The Definition of Points
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From: Tony Orlow on 31 Mar 2007 18:30 cbrown(a)cbrownsystems.com wrote: > 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; Please elucidate on the untruth of the statement. It should be easy to disprove an untrue statement. 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. > There have always been religious and political pressures on this area of exploration. >> 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 Okay you have two sequences. > > 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 > Yes, I left out some details.
From: Tony Orlow on 31 Mar 2007 18:31 Lester Zick wrote: > 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~~ Welcome back to your mother-effing thread. :) E R. 01oo
From: Tony Orlow on 31 Mar 2007 18:37 Virgil wrote: > 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. > > No, that's true, The ordinals don't make a set. They're more like a mob, or an exclusive club with very boring members, that forget what their picket signs say, and start chanting slogans from the 60's. > >> 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. > > Yes, that's weak, but it's order. > > 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 > Oh no :o! How can I ever face my family again? >>> 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. I didn't mean to imply it was actually nice.... ;) haha Thanks anyway. Tony
From: Bob Kolker on 31 Mar 2007 18:42 Tony Orlow wrote:>> > > Measure makes physics possible. On compact sets which must have infinite cardinality. The measure of a dense countable set is zero. Bob Kolker
From: Tony Orlow on 31 Mar 2007 18:45
Virgil wrote: > 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? The pail is fine for minnows and tadpoles. Us crabs are always climbing out of the pail. Ah! I got your toe again! Nyah! >>> 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? > The one that allowed me to circumcise you up to the elbow. :) > > 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. Why? What have I defined, if not a sequence? Is there a word for it? It must "exist", if I assert so. Thanks. Tony |