The Definition of Points
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From: Lester Zick on 2 Apr 2007 11:21 On Sun, 01 Apr 2007 13:01:05 -0600, Virgil <virgil(a)comcast.net> wrote: >> Don't be a boor. > >Does TO wish to reserve that tight to himself alone? No. TO is pretty loose when it comes to that. You're welcome to be a boor. ~v~~
From: Mike Kelly on 2 Apr 2007 11:39 On 2 Apr, 15:49, Tony Orlow <t...(a)lightlink.com> wrote: > Mike Kelly wrote: > > On 1 Apr, 04:44, Tony Orlow <t...(a)lightlink.com> wrote: > >> cbr...(a)cbrownsystems.com wrote: > >>> On Mar 31, 5:45 pm, Tony Orlow <t...(a)lightlink.com> wrote:> > >>>> Yes, NeN, as Ross says. I understand what he means, but you don't. > >>> What I don't understand is what name you would like to give to the set > >>> {n : n e N and n <> N}. M? > >>> Cheers - Chas > >> N-1? Why do I need to define that uselessness? I don't want to give a > >> size to the set of finite naturals because defining the size of that set > >> is inherently self-contradictory, > > > So.. you accept that the set of naturals exists? But you don't accept > > that it can have a "size". Is it acceptable for it to have a > > "bijectibility class"? Or is that taboo in your mind, too? If nobody > > ever refered to cardinality as "size" but always said "bijectibility > > class" (or just "cardinality"..) would all your objections disappear? > > Yes, but my desire for a good way of measuring infinite sets wouldn't go > away. You seem to be implying that the existence and acceptance of cardinality as one way of measuring infinite sets precludes the invention of any other. This is patently false. There is an entire branch of mathematics called "measure theory" which, roughly speaking, examines various ways to measure and compare infinite sets. Measure theory builds upon set theory. Set theory doesn't preclude mesure theory. Of course, if *your* ideas were to be formalised then first of all you'd have to pull your head out of.. the sand, accept that you've made numerous egregiously erroneous statements about standard mathematics, learn how to communicate mathematically and learn how to formalise mathematical ideas precisely. Look at NSA and the Surreal numbers if you need evidence that non-standard ideas can be expressed clearly and coherently within an existing framework of mathematical expression. You may be a lost cause though. You've spent, what, three years blathering on Usenet and your mathematical understanding and maturity hasn't improved a jot. It seems like you genuinely don't want to learn. Is ranting incoherently just your way of blowing off steam? > >> given the fact that its size must be equal to the largest element, > > > That isn't a fact. It's true that the size of a set of naturals of the > > form {1,2,3,...,n} is n. But N isn't a set of that form. Is it? > > It's true that the set of consecutive naturals starting at 1 with size x has largest element x. No. This is not true if the set is not finite (if it does not have a largest element). It is true that the set of consecutive naturals starting at 1 with largest element x has cardinality x. It is not true that the set of consecutive naturals starting at 1 with cardinality x has largest element x. A set of consecutive naturals starting at 1 need not have a largest element at all. Do you see that changing the order of words in a statement can change the meaning or that statement? Do you see that one statement can be true, and another statement with the same words in a different order can be false? > Is N of that form? N is a set of consecutive naturals starting at 1. It doesn't have a largest element. It has cardinality aleph_0. -- mike.
