An uncountable countable set
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From: Virgil on 20 Aug 2006 10:46 In article <ec9ot6$5a1$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > Virgil wrote: > > In article <ec8hsd$r5t$1(a)ruby.cit.cornell.edu>, > > Tony Orlow <aeo6(a)cornell.edu> wrote: > > > >> Dik T. Winter wrote: > >>> In article <ec7gek$frg$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: > >>> > Inductively provable: Any set of consecutive naturals starting at 1 > >>> > always has its largest element equal to size of the set. > >>> > >>> What is the size of the empty set? What is the largest element of the > >>> empty set? > >> The empty set is not a set of consecutive naturals starting at 1, > >> because it does not include 1 as an element, and therefore it does not > >> stand as a counterexample. However, even in your von Neumann ordinals, > >> the empty set is 0, and contains zero elements, and could be considered > >> the 0th in this sequence of sets. You fail to address the inductive > >> proof that the non-empty sets of this series, of which the naturals is > >> the limit, always have this property. > > > > TO fails to note that induction does not apply. > > Virgil somehow fails to see that it does. > > > The form of induction is: > > Let F(x) be a predicate whose argument, x, is a member of a minimal > > inductive set S.* > > If F(x) is true for the first member of S > > and > > if whenever F(x) is true then F(successor(x)) is also true. > > then > > the principle of induction says that F(x) is true for each member of S. > > Right. Consider S to be the set of all sets of consecutive naturals > starting at 1. Is N a member of this set? You bet it is. Actually, I would bet against N being a member of a subset of N, myself. N is usually used to represent the set of all naturals. By requiring N to be a member of S, TO is asserting that there is a set which is a member of one of its own subsets. Either that or TO is using N in an unconventional way to mean something other than the set of all naturals. In which case he should explain what he means by it. > Is this an > inductive set? As described, it may not be acceptable as a set at all. > S is the set of all such sets. N is a member of S. So, it says something > about N. Then TO IS claiming that N is a member of a subset of itself! > > > > > In particular it does not say that F(S) is true. > > > > No, it says that F(N) is true, since N e S. Only if N e S e N e S e N e... > Yes, and it's inductively provable likewise that the size of a successor > ordinal is one greater than its max element. That would make aleph_0 1 > greater than the max finite natural. There goes TO's delusion again. What ever makes TO think that a process which cannot not end must end? >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. > > > > > There is no equivalent model in which the ordinal of each element is > > equal to the element itself as that would require sets to be members of > > themselves, which is impossible for ordinals. > > Don't confuse max with size. Relate them formulaically. Does TO mean that a maximum does not have to be larger than those things of which it is allegedly the maximum? If not, then what does TO mean by "max"?
From: Tony Orlow on 20 Aug 2006 10:51 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. Big'un is the number of reals in the unit interval, and the number of unit intervals on the infinite real line. Where you claim to compress or stretch the inherent density of the real points on the line, you make proper measure impossible. That is why set theory stands in the way of properly integrating measure. Whenever the set of points becomes infinite, which is generally case in any geometrical object worth studying, set theory tosses these monkey wrenches into the works. Bijection alone does not indicate equal size. The mapping functions must be taken into account ala IFR, and the unit infinity declared and used formulaically. :) TO
From: Virgil on 20 Aug 2006 10:56 In article <ec9p2c$5a1$2(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > Virgil wrote: > > In article <ec8if0$rtd$1(a)ruby.cit.cornell.edu>, > > Tony Orlow <aeo6(a)cornell.edu> wrote: > > > > > >> Oh, how I love being compared to JSH! Shall I begin ranting about the > >> conspiracy against me, and how you will all suffer for your mathematical > >> sins against my superior intellectual achievements? No, I'll save that > >> for when I'm actually schizophrenic, thanks. ;) > > > > Tomorrow? > >> Indeed, I agree with WM's logic concerning the identity relationship > >> between element count and value in the naturals. > > > > Except that ordinal numbers cannot work that way, > > as it would require each ordinal to be a member of itself, > > which is not possible for ordinals. > > Ordinals and cardinals are schlock. TO's set theory is the only schlock here. The properties of ordinals and cardinals have been carefully worked out in ZF, ZFC and NBG. TO does not have any axiomatic system on which to base his claims but relies only on his demonstrably unreliable intuition as his only mechanism of prof. Mathematically speaking, that is not only schlock but tref. > They're not part of any theory I ascribe to. TO does not have a "theory", he merely has a random collection of intuitions.
