Yes, laughable. > > 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. Yes, for a finite consumer. What if you had access to two universes, and wanted to supply unending beer to both at once? By your logic above, if the person asked for an unending bottle of guinness, and then asked that it be in a pair which is a member of a pair which is a member of a pair, etc, then they are asking for 2^aleph_0 unending bottles. Is that more useful? Is it more? Equating usefulness for finite consumption when discussing infinite sets is like discussing tickertoys at a national security meeting. > > Good luck with your book <g> Why, thank you. I know you mean that with the sincerest of encouragement. > > Brian Chandler > http://imaginatorium.org > |