The Definition of Points
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From: cbrown on 31 Mar 2007 14:40 On Mar 31, 5:30 am, Tony Orlow <t...(a)lightlink.com> wrote: > Virgil wrote: > > In standard mathematics, an infinite sequence is o more than a function > > whose domain is the set of naturals, no two of which are more that > > finitely different. The codmain of such a function need not have any > > particular structure at all. > > That's a countably infinite sequence. Standard mathematics doesn't allow > for uncountable sequences like the adics or T-riffics, because it's been > politically agreed upon that we skirt that issue and leave it to the > clerics. That's false; people have examined all sorts of orderings, partial, total, and other. The fact that you prefer to remain ignorant of this does not mean the issue has been skirted by anyone other than yourself. > However, where every element of a set has a well defined > successor and predecessor, it's a sequence of some sort. > Let S = {0, a, 1, b, 2, c}. Let succ() be defined on S as: succ(0) = 1 succ(1) = 2 succ(2) = 0 succ(a) = b succ(b) = c succ(c) = a Every element of S has a well-defined successor and predecessor. What "sort of sequence" have I defined? Or have you left out some parts of the /explicit/ definition of whatever you were trying to say? Cheers - Chas
From: Lester Zick on 31 Mar 2007 14:43 On 30 Mar 2007 10:23:51 -0700, "MoeBlee" <jazzmobe(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. > >> and >> yet, surreptitiously introduce the notion of order by means of this set. > >"Surreptitiously". You don't know an effing thing you're talking >about. Look at a set theory textbook (such as Suppes's 'Axiomatic Set >Theory') to see the explicit definitions. Kinda like Moe(x) huh. ~v~~
From: Virgil on 31 Mar 2007 14:43 In article <460e5198(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <1175275431.897052.225580(a)y80g2000hsf.googlegroups.com>, > > "MoeBlee" <jazzmobe(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 ZF and in NBG, there is no such thing as a set of all ordinals. In NBG there may be a class of all ordinals, but in ZF, not even that. > > Anyway, my point is that the recursive nature of the definition of the > "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. That is only a partial ordering on sets, and on ordinals is no more than the weak order relation induced by membership, namely: x < y if and only if either x e y or x = y. 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. 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. But your version of "less than" is is only a partial order > > > > On the other hand, Tony Orlow is considerably less of an ignoramus than > > Lester Zick. > > Why, thank you, Virgil. That's the nicest thing you've ever said to me. You would not think so if you knew my opinion of Zick.
From: Lester Zick on 31 Mar 2007 14:44 On Fri, 30 Mar 2007 12:17:31 -0600, Virgil <virgil(a)comcast.net> wrote: >In article <1175275431.897052.225580(a)y80g2000hsf.googlegroups.com>, > "MoeBlee" <jazzmobe(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. > >> > and >> > yet, surreptitiously introduce the notion of order by means of this set. >> >> "Surreptitiously". You don't know an effing thing you're talking >> about. Look at a set theory textbook (such as Suppes's 'Axiomatic Set >> Theory') to see the explicit definitions. > >On the other hand, Tony Orlow is considerably less of an ignoramus than >Lester Zick. Who is considerably less of an ignoramus than you. ~v~~
From: Lester Zick on 31 Mar 2007 14:44
On 30 Mar 2007 11:29:29 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >On Mar 30, 11:17 am, Virgil <vir...(a)comcast.net> 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. > >Of course. I just meant as to which is the primitive, in the sense >that the definitions revert ultimately solely to the membership >relation. How about the "domain of discourse" relation, Moe(x)? ~v~~ |