An uncountable countable set
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From: Tony Orlow on 20 Aug 2006 23:29 Dik T. Winter wrote: > In article <eca31m$itn$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: > > David R Tribble wrote: > ... > > > Not without proof that the formula works for infinite n, anyway. > > > But we're still waiting for Tony to provide that proof. > > > > David, this is not a matter of proof given standard definitions, but a > > redefinition of the principle of induction such that it applies to the > > infinite case. When an inductive proof demonstrates that f(x)>g(x) for > > all x greater than some y, all infinite values falls into that category. > > So, it simply amounts to removing the restriction that induction only > > applies to finite n. Once this restriction is lifted, then we can > > "prove" that, for instance, for all n>2, n^2>2n, and therefore this also > > holds for all infinite n. > > And as 1/n > 1/2n this holds also for infinite n, so 0 > 0? No, Dik. When we declare a unit oo, we treat it as a constant, and its inverse is the unit infinitesimal. If n=oo, then 1/n is iota or alpha (Ross and I will have to debate that ;). And, there are, despite Ross's desire for nilpotency, fractions of infinitesimal intervals, and sub-infinitesimals. Where there are specific infinite values, there are specific nonzero infinitesimal values, via the ring, without the exception for 0, and 1/n>0. :) Tony
From: Virgil on 20 Aug 2006 23:41 In article <ecb0rb$s0m$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > Virgil wrote: > > In article <ec9ssu$9km$1(a)ruby.cit.cornell.edu>, > > Tony Orlow <aeo6(a)cornell.edu> wrote: > > > >> Virgil wrote: > >>> In article <ec9nrs$44k$2(a)ruby.cit.cornell.edu>, > >>> Tony Orlow <aeo6(a)cornell.edu> wrote: > >>> > >>>> Virgil wrote: > >>>>> In article <ec8h1q$prm$1(a)ruby.cit.cornell.edu>, > >>>>> Tony Orlow <aeo6(a)cornell.edu> wrote: > >>>>> > >>>>>> It would seem to me that any good theory > >>>>>> of infinite sets could be applied to infinite sets of points, such as > >>>>>> the reals in (0,1] or those in (0,2], and be able to draw conclusions > >>>>>> such as that there is twice as many points and twice as much space in > >>>>>> the second as in the first > >>>>> That would require each point to occupy enough space that only a finite > >>>>> number of them could fit into any finite interval. > >>>> WRONG! > >>>> > >>>> It requires that some relationship be set up between a particular > >>>> infinity of points and a finite length. What you suggest would be a > >>>> finite set of finite elements, not any kind of infinite set. > >>> If TO insists that infinitely many points can be compressed into a > >>> finite space, then one can just as easily compress twice as many points > >>> into the same space. > >>> > >>> If one takes the points of (0,2] and places each point from x at > >>> position x/2 in (0,1], one has compressed "twice as many" points into > >>> (0,1] as were originally there. Thus there cannot be any fixed > >>> proportionality between the "number of points" in an interval and its > >>> length for intervals of positive length. > >> There is when one declares it. > > > > The problem being that one can declare as many different ones as one > > chooses to declare, and each is just as valid as any other. > > Well, that's why one only decalres one to start with, and sees how that > goes. Rather well, thanx. > > I declare Card(P(N)) to be the number of reals in the unit interval and > > Card(N) to be the number of unit intervals with disjoint interiors in > > the infinite real line. > > > > > TO's definition is either wrong (if his unit intervals are to have > > disjoint interiors) or he has corrupted his own theory by letting the > > number of points in the unit interval equal the number of points in the > > entire real line (if overlapping intervals are allowed, the number of > > unit intervals in the real line equals the umber of points in the real > > line). > > > > As I told you, I entertain a much larger real line, with points actually > infinitely many units from the origin, allowing for infinite reals, > naturals and rationals (eventally even sedonions, prolly). Now TO wants to compress all the members of multi-dimensional real vector spaces on his real line, the complex numbers being of real dimension 2, the quaternions of real dimension 4, octonions 8 and sedonions 16. TO just does not understand how mathematics works. The reals are one dimensional as a real vector space, and every 1 dimensional real vector space is real-vector-space-isomorphic to every other. > > > > >> Where you claim to compress or stretch the inherent density of the real > >> points on the line, you make proper measure impossible. > > > > Is TO going to pontificate on measure theory on the real line? > > What the hell else have I been doing? Oh, well, I guess I've made a few > other points. But, yes, I advocate a means of melding measures. > > > > > If so how is he going to deal with the inevitable non-measurable sets? > > Like the Cantor set, or what? Your ignorance precedes you like a foul miasma. > > > > Cardinality has no problem with them. > > How does it deal with them? Work it out for yourself, TO. it shouldnt take more than a few years, say a decade or so, considering where you are starting from.
