The Definition of Points
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From: Mike Kelly on 14 Apr 2007 07:22 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? > >>> 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.
From: Alan Smaill on 14 Apr 2007 08:56 Lester Zick <dontbother(a)nowhere.net> writes: > On Fri, 13 Apr 2007 16:10:39 +0100, Alan Smaill > <smaill(a)SPAMinf.ed.ac.uk> wrote: > >>Lester Zick <dontbother(a)nowhere.net> writes: >> >>> On Thu, 12 Apr 2007 14:23:04 -0400, Tony Orlow <tony(a)lightlink.com> >>> wrote: >>>> >>>>That's okay. 0 for 0 is 100%!!! :) >>> >>> Not exactly, Tony. 0/0 would have to be evaluated under L'Hospital's >>> rule. >> >>Dear me ... L'Hospital's rule is invalid. > > What ho? Surely you jest! Who, me? > Was it invalid when I used it in college? If you used it to work out a value for 0/0, then yes. > > ~v~~ -- Alan Smaill
From: Tony Orlow on 14 Apr 2007 14:05 Mike Kelly wrote: > On 13 Apr, 19:25, Tony Orlow <t...(a)lightlink.com> wrote: >> Mike Kelly wrote: >>> On 12 Apr, 19:54, Tony Orlow <t...(a)lightlink.com> wrote: >>>> Mike Kelly wrote: >>>>> On 2 Apr, 15:49, Tony Orlow <t...(a)lightlink.com> wrote: >>>>>> Mike Kelly wrote: >>>>>>> On 1 Apr, 04:44, Tony Orlow <t...(a)lightlink.com> wrote: >>>>>>>> cbr...(a)cbrownsystems.com wrote: >>>>>>>>> On Mar 31, 5:45 pm, Tony Orlow <t...(a)lightlink.com> wrote:> >>>>>>>>>> Yes, NeN, as Ross says. I understand what he means, but you don't. >>>>>>>>> What I don't understand is what name you would like to give to the set >>>>>>>>> {n : n e N and n <> N}. M? >>>>>>>>> Cheers - Chas >>>>>>>> N-1? Why do I need to define that uselessness? I don't want to give a >>>>>>>> size to the set of finite naturals because defining the size of that set >>>>>>>> is inherently self-contradictory, >>>>>>> So.. you accept that the set of naturals exists? But you don't accept >>>>>>> that it can have a "size". Is it acceptable for it to have a >>>>>>> "bijectibility class"? Or is that taboo in your mind, too? If nobody >>>>>>> ever refered to cardinality as "size" but always said "bijectibility >>>>>>> class" (or just "cardinality"..) would all your objections disappear? >>>>>> Yes, but my desire for a good way of measuring infinite sets wouldn't go >>>>>> away. >> <snip> >> >>> It'd be nice if some day you learned some math above high school >>> level. Seems a rather remote possibility though because you are >>> WILLFULLY ignorant. >> And you are willfully obnoxious, but I won't take it seriously. > > Look who's talking.. > "Takes one to know one" (sigh) >> I'm not the only one in the revolution against blind axiomatics. > > Blind axiomatics? So you think ZFC was developed by blindly? People > picked the axioms randomly without any real consideration for what the > consequences would be? Please. ZF(C) provides a foundation for > virtually all modern mathematics. This didn't happen by accident. > > What's "blind" about ZF(C)? What great insight do you think is missed > that you are going to provide, oh mighty revolutionary? What > mathematics can be done with your non-existant foundation that can't > be done in ZF(C)? > Axiomatically, I think the bulk of the burden lies on Choice in its full form. Dependent or Countable Choice seem reasonable, but a blanket statement for all sets seems unjustified. Uncountable sets always lead to infinite regression, whether due to being divided into an uncountable number of partitions, or due to having an uncountable partition, when trying to explicitly define a well order. We explored that in Well Ordering the Reals. Maybe you missed that thread, but that's hard to believe... There's the matter of not considering proper subsets to be necessarily smaller, when the subset relation is always as transitive as quantitative inequality. The proper subset always has the same elements, minus some nonzero number. That's less. The proper superset's always more. Any theory that violates that basic principle is rather suspect. x+y=y+x x+0=x x>0->x+y>y x<0->x+y<y ....basic definitions, or parts thereof. Which is suspect to you? >>>>>>>> given the fact that its size must be equal to the largest element, >>>>>>> That isn't a fact. It's true that the size of a set of naturals of the >>>>>>> form {1,2,3,...,n} is n. But N isn't a set of that form. Is it? >>>>>> It's true that the set of consecutive naturals starting at 1 with size x has largest element x. >>>>> No. This is not true if the set is not finite (if it does not have a >>>>> largest element). >>>> Prove it, formally, please, from your axioms. >>> I don't have a formal definition of "size". You understand this point, >>> yes? >> Then how do you presume to declare that my statement is "not