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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~~
From: Lester Zick on 31 Mar 2007 14:45 On Sat, 31 Mar 2007 07:18:15 -0500, 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 >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" 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. 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. > >>>> 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. > >Why, thank you, Virgil. That's the nicest thing you've ever said to me. Virgil speaketh in tongues and truisms. ~v~~
From: Virgil on 31 Mar 2007 14:49
In article <460e5476(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <460d4813(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > > > >> An actually infinite sequence is one where there exist two elements, one > >> of which is an infinite number of elements beyond the other. > > > > Not in any standard mathematics. > > Well, "actually infinite" isn't a defined term in standard mathematics. If it is outside the pale, why bother with it at all? > > > > > In standard mathematics, an infinite sequence is no 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. What sect would have been so foolish as to have ordained TO into its priesthood? However, where every element of a set has a well defined > successor and predecessor, it's a sequence of some sort. Every such "sequence" set must be representable as a function from the from the integers to that set. |