The Definition of Points
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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
From: MoeBlee on 2 Apr 2007 12:51 On Mar 31, 5:39 am, Tony Orlow <t...(a)lightlink.com> wrote: > In order to support the notion of aleph_0, one has to discard the basic > notion of subtraction in the infinite case. That seems like an undue > sacrifice to me, for the sake of nonsense. Sorry. For the sake of a formal axiomatization of the theorems of ordinary mathematics in analysis, algebra, topology, etc. But please do let us know when you have such a formal axiomatization but one that does have cardinal subtraction working in the infinite case just as it works in the finite case. MoeBlee
From: Virgil on 2 Apr 2007 15:51 In article <4611182b(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > It's true that the set of consecutive naturals starting at 1 with size x > has largest element x. Not unless x is less than or equal to some natural. > Is N of that form? It is if x >= aleph_0.
From: Lester Zick on 2 Apr 2007 17:36
On 2 Apr 2007 08:39:25 -0700, "Mike Kelly" <mikekellyuk(a)googlemail.com> wrote: >Is ranting incoherently just your way of blowing off steam? It seems to be yours. ~v~~ |