The Definition of Points
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From: Tony Orlow on 12 Apr 2007 15:28 Virgil wrote: > 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. Prove it, without assuming it. > >> Is N of that form? > > It is if x >= aleph_0. Then aleph_0 e N. Sorry. Tony
From: Tony Orlow on 12 Apr 2007 15:30 Lester Zick wrote: > On Mon, 2 Apr 2007 16:12:46 +0000 (UTC), stephen(a)nomail.com wrote: > >>> 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. > > Is that true? > > ~v~~ Yes, Lester, Stephen is exactly right. I am very happy to see this response. It follows from the assumptions. Axioms have merit, but deserve periodic review. 01oo
From: Tony Orlow on 12 Apr 2007 15:34 Virgil wrote: > In article <461e7764(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Lester Zick wrote: > >>> And what is your definition of "infinite"? >>> >>> ~v~~ >> "greater than any finite" >> > > And is TO's definition of finite "less than infinite"? Hi Virgil. Sweet greetings. Finite is properly defined by bijection. A finite set cannot be bijected with a subset, and a finite real lies between two finite naturals, each the size of a finite set. (no signature)
From: Virgil on 12 Apr 2007 16:32 In article <461e879d(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > MoeBlee wrote: > > 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. > > > > Right. Ordinals are...ordered. Sets aren't. Ordinals have a unique ordering by reason of their being ordinals. Sets in general have all sorts of orderings, but none which is as inherent in their being sets as the ordinal order is in sets being ordinals. > >> > >> Anyway, my point is that the recursive nature of the definition of the > >> "set" > > > > What recursive definition of what set? > > > > Oh c'mon! N. ala Peano? (sigh) What kind of question is that? Does TO seem to thing that N is the only set defineable recursively or that "successor" is the only recursively defineable operations on sets? > > > >> 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'. > > > > It defines order, in general. Only to TO. For everyone else, other properties are required. For example, in addition to transitivity, ((x>y) and (y>x)) -> x = y is a necessary property /every/ ordering. Also there are lots of transitive relations which are not orderings, at least as usually understood. E.g., universal relations, which hold true for all x and y in the relevant set. So that TO's notion of an ordering does not necessarily order anything. > > >> 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. The mechanics of "less than" depends on what standard of measurement one is using, so claiming that one measure measures all is a procrustean fallacy. > There can always be a 1-1 correspondence defined between a set > with no end and its proper subset with no end, even if that > correspondence is so complicated so as to defy all attempts to define > it. Trivially false. Neither the set of reals nor the set of rationals has an end, and the rationals are a proper subset of the reals, but there is no bijection between them. And, given the axiom of choice, any well ordered uncountable set even has well ordered countable subsets with which it does not biject.
From: MoeBlee on 12 Apr 2007 17:04
On Apr 12, 11:16 am, Tony Orlow <t...(a)lightlink.com> wrote: > > And what is your definition of "infinite"? > "greater than any finite" Define 'finite' and 'greater than'. Nevermind, you have no primitives anyway to which ANY of your definitions ultimately revert. MoeBlee |