An uncountable countable set
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From: Virgil on 8 Oct 2006 15:39 In article <1160310273.517860.47380(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > A bijection with N does not define the indexes such that there was only > a unique sequence. Therefore, there is not, as you asserted, one unique > number 0.111... . There are as many such numbers as there are contexts in which that symbol can be interpreted as a numeral. And there are infinitely many such contexts. As a base n fraction, for every natural n > 1, it equals 1/(n-1)
From: Virgil on 8 Oct 2006 15:43 In article <1160330371.949163.273120(a)c28g2000cwb.googlegroups.com>, Han.deBruijn(a)DTO.TUDelft.NL wrote: > Virgil wrote: > > > In article <45286ce5$1(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > > > David R Tribble wrote: > > > > > > What about: > > > > sum{n=0 to oo} (10n+1 + ... + 10n+10) - sum{n=1 to oo} (n) > > > > The left half specifies the number of balls added to the vase, and > > > > the right half specifies those that are removed. > > > > > > Do you mean: > > > sum{n=0 to oo} (10) - sum{n=0 to oo} (1)? > > > That sounds like what you re describing, and termwise the difference is > > > sum(n=0 to oo) (9). That's infinite, eh? > > > > But the sums are not given termwise in the question, but sumwise, so > > cannot be calculated termwise in your answer, but must be done sumwise. > > > > And sumwise they are no different. > > Wrong. Go back to Calculus school. Sumwise they are both undefined. > Precisely! All series that are divergent because of being increasing and unbounded above are essentially the same.
From: Virgil on 8 Oct 2006 15:46 In article <452942bf(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <45251b3e(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> In article <4523c954$1(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> > >>>> David R Tribble wrote: > >>>>> Tony Orlow wrote: > >>>>>>> On the other hand > >>>>>>> I don't know why I said "neither can the reals". In any case, the > >>>>>>> only > >>>>>>> way the ordinals manage to be "well ordered" is because they're > >>>>>>> defined > >>>>>>> with predecessor discontinuities at the limit ordinals, including 0. > >>>>>>> That doesn't seem "real" > >>>>> Virgil wrote: > >>>>>>> In what sense of "real". There are subsets of the reals which are > >>>>>>> order > >>>>>>> isomorphic to every countable ordinal, including those with limit > >>>>>>> ordinals, so until one posits uncountable ordinals there are no > >>>>>>> problems. > >>>>> Tony Orlow wrote: > >>>>>> The real line is a line, with > >>>>>> each point touching two others. > >>>>> That's a neat trick, considering that between any two points there is > >>>>> always another point. An infinite number of points between any two, > >>>>> in fact. So how do you choose two points in the real number line > >>>>> that "touch"? > >>>>> > >>>> They have to be infinitely close, so actually, they have an > >>>> infinitesimal segment between them. :) > >>> But any "infinitesimal segment" within the reals is bisectable. > >> Within the standard reals, it's one number, if it's closer than any > >> finite distance of a that number. > > > > In Standard reals,"infinitesimal", if it means anything, merely means > > very small but not zero. > > In The Robinson, or similar, non-standard models, infinitesimals are > > different from standard numbers but still non-zero. > > In both, they are bisectable, and between two distinct numbers, even > > when only infinitesimally different, there is always another. > > Yes, but that infinitesimal difference doesn't count in the STANDARD > reals, does it? In standard reals there is no such thing as an infinitesimal difference at all, but any difference at all other than zero can be infinitely subdivided. >There is no requirement that any two standard reals > which are infinitesimally different have another between them, because > there RAE no two stanard reals that are infinitesimally different. RIGHT! For once.
From: Virgil on 8 Oct 2006 15:52 In article <4529434c(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <45251bc5(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > >> Uh, no, the very conclusion that the vase empties, when at most one ball > >> is removed at a time, implies that there is a last ball removed > > > > That is TO's assumption, contrary to the facts required by the > > experiment. > > > > Infinite processes can end in finite time or else Zeno's 'paradoxes' > > would prevent all action. > > Zeno's paradoxes involve a continuous motion at finite speed over finite > time. The error is in considering each successively smaller time slice > to be equal and add up to an infinite time. The vase problem is similar, > but it's a paradox, not a fact. It's resolved with infinite series. Webster?s Concise Electronic Dictionary 1. par?a?dox (noun) [par?a?dox?es] statement that seems contrary to common sense yet is perhaps true In this case, the emptiness of the vase at noon may be a paradox and yet is true, at least in the world of mathematics. And the situation cannot arise in any other world.
From: Ross A. Finlayson on 8 Oct 2006 15:55
Virgil wrote: > In article <1160310024.534838.300800(a)i3g2000cwc.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Lester Zick schrieb: > > > > > On Fri, 06 Oct 2006 12:52:43 -0600, Virgil <virgil(a)comcast.net> wrote: > > > > > > >In article <1160122708.108138.77770(a)e3g2000cwe.googlegroups.com>, > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > > > > > >> I am sure, the results "all balls in B" and "all balls not in B" are > > > >> not to be interpreted as an actual contradiction of set theory. It is > > > >> just counter intuitive. > > > >> > > > >> Regards, WM > > > > > > > >It seems to be intuitive in "Mueckenh" 's world, but it is not only not > > > >intuitive, it is not true in any set theory of my acquaintance. > > > > > > So it's counter intuitive. What's the problem? Surely it isn't the > > > first counter intuitive suggestion you've ever run across. > > > > I did not know that the simultaneous existence of A and ~A is not a > > contradiction but only counter intuitive. Now I learned it, and I find > > ZFC is not very attractive to me. > > > > Regards, WM > > Then why not stay just away form it. > > > Those of us who find ZF, ZFC and NBG, etc., worth considering find the > continual preaching of those dogmatists who do not offensive. That's obviously untrue. Congratulations. The realm of sets addressed by ZF without going into logical paradoxes: finite ones, find wide utility in application. So, ZF has been examined by many and by some found lacking. ZF is inconsistent. Cantor's set theory was claimed by him to have a universe in it, and there's a difference between NBG, and NBG with classes, there's no universe in ZF, nor ZFC. Ross |