true"? >> No answer? Do you retract the claim? >>> 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. >> >>> It's rather disingenuous to ask me for a formal proof of something >>> that is couched in your informal terms. >> Don't say "This is not true" if you can't disprove it. >> Please? >>>>> It is true that the set of consecutive naturals starting at 1 with >>>>> largest element x has cardinality x. >>>> Forget cardinality. Can a set of naturals starting with 1 and with size >>>> X possibly have any other maximum value besides X? This is inductively >>>> impossible. >>> (Just to be clear, we're talking very informally here. It's quite >>> obviously the only way to talk to you.) >>> Tony, can you discern a difference between the following two >>> statements? >>> a) A consecutive set of naturals starting with 1 with size X can not >>> have any maximum other than X. >>> b) A consecutive set of naturals starting with 1 with size X has >>> maximum X. >> Yes, the first allows that may be no maximum, but where there is a >> specific size for such a set, there is a specific maximum as well. I am >> not the one having the logical difficulty here. >> >>> Seriously, do you comprehend that they are saying different things? >>> This is important. >> It would be if it had anything substantive to do with my point. Whatever >> the size of a set of consecutive naturals from 1 is, that is its maximal >> element. >> >>> I'm not disputing a) (although you haven't defined "size" and thus >>> it's trivially incorrect). I'm disputing b). I don't think b) follows >>> from a). I don't think that all sets of naturals starting from 1 have >>> a maximum. So I don't think that "the maximum, if it exists, is X" >>> means "the maximum is X". Because for some sets of naturals, the >>> maximum doesn't exist. >> It is inductively provable that for all such sets the maximum is EQUAL >> to the size. Therefore, if one exists, then so does the other, in both >> directions. Do you know what EQUAL means? If a=b, can a exist and b not? >> >>> Do you agree that some sets of consecutive naturals starting with 1 >>> don't have a maximum element (N, for example)? Do you then agree that >>> a) does not imply b)? >> No, b) implies that such sets also do not have a size. Get it? > > OK so all of the above comes down to you demanding that we don't call > cardinality "size". If we don't call cardinality "size" then all your > objections to cardinality disappear. It would also be nice to have an alternative to cardinality, as a definition of infinite size, entertained without so much antagonism, but hey! Whaddya expect from sticklers, mathematikers and logikers? A lot of what Lester does, and remember that he started this thread, is stir the pot. Most are sheep and follow, some are shepherds, leading and chasing, and some really can't follow or lead, but stick around and make a presence anyway, and keep the dogs and rams a'boutin'. So, what's your opinion of infinite-case induction, IFR and N=S^L, and multilevel logics, again? I forget. > > I'm bored of trying to get you to realise that logic doesn't care what > label we give to concepts. We have this definition called > "cardinality" which is to do with which sets are bijectible. Some > people think it seems like a fair notion of "size", but it's > immaterial whether you agree with them or not. Cardinality is still > perfectly well defined. You seem incapable of grasping this point. > Moving on... I grasp the logical deductions. I don't grok the conclusions. Therefore, I question the assumptions. > > At this point you're probably going to say "cardinality works but it's > not sufficient. I want a richer way of measuring infinite sets, so I > can say the evens are half the naturals... blah blah blah". I'm not the first blahmeister on that.... It's bigger than that, anyway, but you'll never grasp that, mike. > > Of course, there is nothing stopping you doing this. Certainly, > cardinality doesn't stop development of other ways of measuring sets > (see measure theory for example [note: I'm not saying measure theory > does what you want. It was an *example* of another way of measuring > sets that isn't precluded by set theory, rather it builds upon it.]). > > Of course, it's not clear WHY you want to develop "Bigulosity". It > tells us NOTHING interesting about sets. It doesn't lead to any new > mathematics. It (pupportedly) matches one persons intuitions better > than cardinality. Woo hoo. > Go live some life and develop some intuition, and come back and tell me about it. I'll listen. >>>>> 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. >>>> Given the definition of the naturals, given any starting point 0, a set >>>> of consecutive