The Definition of Points
in [Math]
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From: Virgil on 30 Mar 2007 14:17 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.
From: Virgil on 30 Mar 2007 14:22 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. 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.
From: MoeBlee on 30 Mar 2007 14:29 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. MoeBlee
From: Virgil on 30 Mar 2007 14:31 In article <460d489b(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Lester Zick wrote: > > On Thu, 29 Mar 2007 09:37:21 -0500, Tony Orlow <tony(a)lightlink.com> > > wrote: > > > >>> Just ask yourself, Tony, at what magic point do intervals become > >>> infinitesimal instead of finite? Your answer should be magnitudes > >>> become infintesimal when subdivision becomes infinite. > >> Yes. > > > > Yes but that doesn't happen until intervals actually become zero. > > > >> But the term > >>> "infinite" just means undefined and in point of fact doesn't become > >>> infinite until intervals become zero in magnitude. But that never > >>> happens. > >> But, but, but. No, "infinite" means "greater than any finite number" and > >> infinitesimal means "less than any finite number", where "less" means > >> "closer to 0" and "more" means "farther from 0". > > > > Problem is you can't say when that is in terms of infinite bisection. > > > > ~v~~ > > Cantorians try with their lame "aleph_0". Better you get used to the > fact that there is no more a smallest infinity than a smallest finite, > largest finite, or smallest or largest infinitesimal. Those things > simply don't exist, except as phantoms. But all other mathematical objects are equally fantastic, having no physical reality, but existing only in the imagination. So any statement of mathematical existence is always relative to something like a system of axioms.
From: Mike Kelly on 30 Mar 2007 15:26
On 30 Mar, 18:25, Tony Orlow <t...(a)lightlink.com> wrote: > Lester Zick wrote: > > On Thu, 29 Mar 2007 09:37:21 -0500, Tony Orlow <t...(a)lightlink.com> > > wrote: > > >>>> If n is > >>>> infinite, so is 2^n. If you actually perform an infinite number of > >>>> subdivisions, then you get actually infinitesimal subintervals. > >>> And if the process is infinitesimal subdivision every interval you get > >>> is infinitesimal per se because it's the result of a process of > >>> infinitesimal subdivision and not because its magnitude is > >>> infinitesimal as distinct from the process itself. > >> It's because it's the result of an actually infinite sequence of finite > >> subdivisions. > > > And what pray tell is an "actually infinite sequence"? > > >> One can also perform some infinite subdivision in some > >> finite step or so, but that's a little too hocus-pocus to prove. In the > >> meantime, we have at least potentially infinite sequences of > >> subdivisions, increments, hyperdimensionalities, or whatever... > > > Sounds like you're guessing again, Tony. > > > ~v~~ > > An actually infinite sequence is one where there exist two elements, one > of which is an infinite number of elements beyond the other. > > 01oo Under what definition of sequence? -- mike. |