The Definition of Points
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From: Tony Orlow on 17 Apr 2007 12:21 Lester Zick wrote: > On Fri, 13 Apr 2007 16:52:21 +0000 (UTC), stephen(a)nomail.com wrote: > >> In sci.math Tony Orlow <tony(a)lightlink.com> wrote: >>> Lester Zick wrote: >>>> On Mon, 2 Apr 2007 16:12:46 +0000 (UTC), stephen(a)nomail.com wrote: >>>> >>>>>> It is not true that the set of consecutive naturals starting at 1 with >>>>>> cardinality x has largest element x. A set of consecutive naturals >>>>>> starting at 1 need not have a largest element at all. >>>>> To be fair to Tony, he said "size", not "cardinality". If Tony wishes to define >>>>> "size" such that set of consecutive naturals starting at 1 with size x has a >>>>> largest element x, he can, but an immediate consequence of that definition >>>>> is that N does not have a size. >>>> Is that true? >>>> >>>> ~v~~ >>> Yes, Lester, Stephen is exactly right. I am very happy to see this >>> response. It follows from the assumptions. Axioms have merit, but >>> deserve periodic review. >>> 01oo >> Everything follows from the assumptions and definitions. > > And since definitions are considered neither true nor false everything > follows from raw assumptions which are considered neither true nor > false. > > ~v~~ Oh come on. Assumptions are considered true for the sake of the argument at hand. That's what an assumption IS. 01oo
From: Tony Orlow on 17 Apr 2007 12:51 Lester Zick wrote: > On Fri, 13 Apr 2007 13:42:06 -0400, Tony Orlow <tony(a)lightlink.com> > wrote: > >> Lester Zick wrote: >>> On Thu, 12 Apr 2007 14:23:04 -0400, Tony Orlow <tony(a)lightlink.com> >>> wrote: >>> >>>> Lester Zick wrote: >>>>> On Sat, 31 Mar 2007 16:18:16 -0400, Bob Kolker <nowhere(a)nowhere.com> >>>>> wrote: >>>>> >>>>>> Lester Zick wrote: >>>>>> >>>>>>> Mathematikers still can't say what an infinity is, Bob, and when they >>>>>>> try to they're just guessing anyway. So I suppose if we were to take >>>>>>> your claim literally we would just have to conclude that what made >>>>>>> physics possible was guessing and not mathematics at all. >>>>>> Not true. Transfite cardinality is well defined. >>>>> I didn't say it wasn't, Bob. You can do all the transfinite zen you >>>>> like. I said "infinity". >>>>> >>>>>> In projective geometry points at infinity are well defined (use >>>>>> homogeneous coordinates). >>>>> That's nice, Bob. >>>>> >>>>>> You are batting 0 for n, as usual. >>>>> Considerably higher than second guessers. >>>>> >>>>> ~v~~ >>>> That's okay. 0 for 0 is 100%!!! :) >>> Not exactly, Tony. 0/0 would have to be evaluated under L'Hospital's >>> rule. >>> >>> ~v~~ >> Well, you put something together that one can take a derivative of, and >> let's see what happens with that. > > Or let's see you put something together that you can't take the > deriviative of and let's see how you managed to do it. > > ~v~~ Okay. What's the derivative of 0? 01oo
From: Tony Orlow on 17 Apr 2007 13:02 Brian Chandler wrote: > Tony Orlow wrote: >> MoeBlee wrote: >>> On Apr 13, 10:38 am, Tony Orlow <t...(a)lightlink.com> wrote: >>>> MoeBlee wrote: >>>>> Zermelo's motivation was to prove that for every set, there exists a >>>>> well ordering on it. >>>> I am not sure how the Axiom of Choice demonstrates that. >>> You don't know how the axiom of choice is used to prove that for every >>> set there exists a well ordering of the set? Virtually any set theory >>> textbook will give a cycle of proofs showing equivalence of (not >>> necessarily in order) the axiom of choice (in its various >>> formulations), Zorn's lemma, the well ordering theorem, the numeration >>> theorem, etc. >>> >>> Among those textbooks I recommend Stoll's 'Set Theory And Logic' as it >>> accomplishes some of the proofs without using the axiom schema of >>> replacement while other textbooks do use the axiom schema of >>> replacement for certain of the proofs, though, I don't recommend >>> Stoll's book for an overall systematic treatment since it jumps around >>> topics too much and doesn't have the kind of "linear" format that >>> Suppes does so well. >>> >>> Anyway, even if you don't know the details of the proofs, don't you at >>> least have an intuition how a choice function would come in handy >>> toward proving the well ordering theorem? >>> >>> MoeBlee >>> >> Hi MoeBlee - >> >> I had said that, hoping you might give some explanation, but you didn't >> really. However, before I sent that response I reminded myself on >> Wikipedia exactly what AC says, and looked at the definitions of >> Dependent Choice and Countable Choice as well, and descriptions of the >> relationships between them. I didn't find anything objectionable in ACC >> or DC. I think it is the broad statement of AC that any set is well >> orderable that offends my sensibilities. > > What a good job then, that mathematics is not about "your > sensibilities". If you accept (use, adopt, whatever) the Axiom of > Choice then there is a proof that any set has a well-ordering. > Sensibilities don't come into it. > Sensibilities indeed play a role in what assumptions we do or don't accept to begin with. The relegation of mathematics to the purely deductive realm is a copout. Where a lot of intuition went into the formulation of ZFC, intuition may also reject it. >> Here's my intuition. If you have a set, and can partition it into >> mutually exclusive subsets, within which and between which exists an >> order, then you can linearly arrange the elements of the set by choosing >> the first element of the first partition, and repeatedly choosing