The Definition of Points
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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.
From: Lester Zick on 31 Mar 2007 14:53 On Fri, 30 Mar 2007 11:50:10 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >>And the only way we can address >> relations between zeroes and in-finites is through L'Hospital's rule >> where derivatives are not zero or in-finite. And all I see you doing >> is sketching a series of rules you imagine are obeyed by some of the >> things you talk about without however integrating them mechanically >> with others of the things you and others talk about. It really doesn't >> matter whether you put them within the interval 0-1 instead of at the >> end of the number line if there are conflicting mechanical properties >> preventing them from lying together on any straight line segment. >> >> ~v~~ > >Well, if you actually paid attention to any of my ideas, you'd see they >are indeed mostly mechanically related to each other, but you don't seem >interested in discussing the possibly useful mechanics employed therein. Only because you don't seem interested in discussing the mechanics on which the possibly useful mechanics employed therein are based, Tony. I'm less interested in discussing one "possibly useful mechanics" over another when there is no demonstrable mechanical basis for the "possibly useful mechanics" to begin with. You claim they're "mostly mechanically" related but not the mechanics through which they're "mostly mechanically related" except various ambiguous claims per say. ~v~~
From: Virgil on 31 Mar 2007 14:54 In article <460e56a5(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > 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. > > Sure, but the question is whether any such assumption of existence > introduces nonsense into your system. It has in each of TO's suggested systems so far. > With the very basic assumption > that subtracting a positive amount from anything makes it less That presumes at least a definition of "positive" and a definition of "amount" and a definition of "subtraction" and a definition of "less" before it makes any sense at all.
From: Lester Zick on 31 Mar 2007 14:55
On Fri, 30 Mar 2007 12:06:42 -0500, 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: >> >>>>> You might be surprised at how it relates to science. Where does mass >>>>> come from, anyway? >>>> Not from number rings and real number lines that's for sure. >>>> >>> Are you sure? >> >> Yes. >> >>> What thoughts have you given to cyclical processes? >> >> Plenty. Everything in physical nature represents cyclical processes. >> So what? What difference does that make? We can describe cyclical >> processes quite adequately without assuming there is a real number >> line or number rings. In fact we can describe cyclical processes even >> if there is no real number line and number ring. They're irrelevant. >> >> ~v~~ > >Oh. What causes them? Constant linear velocity in combination with transverse acceleration. ~v~~ |