An uncountable countable set
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From: Virgil on 17 Sep 2006 15:51 In article <450d5757(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Without the additional > structure which Aatu suggests, bijection may show some kind of > equivalence, but it cannot be considered any kind of exact analog for > the size of finite sets. You're trying to extract measure from something > with no measure in it, like blood from a stone. On the contrary, Cantor was trying to devise a measure which was entirely independent of every property of the members of each set other than their being distinguishable from each other. TO is measuring order relations, not sets.
From: Virgil on 17 Sep 2006 15:52 In article <450d5880(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Aatu Koskensilta wrote: > > Virgil wrote: > >> In article <Yz2Pg.13567$VX1.6175(a)reader1.news.jippii.net>, > >> Aatu Koskensilta <aatu.koskensilta(a)xortec.fi> wrote: > >> > >>> This is not to say that other notions of "size" as applied to sets > >>> are ignored; the idea that there are twice as many naturals as there > >>> are odd naturals can be captured mathematically, although this notion > >>> is less general and applies only in case the sets in question are > >>> equipped with additional structure. > >> > >> Exactly the point that Tony Orlow rejects. > > > > Quite possibly. Since he's an obvious crank there's really very little > > point in caring about what he thinks or rejects, and even less point in > > engaging him in endless "debates". Of course, this is USENET and there's > > very little point to anything in any case; hence my few observations on > > the rhetorical tactics in these debates. > > > > Okay, well, I reject that. > > Tony As usual, TO and truth do not get on well together.
From: Virgil on 17 Sep 2006 15:57 In article <450d5b3b(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > There IS no LUB on the finites, omega notwithstanding. There is a LUB of finite ordinals, the first limit ordinal. That TO denies it is, if anything, evidence in its favor.
From: Virgil on 17 Sep 2006 16:08 In article <450d5c83(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > Since Aleph_0 is not a real number, neither would it have a real > > number reciprocal. The set of non-zero real numbers is a group, so that > > only real numbers can have real number multiplicative inverses. > > It's not a real number at all, in the sense of not being a number at > all. It's not a count of anything and it doesn't have any valid > arithmetic. It can't be expressed except by invoking it's holy name. If > it's not greater than any finite number, then it's not a number. Mathematics has definitions for all sorts of specific types of "numbers", such as naturals or reals, but none for "number" in general. So that "not a number" is, at best, ambiguous, and at worst, meaningless. > > Because aleph_0 ISN'T a number. Aleph_0 is a well defined CARDINAL number. > It's a phantom with a name and cult that > worships it. As it has a definition and an instanciation (as the set of all naturals), TO is WRONG! AGAIN! AS USUAL!!! > > "No Largest Finite!!! (GONGGG!!!) Huyah huyah huyah Ommmmmmmm......ega!" > > Chant it, Baby. You'll find it one day. You're potentially infinite. TO's potentiality is infinitesimal, which makes him mistake that of anyone of normal talents as larger than life..
From: Virgil on 17 Sep 2006 16:20
In article <450d5dd8(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > > > The distance between any two "points" on the real line is a real number, > > the (absolute) difference between the real numbers for these points. > > And there is no such difference which is infinite on the entire real > line. How then do you fit an infinite number of unit intervals in that > space? With a shoehorn? > > > > What TO means by "the count of naturals" only he knows, but the > > cardinality of the set of naturals exists. > > > > Right, because cardinality is only a number in the finite case, and > otherwise is an amorphous equivalence class. In most systems,nowadays, such as ZF or NBG, one chooses a specific representative of each cardinality, for example {} for 0 and N for aleph_0, and does not worry about equivalence classes at all. And a set has that cardinality if and only if it bijects with that representative. > > > There is a standard measure for the distance between any two numbers on > > the real line which is given by the absolute difference between those > > real numbers. But that in no way gives any measure for the line in its > > entirety, because it has no end point numbers from which to take an > > absolute difference. > > > > If TO's mind is too perplexed to see that, he needs a shrink. > > > > Yeah, a shrink will help me with infinity. Good suggestion. > > Either you have a complete real line or you do not. We have a complete set of real numbers. >If you are comparing > infinite sets over this entire possible interval Intervals imply order relations, so that TO is only measuring order relations, not ordinary sets. > > > > > > >> My logic is clear and simple. > > > > And wrong! > > Define "wrong". TO is measuring order relations, Cantor is measuring sets. > I thought for you it was all about internal consistency Since TO cannot measure anything set without having a particular order relation on it, and gets different measures for different order relations on the same set, what TO is measuring is order relations, not their underlying sets. Cantor, on the other hand, IS measuring the underlying sets. |