From: Tony Orlow on 20 Aug 2006 10:58 Virgil wrote: > In article <ec9o8p$4ne$1(a)ruby.cit.cornell.edu>, > Tony Orlow <aeo6(a)cornell.edu> wrote: > >> Virgil wrote: >>> In article <ec8hlf$qq8$1(a)ruby.cit.cornell.edu>, >>> Tony Orlow <aeo6(a)cornell.edu> wrote: >>> >>> >>>> If the formula applies to an infinite number of finites, then does the >>>> sum of the aleph_0 finite naturals equal (aleph_0^2+aleph_0)/2? >>> >>> Only if (aleph_0^2+aleph_0)/2 = aleph_0. >> Which it does, in standard theory. It is not until you get to 2^aleph_0 >> that you are able to distinguish aleph_0 from a larger infinity. >> >>> But even then, one must have some prior definition of addition of >>> infinite cardinalities, and some prior definition multiplication of >>> infinite cardinalities (or at least squarings of them), and some prior >>> definition of division infinite cardinalities by finite cardinalities >>> (or at least halving of infinite cardinalities), none of which TO has >>> provided, so his "formula" is all nonsense. >>> >> We have the prior definitions of all those operations with regards to >> variables in general. > > On the contrary, each operation has been defined only for particular > domains, none of which domains include any infinite quantities. > In fact, what we call a single operation is often several operations. > For example addition has one definition for naturals, a different > definition for rationals, yet a different one for reals and another > different one for complex numbers. Balderdash! Addition is defined geometrically and consistently as the translation of the beginning of a segment representing one quantity to the end of a segment representing the other quantity, the sum being the position of the first one's new endpoint with respect to the origin. Complex numbers reside on a plane, so you must do this with each of two components, but is the same geometrical operation. > > So definitions of operations are valid only for their original domains > of definition and new definitions are required for different domains of > definition. Bull. The rules for arithmetic manipulation can be applied without problems (for anything but transfinite mathology) to variables of infinite value. > > Anyone wishing to extend an operation only defined on one domain to a > larger domain must give a clear definition of what that extended > definition is to mean. TO has not done this. > The domain is the real line. If f(n)>g(n) for all n>m, then this result applies to ALL values greater than m, including infinite ones. There is nothing fancy going on here. It's straight-up algebra and inductive proof extended. TO > >> All that's additionally required is allowing the >> variables to assume infinite values. > >>> Only a nutcase would suppose that an infinite cardinal need behave >>> entirely like a finite cardinal anyway. >> Transfinite cardinals and limit ordinals don't behave like numbers or >> quantities at all, but it's not nutty to expect numbers to act like >> numbers. > > it is no more nutty to expect infinite ordinals or cardinals to behave > differently from finite ordinals or cardinals that to expect even > naturals to behave differently from odd naturals. > > >> It's nutty to allow them to act like potions and incantations. > > Then TO is confessing his nuttiness, as that is precisely what he is > doing. > > > >> Yeah, that would the the lst natural. We all know THAT doesn't exist. >> Or, is it aleph_0? Same thing anyway. ;) > > > And here I though the first natural was 0 or 1, depending on whose > model one was following.
From: David R Tribble on 20 Aug 2006 10:58
Tony Orlow wrote: > By the way, hello! Nice to see you all again, I think. I admit it, I missed your postings. I enjoy a good laugh, and you manage to provide us with at least one good giggle each day. That's true, isn't it, Virgil? |