An uncountable countable set
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From: Virgil on 16 Sep 2006 00:06 In article <450b4893(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > >>>>>> If the string is unbounded but finite Which is impossible. > > > > By whom? Nobody knows what you mean by those terms. > > > >>>> One with all finite bit positions but no greatest. > >>> So... why does 2) follow from 1? > >> Because then the string is considered finite. Not by anyone of sense. > >>> 1 + 2 + 4 + 8 + 16 + 32 + .... > >>> > >>> but there is no such natural. This sum is divergent. > >>> > >> The sum diverges to an infinite value as n APPROACHES oo. It is finite > >> for all finite n. "It" does not even exist for finite n. The 'partial sums' exist for finite n, but they are not the infinite series itself but only miniscule bits off the front end of it. > If all n are finite, it's finite. It's not infinite > until n is infinite. False in all standard math, and TO has no system of his own in which it is true. > > The sum doesn't sum to a finite value? Right? > > That depends whether it ever gets to infinite n. The sum is finite for > all finite n. The SUM, 1 + 2 + 4 + 8 + 16 + ..., is not defined as a value for finite n or for infinite n. There are partial sums that are defined, but they are quite something else. An infinite series may sometime have an associated value, called its limit when it exists, or may not, as in the above case. But a given series is just that series, and not anything else.
From: Virgil on 16 Sep 2006 00:20 In article <450b4aff(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > >> Randy Poe wrote: > >>> Tony Orlow wrote: > >>>> Haha!! Good one. "There are the same number of primes as there are > >>>> naturals, a proper superset." Good answer. > >>> Obviously you don't believe that. > >> Good guess. > >> > >>> Here's an equivalent statement: If n is a natural, > >>> there is an n-th prime. > >>> > >>> Do you think that's false? Do you think that there > >>> is a natural n such that there is an (n-1)th prime, > >>> but no n-th prime? > >> No, I don't think that, but I do recognize that the set of primes is a > >> subset of the naturals which proportion thereof has a limit of 0 as the > >> range of the naturals approaches oo. > > > > So, uh, do you think that there is always an nth natural? And always an > > nth prime? So arne't there, like, the same number of both, given the > > equivalence between the two ideas? Why do you *always* avoid answering > > the most pertinent questions directly? > > I have answered that, directly, several times. No, I do not consider > simple bijection to be grounds for considering two sets equal, when the > value range is not determined. Since "value ranges" do not exist for unbounded sets, TO is basing is theories on unreality. > If you speak of all reals or all > naturals, that covers the same real range called "all", so if one set is > denser than another, and covers the same range, it's a bigger set. It's > that simple. If you claim the naturals and evens, for instance, have the > same count, that in essence says the evens have twice the range, since > every even is twice its corresponding natural. That's wrong. It may be wrong in TO's mind, but it is right everywhere else. Cardinality is self-consistent. > > > > >> Do you honestly think bijection works as an infinite analog to equality? Bijection is the universal test for equal cardinality for finite sets. Cantor makes a good case for using it for infinite sets. > >> 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? Are they bijectable? if so they have the same cardinality. If TO wants to measure size some other way, he has yet to come up with anything as demonstrably self-consistent as cardinality. > > > > Doesn't matter if you feel that way. All cardinality says in set theory > > is "sets with this cardinality are bijectable with one another". > > Yes, it's the claim of mathematicians that this is a valid extension of > the finite meaning of "set size" which is so irksome. It works. Go ahead and have > your broad equivalence classes based on nothing other than bijections, > but stop pretending that hocus pocus applies to anything outside of > Cantor's Garden O' Tricks, because it doesn't. Until the concept of > measure is applied to infinite sets of reals, they cannot be properly > compared. TO has no idea of measure theory, as it rejects most of TO's requirements. For one thing, in measure theory, proper subsets often have the same measure as their supersets. > > Most people intuitively think that if two sets can have all their > > elements paired up then they have "the same number of" elements. You > > don't. So what? Set theory doesn't "care" if that's what you think > > "bijectable" means or not. > > > > Some people do, and many are turned off by the hat tricks of > transfinitology. It's not correct that you can cut a ball into five > pieces and reassemble it into two solid balls of the original size Depends on the kind of "knife" one uses. > and > it's not true that you can put ten balls in a vase and take one out, > over and over, and ever get an empty vase. But it is true that if one puts a lot of balls in a vase and takes all of them out again one can end with an empty vase. Which is what happens. > If you don't think people > "care" whether what they think makes sense or not, then I guess you > don't care whether what you think makes sense. Since mathematicians' rules about what "makes sense" are much stricter that TO's, I will go with the stricter rules.
From: Virgil on 16 Sep 2006 00:26 In article <450b4c1c(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <450b0042(a)news2.lightlink.com>, > > Until TO proves his trichotomy, and connectivity, he has neither. Note > > that one can prove the density and "continuity" of the reals (contunuity > > being the LUB and GLB properties. > > > > Accordingly TO must show that any non-empty set of his extended reals > > which is bounded above must have a least upper bound. > > > > The set of standard reals is bounded above if there are any infinite > > reals, so must have a LUB, so what is it TO? > > I have already said I have no least infinity. Then there is a gap between the finite reals and TO's supposed infinite numbers. > You're the one with an LUB > of aleph_0, That is only for the cardinality of set of finite naturals, and does not work for the cardinality of the set of finite reals. > even though the same logic applies to that fallacy as to the > largest finite, unless one disregards that removing elements makes a set > smaller. What you are saying applies only within any finite neighborhood > on the real line. GIGO again. > > > > What "system" is that? > > Bigulosity. AS it has never been presented as a complete system, it is mere hand waving.
From: Virgil on 16 Sep 2006 00:29 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?
From: Mike Kelly on 16 Sep 2006 08:10
Tony Orlow wrote: > Virgil wrote: > > In article <45098084(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Dik T. Winter wrote: > >>> In article <4506d1ae(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> > >>> writes: > >>> > David R Tribble wrote: > >>> ... > >>> > > Start with a simple proof: Take the set of naturals, N = {0,1,2,3,...} > >>> > > and remove one element to get set S = {1,2,3,...}. Now show that > >>> > > the "T-size" of N is exactly one less than the T-size of S. In other > >>> > > words, find a way to show that every counting of S versus every > >>> > > counting of N always leaves one element of N (0) left over. > >>> > > >>> > Use IFR. N maps to S using f(n)=n+1. The inverse of that function is > >>> > g(x)=x-1. So, over the range of 0 to N, |S|=|N|-1. Map N to the Evens E > >>> > using f(n)=2n. The inverse function is g(x)=x/2, so over the range of N, > >>> > the evens have |N|/2 elements. Isn't that intuitively satisfying? And > >>> > gee, it works for finite sets accurately too!! > >>> > >>> How many elements has the set of primes? > >> There is no well-known function that maps n to the nth prime, so IFR > >> doesn't apply. Do you have an inverse function that specifies the nth > >> prime for all neN? Didn't think so. > > > > Then there are sets whose size TO cannot measure, which makes his > > "measure" less useful that cardinality, at least cardinality in ZFC and > > NBG. > > Haha!! Good one. "There are the same number of primes as there are > naturals, a proper superset." Good answer. It's almost as bad as, "There > are an infinite number of rationals between any two naturals, but there > are the same number of each." What a "theory"! It's mathematical > creationism. It occurs to me that your MO - decide your conclusions based on what you want to be true, then try to (poorly) justify them - is a rather closer analogue to creationism. -- mike. |