The Definition of Points
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From: Tony Orlow on 31 Mar 2007 08:18 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. :D
From: Tony Orlow on 31 Mar 2007 08:30 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. > > 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. However, where every element of a set has a well defined successor and predecessor, it's a sequence of some sort. Is the 2-adic really a pair of countable sets, counting upward from ....000 and downward from ..111? It would appear so, at first glance, but those two sequences count toward each other. They're really two elements of a sequence, infinitely far apart, in a single sequence. The problem is, we can't express any number halfway between the two. The T-riffics are supposed to address the issue of that point of intersection by declaring a bit at oo. In that system we count up from 000...000 to 111...111, the midpoint obviously being the transition from 011...111 to 100...000. The ellipses may represent a finite or an infinite number of bits, without affecting the arithmetic. We can say that left most bit is number oo, and we're counting from 0 to 2^(oo+1)-1. In that case, the sequence is definitely uncountable, even though the set or T-riffic numbers we can represent with finite strings is countable. Tony
From: Tony Orlow on 31 Mar 2007 08:39 Virgil wrote: > 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. Sure, but the question is whether any such assumption of existence introduces nonsense into your system. With the very basic assumption that subtracting a positive amount from anything makes it less, the inductive logic that proves there is no largest finite can be applied to prove there is no smallest infinite: Assume we are working with positive numbers. Subtract a finite number from an infinite number. The result must be an infinite number, because if it were finite, then its addition to the finite number we subtracted would not yield the original infinite number. The result must be smaller than the original infinite number, because we have subtracted a positive amount from it. Therefore, for any infinite number, one can produce a smaller infinite number, and there is thus no smallest infinite number. In order to support the notion of aleph_0, one has to discard the basic notion of subtraction in the infinite case. That seems like an undue sacrifice to me, for the sake of nonsense. Sorry. Tony
From: Tony Orlow on 31 Mar 2007 08:41 Mike Kelly wrote: > 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. > A set where each element has a well defined unique successor within the set. Good enough? tony.
From: Tony Orlow on 31 Mar 2007 08:48
stephen(a)nomail.com wrote: > In sci.math Virgil <virgil(a)comcast.net> 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. > > It is not even true in Tony's mathematics, at least it was not true > the last time he brought it up. According to this > definition {1, 2, 3, ... } is not actually infinite, but > {1, 2, 3, ..., w} is actually infinite. However, the last time this > was pointed out, Tony claimed that {1, 2, 3, ..., w} was not > actually infinite. > > Stephen No, adding one extra element to a countable set doesn't make it uncountable. If all other elements in the sequence are a finite number of steps from the start, and w occurs directly after those, then it is one step beyond some step which is finite, and so is at a finite step. Try (...000, ..001, ...010, ......, ...101, ...110, ...111) Tony |