From: stephen on 2 Apr 2007 12:12 In sci.math Mike Kelly <mikekellyuk(a)googlemail.com> wrote: > On 2 Apr, 15:49, Tony Orlow <t...(a)lightlink.com> wrote: >> Mike Kelly wrote: >> >> > That isn't a fact. It's true that the size of a set of naturals of the >> > form {1,2,3,...,n} is n. But N isn't a set of that form. Is it? >> >> It's true that the set of consecutive naturals starting at 1 with size x has largest element x. > No. This is not true if the set is not finite (if it does not have a > largest element). > It is true that the set of consecutive naturals starting at 1 with > largest element x has cardinality x. > It is not true that the set of consecutive naturals starting at 1 with > cardinality x has largest element x. A set of consecutive naturals > starting at 1 need not have a largest element at all. To be fair to Tony, he said "size", not "cardinality". If Tony wishes to define "size" such that set of consecutive naturals starting at 1 with size x has a largest element x, he can, but an immediate consequence of that definition is that N does not have a size. Stephen
From: Mike Kelly on 2 Apr 2007 12:24 On 2 Apr, 17:12, step...(a)nomail.com wrote: > In sci.math Mike Kelly <mikekell...(a)googlemail.com> wrote: > > > On 2 Apr, 15:49, Tony Orlow <t...(a)lightlink.com> wrote: > >> Mike Kelly wrote: > > >> > That isn't a fact. It's true that the size of a set of naturals of the > >> > form {1,2,3,...,n} is n. But N isn't a set of that form. Is it? > > >> It's true that the set of consecutive naturals starting at 1 with size x has largest element x. > > No. This is not true if the set is not finite (if it does not have a > > largest element). > > It is true that the set of consecutive naturals starting at 1 with > > largest element x has cardinality x. > > It is not true that the set of consecutive naturals starting at 1 with > > cardinality x has largest element x. A set of consecutive naturals > > starting at 1 need not have a largest element at all. > > To be fair to Tony, he said "size", not "cardinality". If Tony wishes to define > "size" such that set of consecutive naturals starting at 1 with size x has a > largest element x, he can, but an immediate consequence of that definition > is that N does not have a size. > > Stephen Well, yes. But Tony wants to use this line of reasoning to then say "and, therefore, if N has size aleph_0 then aleph_0 is the largest element, which is clearly bunk". This is where his claims that "aleph_0 is a phantom" come from. But, obviously, this line of reasoning doesn't apply to all notions of "size". This isn't quite quantifier dyslexia, but it's related I guess. Tony describes one notion of size where N doesn't have a size. Then he wants to point to cardinality, which is another notion of size, and triumphantly exclaim "Look, cardinality gives a size to N! Cardinality is bunk! Aleph_0 is a phantom!". Who he thinks he is fooling is beyond me. -- mike.
From: MoeBlee on 2 Apr 2007 12:47
On Mar 31, 5:18 am, Tony Orlow <t...(a)lightlink.com> wrote: > Virgil wrote: > > In article <1175275431.897052.225...(a)y80g2000hsf.googlegroups.com>, > > "MoeBlee" <jazzm...(a)hotmail.com> wrote: > > >> On Mar 30, 9:39 am, Tony Orlow <t...(a)lightlink.com> wrote: > > >>> They > >>> introduce the von Neumann ordinals defined solely by set inclusion, > >> By membership, not inclusion. > > > By both. Every vN natural is simultaneously a member of and subset of > > all succeeding naturals. > > Yes, you're both right. Each of the vN ordinals includes as a subset > each previous ordinal, and is a member of the set of all ordinals. In the more usual theories, there is no set of all ordinals. > In > this sense, they are defined solely by the "element of" operator, or as > MoeBlee puts it, "membership". Members are included in the set. Or, > shall we call it a "club"? :) > > Anyway, my point is that the recursive nature of the definition of the > "set" What recursive definition of what set? > introduces a notion of order which is not present in the mere idea > of membership. > Order is defined by x<y ^ y<z -> x<z. Transitivity is one of the properties of most of the orderings we're talking about. But transitivity is not the only property that defines such things as 'partial order', 'linear order', 'well order'. > This is generally > interpreted as pertaining to real numbers or some subset thereof, but if > you interpret '<' as "subset of", then the same rule holds. Yes, the subset relation on any set is a transitive relation. > I suppose > this is one reason why I think a proper subset should ALWAYS be > considered a lesser set than its proper superset. It's less than the > superset by the very mechanics of what "less than" means. A proper subset is less than a proper superset of it, in the sense that the proper superset has all members of the proper subset plus at least one more. It is not always the case though that a set is not 1-1 with some proper subset of itself. In the finite, the two aspects coincide, but not in the infinite. That's just the way it is in the more usual set theories. That does not stop you from formulating a different theory though. MoeBlee |