The Definition of Points
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From: MoeBlee on 20 Apr 2007 12:59 On Apr 20, 9:13 am, Tony Orlow <t...(a)lightlink.com> wrote: > MoeBlee wrote: > As it was explained to me, because every natural is finitely far from > the lest in w, there is no infinite descending chain of predecessors one > can define. I wouldn't put it that way myself, but okay. Yes, the converse of the membership relation on any natural number is finite. > In other words, between any two limit ordinals can only a > countable number of elements. "In other words." I love it! In 1964 Lyndon Johnson defeated Barry Goldwater in the U.S. presidential election. In other words, Mickey Rooney is a spy for an ancient species of shapeshifters from another galaxy. > Now, can a well order have an infinite > descending chain of limit elements? I asked you already to define your terminology: Define 'infinite descending chain'. No, nevermind. Better you should go back even further to understand that 'successor' is defined by 'e'. > If so, then an uncountable set could > be partitioned into an uncountable set of countable partitions, "partitioned into an uncountable set of countable partitions"? I'm not sure you're saying what you mean to say. > and be > well ordered, without a doubt, and it would already have been done. > I went and looked those things up a bit, except for Zorn's lemma, which > I just googled. I'm doing some homework here and there, though kind of > casually. I'ts better than nothing, but barely productive - hit and miss, smatterings of this and that. A proper understanding requires systematically working through the definitions and theorems. I don't know why you would settle for a haphazard, quite sloppy, thrown together hodgepodge of bits of information. I'd think your intellectual curiosity would demand that you have the real thing - a real solid and coherent grasp of the subject, even if your purpose in obtaining that understanding is to act as a critic. And even if not motivated by true intellectual curiosity, I'd think at least your vanity would require that you be able to talk about this subject with at least the authority of having grasped it at a first year level. I mean, you pretend to go toe to toe with your interlocuters here and it's nothing but a joke as you come so completely intellectually disarmed and you stay that way for years! MoeBlee
From: David R Tribble on 20 Apr 2007 13:12 Stephen wrote: >> You cannot define the sqrt(-1) geometrically. You are never going >> to draw a line with a length of i. > David R Tribble wrote: >> That's not entirely true. It's all a matter of definitions. For >> example, suppose I define "1" to be the length of a line one cm long >> drawn in a north/south direction. I can further define, arbitrarily >> mind you, that "i" is the length of a line one cm drawn in an >> east/west direction. But we both have to agree on these definitions, >> of course, if they are going to make any sense. > Lester Zick wrote: > Well I daresay we can find crackpots most anywhere who'll agree with > your definitions, David. By the way why exactly do you define "1" to > be a line of length "1"? I mean surely there must be other numbers you > could use? Don't be shy. Why not define "1" as the square root of 00? If you read carefully, you'll see that I defined the unit "1" to be a line of length 0.3937.
From: David R Tribble on 20 Apr 2007 13:17 Mike Kelly wrote: >> OK, whatever. You don't think Omega is a "quantity". You haven't >> defined "quantity" in any unambiguous way but I'm perfectly happy to >> believe you don't think the set of all finite ordinals is a >> "quantity". So what? I don't think omega sounds much like a >> "quantity", either. So? This doesn't change anything about my >> understanding of set theory. It doesn't make me say silly things like >> "aleph_0/omega is a phantom". Who cares if they're called "numbers"? > Tony Orlow wrote: > A raw quantity is a point on the real line. Points to the right > represent greater quantities than points to the left. When we add a > negative quantity, we indicate a point to the left of our starting > point, which is a different point on the real line, representing a > lesser quantity. That's one geometric representation of the reals, yes. What makes you think it's the only one? > If omega, or aleph_0, lies anywhere on this infinite > line containing all quantities, ... It doesn't. > If a set size is defined as "an integral quantity" of elements, ... It's not.
From: MoeBlee on 20 Apr 2007 13:19 On Apr 20, 9:24 am, Tony Orlow <t...(a)lightlink.com> wrote: > MoeBlee wrote: > So, defining N doesn't involve a successor relation between two > elements, as well as a member relation between an element and a set? The successor operation is DEFINED in terms of the membership relation. EVERYTHING in set theory is defined in terms of the membership relation. The only non-logical primitives of set theory are '=' and 'e' (and we could even define '=' in terms of 'e' if we want to set it up that way). There is NO formula of set theory that doesn't revert to a formula in the primitive language with just 'e' and '=' (or even just 'e') as the ONLY non-logical symbols. I've been telling you this for probably over a year now. Why don't you understand this? > When you define N, doesn't the rule E x e N -> E succ(x) e N define a > relation between two elements, as well as between those elements and N? "Define a relation". From N of course we can define the successor relation on N. So what? That doesn't refute that every definition in set theory ultimately reverts to the membership relation. I'll say it YET AGAIN: The only non-logical primitives of set theory are '=' and 'e' (and we could even define '=' in terms of 'e' if we want to set it up that way). There is NO formula of set theory that doesn't revert to a formula in the primitive language with just 'e' and '=' (or even just 'e') as the ONLY non-logical symbols. Over a year I've been telling you that over and over and over. But it seems that as far as you're concerned, such information is just a random collection of characters appearing on a computer monitor. MoeBlee > > TOEknee- Hide quoted text - > > - Show quoted text -- Hide quoted text - > > - Show quoted text -
From: cbrown on 20 Apr 2007 13:39
On Apr 20, 9:59 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Apr 20, 9:13 am, Tony Orlow <t...(a)lightlink.com> wrote: > > > MoeBlee wrote: > > As it was explained to me, because every natural is finitely far from > > the lest in w, there is no infinite descending chain of predecessors one > > can define. > > I wouldn't put it that way myself, but okay. Yes, the converse of the > membership relation on any natural number is finite. > > > In other words, between any two limit ordinals can only a > > countable number of elements. > > "In other words." I love it! > > In 1964 Lyndon Johnson defeated Barry Goldwater in the U.S. > presidential election. In other words, Mickey Rooney is a spy for an > ancient species of shapeshifters from another galaxy. > Well, at least he /should/ be. Just check out his eyes. Cheers - Chas |