From: Virgil on 20 Aug 2006 23:46 In article <ecb18f$sgk$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > Virgil wrote: > > In article <ec9th3$a4n$2(a)ruby.cit.cornell.edu>, > > Tony Orlow <aeo6(a)cornell.edu> wrote: > > > > > >> Well, then, since the size of the set in width is always equal to the > >> max of the set in height, then it would seem that aleph_0 is the LUB on > >> the set size as well, but that the set never actually achieves this > >> size. Once it does have some aleph_0th element, that element would have > >> to be.....aleph_0! But that's not allowed in the set, so how can it be > >> the size of the set? > > > > The same way that the size of the open interval from a to b, with a < b, > > is of length equal to the difference b-a even though neither a nor b is > > a member of the set. > > Oh yes! You mean the way sum(n=1->oo: 9*10^-n) has a LUB of 1, but never > reaches it. Well, good point. Very good. That's true. ;) It is only the partial sums, a_k = sum(n=1->k: 9*10^-n), k in N, which never reach 1. The limit actually is EXACTLY 1, by reason of how limits are defined.
From: Virgil on 21 Aug 2006 00:06 In article <ecb1o9$h$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > Virgil wrote: > > Is TO off on his "largest natural" kick again? > > > > Because it's a set of consecutive naturals starting at 1. It doesn't matter where it starts, the issue is whether it ends with a "largest natural". In standard mathematics it does not. TO seems to switch postions erratically on the issue. > >>> > >>> There goes TO's delusion again. What ever makes TO think that a process > >>> which cannot not end must end? > >>> > >> If it doesn't end what makes Virgil think he can tack a count on it? > > > > I can "tack on " bijections of endless sets by defining those bijections. > > And in standard mathematics that is one way of counting. > > > > But since there is no end to counting, there is no particular count. If one can prove that some function is a bijection between some ordinal and the set to be counted, the counting has been completely achieved by that proof, at least for those not off in TO's never never land of TOmania. > > > > >> am not saying there IS a max finite ordinal. I am saying that that's > >> what the assumption of aleph_0 implies when one is allowed to apply > >> inductive proof to the infinite case. Which is sufficient justification for refusing to allow TO's extension of induction beyond standard induction. > > > > The only acceptable forms of induction in ZF or NBG are finite > > induction and transfinite induction, neither of which has TO shown he > > knows how to apply. > > But I am explicitly suggesting the alternative, that if f(x)>g(x) for > all x>y, then it is true for all infinite x. This leads to a clearer set > of conclusions. TO's alternative also leads to inconsistencies with the status quo ante, which is sufficient reason for rejecting it. > > >>>> The only way around this is the > >>>> declaration of the limit ordinals, but that's just a philosophical > >>>> monkey wrench. > >>> it certainly throws a monkey wrench in TO's gearbox, but ZF, ZFC and NBG > >>> are unaffected by non-existence of TO's alleged maximum finite natural. > >> It creates the problem. > > > > Not in ZF, ZFC or NBG, and problems occurring in the isolation of > > TOmania are irrelevant to mathematics. > > > > Very relevant to mathematical truth As TO has shown no talent for recognizing truth, and even seems to have a talent for avoiding it, his judgements on where mathematical "truth" lies are, at best, untrustworthy. > > > >> Nowhere have I ever claimed there IS a max > >> finite natural. I am saying that's the implication of your theory, > >> unless you throw out induction when n is infinite. > > > > For those cases we can sometimes invoke transfinite induction, but it > > doesn't give any of the results that TO wants. > > No, transfinitology doesn't satisfy my spiritual needs. For spiritual needs, one needs something a little less literal minded than mathematics. > > What does one do for sets which are not ordered, or ordered sets which > > have no max?, both of which are abundant. > > Examples? Symbolic sets (languages) need not be ordered to be measured, > necessarily, and sets which have no distinct range have no distinct > size, unless the elements themselves have no relative measure. The multidimensional point sets of multidimnsonal spaces, such as real vector spaces, are not ordered and are not orderable in any way consistent with their geometric properties. Nevertheless they are quite important mathematical sets. But TO has to little mathematics to think of them.
From: Virgil on 21 Aug 2006 00:17
In article <ecb69r$88o$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > When you prove this inductively, how much larger is the size of the set > than its maximum element? Is it always 1 larger than the max element? If > so, then how does the set size at some point jump to infinity, while all > elements remain finite? Is there a largest finite, and beyond it a > point, beyond which everything is infinite? This is precisely why TO's extension of induction is not allowed in standard set theory. What is true for the finite sets does NOT have to be true for the infinite ones. |