naturals of size y has maximum element x+y. >>> x+y? Typo I guess. >> Yes, I was going to say "starting point x", then changed that part and >> not the other (which would have needed a "-1", anyway). >> >>>> Does the set of naturals have size aleph_0? If so, then aleph_0 is the maximal natural. >>> 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. > > Sigh. But some people do. And some people don't. Some people don't > care, because "size" is inherently vague. It's no problem whatsoever > to use different definitions as we like, so long as we are always > clear which definitions we are using. Unless we have squirrels living > in our head and can't distinguish between how we label a thing from > the thing itself. Then we run into problems. > Don't claim it's the "correct" version of infinite "size" and I won't tell you to "shut up". Deal? >>> Under some definitions of "size" your statement is true. Under others >>> (such as cardinality) it isn't. So you can't use your statement about >>> SOME definitions of size to draw conclusions about ALL definitions of >>> size. Not all sets of naturals starting at 1 have a maximum element >>> (right?). Your statement is thus obviously wrong about any definition >>> of "size" that gives a size to non-finite sets. >> It's wrong in any theory that gives a size to any countably infinite >> set > > Yes. Well done. > Tanx >> , except as a formulaic relation with N. > > Que? > That is to say, there is no fixed size to any countably infinite set. But, the size achieved per iteration, or the Big O value, if you will, can be expressed formulaically so that, if you know the size of one set, you can tell the size of the other, at any point in their mutual iterations. We don't get into trouble here, because we don't try to define the boundary between finite and infinite as a definite location, but only relate the size of two structures defined in bijection with N. >>> I find it hard to beleive you don't understand this. Indicate the >>> point(s) where you disagree. >>> a) Not all consecutive sets of naturals starting from 1 have maximum >>> elements. >> agree >>> b) Some notions of "size" give a "size" to sets of naturals without >>> maximum elements. >> disagree, personally. I can't accept transfinite cardinality as a notion >> of "size". > > Blah blah blah. Labels aren't important. > It's not the "labels", but the essential principles that are sacrificed, such as, "a set plus additional elements" is not "a greater set". That's a very basic concept, and not one easily relinquished by most that have thought about it before being "educated". So, blah, blah, blah to you. >> c) Some notions of "size" give a "size" to sets of consecutive >>> naturals starting from 1 without a maximum element. >> same >>> d) The "size" that these notions give cannot be the maximum element, >>> because those sets don't *have* a maximum element. >> agree - that would appear to be the rub >> >> >>> e) Your statement about "size" does not apply to all reasonable >>> definitions of "size". In particular, it does not apply to notions of >>> "size" that give a "size" to sets without a largest element. >> It does not apply to transfinite cardinality. The question is whether I >> consider it a "reasonable" definition of size. I don't. > > Who cares? Apparently, you, and others. So what? :) Set theory doesn't claim "cardinality is a reasonable > definition of size". It uses cardinality to denote which sets are > bijectible. That's all. You don't have to "consider cardinality a > reasonable definition of size" to use it in set theory. > > Is your only objection to cardinality is that some people call it > "size"? > It's mostly that it sucks. It predicts the nonzero possibility of reorganizing a solid finite sphere into two solid ones of the same size. In other words it magically makes space and matter by dictate. It says that, even though only half the integers are even, there are as many even integers as integers. Can we please not totally corrupt logic itself? Not all logic is deductive. The other half is inductive, and I don't mean the misnomered version of deductive proof... >>>>> Do you see that changing the order of words in a statement can change >>>>> the meaning or that statement? Do you see that one statement can be >>>>> true, and another statement with the same words in a different order >>>>> can be false? >>>> This is not quantifier dyslexia, and I am not interested in entertaining >>>> that nonsense, thanx. >>> It is doublethink though. You are simultaneously able to hold the >>> contradictory statements "Not all sets of naturals have a largest >>> element" with "All sets of naturals must have a largest element" to be >>> true, >> No, "All