The Definition of Points
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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: David R Tribble on 20 Apr 2007 13:19 "G.E. Ivey wrote: >> I believe it was Hilbert who said that "If you replace >> points and lines by beer steins and tables, every statement should still be >> true." > Michael Press wrote: > Ever pair of tables has a beer stein in common? > No wonder he had so many problems. I think you mean "every pair of beer steins has a table in common". Also, "two parallel tables have no beer steins in common".
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
From: Virgil on 20 Apr 2007 14:56
In article <4628df20$1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > > I do say so. Does this mean you're going to stop claiming that > > relations in set theory aren't based solely on 'e'? > > > > -- > > mike. > > > > No, not when sequences are defined using a recursively defined successor > function, which is a relation between two elements, as opposed 'e', a > relation between an element and a set. The combination of the two is > what produces an infinite set, no? So that x -> x u {x} does not depend on 'e' ? On can define successor entirely in terms of 'e'. Successor, in the x -> x u {x} sense, depends only on singleton sets, subsets and unions, all of which are defined strictly in terms of 'e', so what is left? Nothing. |