The Definition of Points
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From: Tony Orlow on 2 Apr 2007 10:49 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. >> 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. Is N of that form? > > -- > mike. >
From: Lester Zick on 2 Apr 2007 11:19 On Sun, 01 Apr 2007 13:21:29 -0600, Virgil <virgil(a)comcast.net> wrote: >> Is the true for R or N? No. > >Can TO prove that? Prove that what? ~v~~
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 |