The Definition of Points
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From: Virgil on 1 Apr 2007 15:21 In article <460fcfe5(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > cbrown(a)cbrownsystems.com wrote: > > On Mar 31, 8:38 pm, Tony Orlow <t...(a)lightlink.com> wrote: > >> cbr...(a)cbrownsystems.com wrote: > >>> On Mar 31, 5:33 pm, Tony Orlow <t...(a)lightlink.com> wrote: > >>>> Mike Kelly wrote: > > > >>>>> What does it mean for an ordering to be "discrete" or "linear"? What > >>>>> does it mean for something to "occur in" an ordering? > > > >>>> Linear means x<y ^ y<z ->x<z > >>> Funny; everyone else calls that "a transitive relation". > >> Yes, is that unrelated? > >> > > > > It's certainly /necessary/ for "<" to be a partial order; but it's not > > sufficient. Just like it's necessary for my car to have gasoline to > > run; but not sufficient. A partial order is transitive; but not every > > transitive relation is a partial order. > > > > See: > > > > http://en.wikipedia.org/wiki/Partial_order > > > > It's just three simple rules, man. Sheesh! > > > > So, add a<b v b<a v a=b, and make it a total order. If we're trying to > address the real line and its subsets, we can assume total order. Those who know what a total order is and what it requires may make such assumptions, if they are careful, but TO assumes one property implies all the others, which is not the case. > > > > Right. That's why it's called a /partial/ order, and not a /total/ > > order; there are elements which are incomparable - i.e., they cannot > > be compared in the ordering. > > > > Is the true for R or N? No. Can TO prove that? > > >> That's why > >> there's a parallel route, and so the diagram is "nonlinear". It could be > >> all on a line, but there would be several possible ordering given the > >> stated relationships. > >> > > > > And sometimes, depending on the ordering, there is no particularly > > useful way to extend that ordering. > > > > Consider the subsets of {a, b, c}, ordered by inclusion. I can say > > that every subset A <= {a, b, c} in this ordering; and I can say that > > {} <= A for every subset A; but some subsets can;t be compared in this > > ordering; for example, {a,b} and {b,c}. > > > > That's a perfectly reasonable state of affairs; not every partial > > order is somehow "required" to be a particular canonical total order. > > > > There is an obvious and useful way to provide total order on the power > set, and that is to assign the value 2^n to the element n (starting at > 0), and order according to the sum of those values corresponding to the > elements of the subsets. And how does this work for the power set of infinite sets, i which every infinite subset corresponds to a divergent series of powers of 2? > For a set of size n, your subsets are numbered > 0 through 2^n-1. Power set is the same as binary. Only for comparing finite sets. > >> What I said was that a discrete order will have pairs of elements which > >> have no elements between them, whereas a continuous order will not. But, > >> I'm sure I'm wrong. :) > >> > > > > Not so much wrong as inconsistent to the point of incomprehensibility. > > There are /no/ two real numbers x, y in the set [0,1) union (1, 2] > > with x < y such that there exists no element z in [0,1) union (1,2] > > with x < z < y. 1 is /not/ an element of [0,1) union (1,2]. > > > > Um, you're being inconsistent, or at least unclear. Alright, let's see. > I defined a discrete order as one where there exist pairs of elements > not separated by any intermediate elements. It is a good deal more than that. The set S is discretely ordered if and only if for /every/ member except a largest member there is a unique next larger and for every member except a smallest there is a unique next smaller member. Discreteness is global property of the set, in a sense the extreme opposite of denseness. > You offered [0,1)u(1,2] as a > counterexample. Can you name two elements in the union which do not have > an element between them? Try again. AS it was only a counter-example to continuity, it is till valid, That union os not continuous since it does not contain the LUB of [-1,0). > > > > So when you say "discrete partions of R", you either mean something / > > different/ from saying "the ordering on that partition is discrete", > > or else you don't have a good sense of what you really mean yourself > > when you say "discrete". > > Funny. But it also shows that you can't actually answer: what "time" > > does 2^pi "happen" in the function f: R->R defined by f(x) = 2^x? > > > > What is your point? It happens after 2^3 and before 2^4. Is that 2^pi AM or 2^pi PM? Standard or Daylight? And in what time zone?
From: Virgil on 1 Apr 2007 15:26 In article <460fd109(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <460f22e6(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >>> The obvious question is why haven't /you/ studied them; instead of > >>> making vague and uninformed statements about them (regardless of what > >>> you choose to call these ordered sets). > >>> > >> The question is, "is there an acceptable term with which to refer to > >> such uncountable linearly ordered sets?" > > > > The set of real numbers, whether with or without infinitesimals is an > > uncountable linearly ordered set, but of course not discretely ordered. > > > > And I cannot believe that TO, who is usually quite inventive, if not > > always accurate, cannot create one. > > I did, the H-riffics. > > > > >>>>>> However, where every element of a set has a well defined > >>>>>> successor and predecessor, it's a sequence of some sort. > > > > Not necessarily. If a set is partitioned into two or more subsets each > > with such an order on it, but with no order between partitions, then the > > set itself is not even an ordered set even though every member has a > > well defined predecessor and successor. > > It's certainly easy enough to order the partitions, though not without > infinite descending sequences either within or between the partitions, > in an uncountable set, as far as I can see. There is a distinction between "ordered" and "orderable" that is essential here. If two or more partitions are merely orderable, then they are orderable in more than one way and the consequences of different orderings are different. So that in the absence of any particular ordering of partitions, the set itself is NOT ordered even though each partition may be internally ordered. So that TO cannot see far enough.
From: Virgil on 1 Apr 2007 15:31 In article <460fd453(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > If the size of N is Q, then Q is the last element of N. It doesn't exist. By that argument, wouldn�t the size of the set of negative integers be -1. >
From: Lester Zick on 1 Apr 2007 15:56 On Sat, 31 Mar 2007 17:31:58 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >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. :) What's interesting here, Tony, is the sudden explosion of interest in a thread you commented only the other day appeared moribund. I mean 200+ posts on any given Sunday may well be a record. I think the trick is that you have to confine posts pretty much to a few sentences so mathematikers can read and respond to them whilst moving their lips. I often suspected mathematikers only had verbal IQ's about room temperature and the retention capacity of orangutans and now we have empirical evidence to that effect. Probably why they're modern mathematikers to begin with because their intellectual skills appear fairly well limited to memorizing and repeating slogans. ~v~~
From: Lester Zick on 1 Apr 2007 18:38
On Sat, 31 Mar 2007 17:48:57 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> 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. > >But I am. I've asked that you fill in those true/false entries in the >table I gave you, so we can see what relation you're employing. That's >an effort in "discussing" the "mechanics". What mechanics? True/false tabular mechanics? How about a definite "maybe". All I can see you're doing, Tony, is listing the details of what you consider true/false values and conjunctions in greater detail whereas I'm interested in ascertaining why and how things are true or false to begin with. >> 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~~ > >Pro say, to be exact. How many inputs, how many outputs, and what >mapping, what relation? Them's mechanics. So, expliculate. They're mechanics? How can you tell? What is it that makes them mechanics? You provide no evidence they're exhaustive or anything you say is necessarily true to the exclusion of other possibilities. You just say they are or aren't whatever you want according to whatever assumptions you want to make. I don't see that as mechanics. ~v~~ |