sets of naturals WITH A SIZE must have a largest element", or >> more specifically, "All sets of consecutive naturals starting from 1 >> have size and maximal element equal." Equal things either both exist, or >> both don't. > > OK, so let's call cardinality "bijection class" or something. Now, you > have no objections? > ] Do you have any objection to Bigulosity? >>>>>> Is N of that form? >>>>> N is a set of consecutive naturals starting at 1. It doesn't have a >>>>> largest element. It has cardinality aleph_0. >>>> If aleph_0 is the size, then aleph_0 is the maximal element. aleph_0 e N. >>> Wrong. If aleph_0 is the "size", AND the set HAS a maximal element >>> then aleph_0 is the maximal element. But N DOESN'T have a maximal >>> element so aleph_0 can be the size without being the maximal element. >>> (speaking very informally again as Tony is incapable of recognising >>> the need to define "size"..) >> I defined formulaic Bigulosity long ago. > > Bullshit. Vague mumblings are not definitions. > How eloquent! >> I've also made it clear that Idon't consider transfinitology to be a valid analog for size in the >> infinite case. > > Who cares? Why should anyone care what you "consider" if it's just an > aesthetic preference? > It's not a matter of fashion. >> I've offered IF and N=S^L in the context of infinite-case >> induction, which contradicts your little religion, and may seem >> offensive, > > Religion? Unjustified catechisms? No, not a religion..... No, I am quite comfortable with the idea of adopting > whatever axiom system seems interesting or useful for intellectual > exploration or practical application. I have nothing invested in ZFC > other than a recognition that it is coherent and useful (maybe even > consistent!). You are the one who is chronically hung-up over the fact > that your intuitions sometimes get violated. > >> but is really far less absurd > > Your ideas violate my intuition! Bigulosity seems very absurd to me. > Now what? > Now, you scroll up, and you look at the very first paragraph that you wrote in this last response, and you think about whether intuition played a part in the formulation of ZF(C), and what role intuition plays in general, and what intuition really is to begin with... :) >> and paradox-free. :) > > You think countable sets can have different sizes. That's a paradox to > me. Some never-ending sequences end before other never-ending > sequences? Ahaha, most amusing.. > That's due to your concept of infinite size, ala Galileo, and your concept of consequences as always being countable. I know you'ld never want to learn anything from some old loner thinker like me, but you might as well taste the soup. >>>> Or, as Ross likes to say, NeN. >>> Here's a hint, Tony : Ross and Lester are trolls. They don't beleive a >>> damn word they say. They are jerks getting pleasure from intentionally >>> talking rubbish to solicit negative responses. Responding to them at >>> all is pointless. Responding to them as though their "ideas" are >>> serious and worthy of attention makes you look very, very silly. >> Yes, it's very silly to entertain fools, except when they are telling >> you the Earth is round. One needn't be all like that, Mike. When you >> argue with a fool, chances are he's doing the same. > > Nope. Ross and Lester are trolls. They are laughing at you when you > agree with their fake online personas. Continue wasting your time on > them if you like. > > -- > mike. > I don't believe that's a true reading, mike. What do you actually know about them, and upon what logic, or intuition, do you so confidently base that very personal opinion? tony.
From: Lester Zick on 14 Apr 2007 14:13 On Sat, 14 Apr 2007 13:56:37 +0100, Alan Smaill <smaill(a)SPAMinf.ed.ac.uk> wrote: >Lester Zick <dontbother(a)nowhere.net> writes: > >> On Fri, 13 Apr 2007 16:10:39 +0100, Alan Smaill >> <smaill(a)SPAMinf.ed.ac.uk> wrote: >> >>>Lester Zick <dontbother(a)nowhere.net> writes: >>> >>>> On Thu, 12 Apr 2007 14:23:04 -0400, Tony Orlow <tony(a)lightlink.com> >>>> wrote: >>>>> >>>>>That's okay. 0 for 0 is 100%!!! :) >>>> >>>> Not exactly, Tony. 0/0 would have to be evaluated under L'Hospital's >>>> rule. >>> >>>Dear me ... L'Hospital's rule is invalid. >> >> What ho? Surely you jest! > >Who, me? > >> Was it invalid when I used it in college? > >If you used it to work out a value for 0/0, then yes. Well the problem is that you didn't claim my application of L'Hospital's rule was invalid. You claimed the rule itself was invalid. So perhaps you'd like to show how the rule itself is invalid or why my application of the rule is? ~v~~
From: Lester Zick on 14 Apr 2007 14:16
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~~ |