the >> first unchosen element from the next partition, returning to the first >> partition when the last is used, and skipping any partitions from which >> all elements have been chosen. > > What if any of thes sets do not have either a first member (within the > ordering) or a last member? If the set is ordered to begin with, such that x<>y -> x<y v y<x, then any subset is also ordered accordingly. > Do your intuitions perhaps tell you that there always _is_ a "first" > member and a "last" member? Do you notice any similarity between this > particular "intuition" and the thing you were claiming to be proving > in the first place? > > > Does a well orderable set have a first member? What was I proving to begin with? Did you think this statement of intuition was a proof? > Where we have a countably infinite set, >> we may choose a finite number of partitions, at least one of which has a >> countably infinite number of elements, or we may choose all finite >> partitions, but a countably infinite number of them. In such a case, >> indeed, it's possible to define a well order. > > If by "countable" and "well order" you mean what mathematicians mean, > then this is a bizarre circumlocution, since a countable set has a > well-ordering more or less by definition. I suppose you mean something > subtly different by these terms - something you yourself understand > perfectly, yet are unable to convey to anyone with a grasp of actual > mathematics. > > <snip> > > Brian Chandler > http://imaginatorium.org > I am saying it's obvious that any countable set has a well ordering. It is not obvious for uncountable sets. Nice snip of the actual import of my post, BTW. Thanx. Tony Orlow http://realitorium.net ;)
From: Tony Orlow on 17 Apr 2007 13:09 Mike Kelly wrote: > On 13 Apr, 20:51, Tony Orlow <t...(a)lightlink.com> wrote: >> MoeBlee wrote: >>> On Apr 13, 11:25 am, Tony Orlow <t...(a)lightlink.com> wrote: >>>>> It's very easily provable that if "size" means "cardinality" that N >>>>> has "size" aleph_0 but no largest element. You aren't actually >>>>> questioning this, are you? >>>> No, have your system of cardinality, but don't pretend it can tell >>>> things it can't. Cardinality is size for finite sets. For infinite sets >>>> it's only some broad classification. >>> Nothing to which you responded "pretends" that cardinality "can tell >>> things it can't". What SPECIFIC theorem of set theory do you feel is a >>> pretense of "telling things that it can't"? >> AC > > And what AC have to do with cardinality? What do any of the axioms of ZFC have to do with cardinality? Extensionality. But, cardinality is a Galilean, read "primitive", extension of Extensionality. > >>>>> It has CARDINALITY aleph_0. If you take "size" to mean cardinality >>>>> then aleph_0 is the "size" of the set of naturals. But it simply isn't >>>>> true that "a set of naturals with 'size' y has maximum element y" if >>>>> "size" means cardinality. >>>> I don't believe cardinality equates to "size" in the infinite case. >>> Wow, that is about as BLATANTLY missing the point of what you are in >>> immediate response to as I can imagine even you pulling off. >> What point did I miss? I don't take transfinite cardinality to mean >> "size". You say I missed the point. You didn't intersect the line. > > The point that it DOESN'T MATTER whther you take cardinality to mean > "size". It's ludicrous to respond to that point with "but I don't take > cardinality to mean 'size'"! > > -- > mike. > You may laugh as you like, but numbers represent measure, and measure is built on "size" or "count". If I say there are countably infinitely many possible lemurs in the future, and countably infinitely many possible mammals in the future, are there equally infinitely many possible lemurs as mammals? It's unlikely that there will ever be a single mammalian species in the universe, much less that it be lemurs. Lemurs will always be a proper subset of mammals. There will always be more mammals than lemurs, until there are none of each. I guess 0 is countable, but not infinite... tony.
From: Tony Orlow on 17 Apr 2007 13:12
Lester Zick wrote: > On Fri, 13 Apr 2007 16:11:22 -0400, Tony Orlow <tony(a)lightlink.com> > wrote: > >>>>>>>>> Constant linear velocity in combination with transverse acceleration. >>>>>>>>> >>>>>>>>> ~v~~ >>>>>>>> Constant transverse acceleration? > >>>>>>> What did I say, Tony? Constant linear velocity in combination with >>>>>>> transverse acceleration? Or constant transverse acceleration? I mean >>>>>>> my reply is right there above yours. >>>>>>> >>>>>>> ~v~~ >>>>>> If the transverse acceleration varies, then you do not have a circle. >>>>> Of course not. You do however have a curve. >>>>> >>>>> ~v~~ >>>> I thought you considered the transverse acceleration to vary >>>> infinitesimally, but that was a while back... >>> Still do, Tony. How does that affect whether you have a curve or not? >>> Transverse a produces finite transverse v which produces infinitesimal >>> dr which "curves" the constant linear v infinitesimally. >>> >>> ~v~~ >> Varying is the opposite of being constant. Checkiddout! > > I don't doubt "varying" is not "constant". So what? The result of > "constant" velocity and "varying" transverse acceleration is still a > curve. > > ~v~~ I asked about CONSTANT transverse acceleration. Oy! More exactly, linearly proportional velocity and transverse acceleration produce the circle. It can speed up and slow down, as long as it changes direction at a rate in proportion with its change in velocity. Close your eyes, and watch.... 01oo |