The Definition of Points
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From: MoeBlee on 17 Apr 2007 15:29 On Apr 17, 11:22 am, Tony Orlow <t...(a)lightlink.com> wrote: > MoeBlee wrote: > > On Apr 14, 7:08 pm, Tony Orlow <t...(a)lightlink.com> wrote: > > >> Maybe I don't understand Robinson as well as I should, but it seems to > >> me the basis of his analysis was semantic, regarding statements that > >> would be considered true of *N if true of N. But, do go on... > > > "But do go on..." Hey, smartass, YOU'RE the one who waves non-standard > > analysis like banner when you don't know ANYTIHNG about it. > > > If you want to know anything about non-standard analysis, then you'd > > start by learning basic predicate calculus, set theory, and > > mathematical logic (personally, I'd recommend that order). > > >>> The MAJOR point - the hypothetical nature of mathematical reasoning > >>> (think about the word 'if' twice in the poster's paragraph) and the > >>> inessentiality of what words we use to name mathematical objects and > >>> their properties. > >> I guess, by "inessentiality", you mean any attribute that one could > >> assign to any object... IF that's what you mean, or not.... > > > No, smartass, months ago, several times, I explained in detail to you, > > and with respect to the technicalities, what I mean. > > >>> I've been trying to get you to understand that for about two years > >>> now. > >> Perhaps what you mean is exactly what I am trying to get across to > >> Lester. If the truth table is the same, then it's the same logical > >> function. It doesn't matter what the parameters are, it always works the > >> same way. The pattern defines the relationship. > > > That's getting closer. I won't quibble with it. But I'll just say that > > in mathematical logic we have even more precise ways of saying it. > > >> On the subject of ifs, "if this then that" means logical implication, or > >> causality, and sometimes it's hard to tell which is meant, or if they're > >> being confused. Is that what you "mean"? > > > No. > > >>>> I don't take transfinite cardinality to mean > >>>> "size". You say I missed the point. You didn't intersect the line. > >>> You just did it AGAIN. We and the poster to whom you responded KNOW > >>> that you don't take cardinality as capturing your notion of size. The > >>> point is then just for your to recognize that IF by 'size' we mean > >>> cardinality, then certain sentences follow and certain sentences don't > >>> follow and that what is important is not whether we use 'size' or > >>> 'cardinality' or whatever word but rather the mathematical relations > >>> that are studied even if we were to use the words 'schmize' or > >>> 'shmardinal' or whatever. > > > THAT, to which you did not respond, is, at least at this informal > > level of discussion with you, good enough for a start as to what I > > mean. > > > MoeBlee > > Why would I respond to that kind of abuse? "Abuse". Come on. > If I am a smartass, well, > I've got more smarts than you have in your little finger. (Did that come > out right?) ;) I've never questioned your intelligence. I have no idea as to our comparative intelligence, and not much interest in it. I guess that most people who have an interest in mathematics are more intelligent, or at least more adept at mathematical problem solving, than I am. I surmise that your problem is not intelligence but something else altogether. I don't necessarily subscribe to such rubrics as 'narcissistic personality disorder', but I'd bet that you'd be get a high evaluation for it. It strikes me that whatever it is, there is something in your inflated sense of your self that blocks you from appreciation that to understand the subject of mathematics and logic requires learning it the way everyone else learns it - textbook by textbook, chapter by chapter, defintion by defintion, theorem by theorem, proof by proof. MoeBlee
From: Mike Kelly on 17 Apr 2007 16:06 On 14 Apr, 19:05, Tony Orlow <t...(a)lightlink.com> wrote: > 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. ZF and ZF+C are equivalent in consistency. I don't see what the problem is with using Choice when appropriate. As long as we are clear about what theorems can be proved without Choice, which can only be proved assuming a Choice function on whatever sets then what is the issue? > 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... Never read it; I'm sure I missed a lot. > 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? The whole premise. Set theory doesn't say anything about "consider proper subsets to not necessarily be smaller". It says they are sometimes bijectible. You refuse to accept this. I don't know why. > >>>>>>>> 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? Huh? I addressed this when I said "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." <snip> > > 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, Why? Anyway if you find it "nice", go ahead and make some clear definitions of what you are talking about and show why they are interesting or useful. Nobody is stopping you, other than yourself and your refusal to learn to reason and communicate. >entertained without so much antagonism, but hey! Antagonism for the way you present your ideas, and your thoroughly misguided critiques of current stuff. Not antagonism for the possiblity of new ideas. It's quite obvious that mathematicians are open to new ideas. Or where did everything people are working on now come from? >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'. Some are mathematicians. Some are trolls. Some are cranks. Lester is a troll. You are a crank. Lester chooses to be a troll. He gets pleasure from annoying people. He is not interested in actually discussing mathematics. You choose to be a crank, by your willful avoidance of learning. It's possible you might change and learn some mathematics some day. But why? Clearly for some ignorance is bliss. Being a mathematician isn't about being a sheep or following. I don't expect you to believe this but oh well. > So, what's your opinion of infinite-case induction, You've never clearly explained what this is supposed to be. Maybe you think you have, in your head. But nobody else understands. Maybe you think we're lying, and we do get it really, but we're just too scared of breaking from mathematical orthodoxy to acknowledge your brilliant ideas. But then you're sounding like a crazy JSH type. > IFR and N=S^L, Banal. > multilevel logics Don't know what this means. > >, 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. But you don't have a problem with the ACTUAL conclusions of set theory such as "the evens and the naturals are bijectible". You have problems with what you hallucinate the conclusions of set theory to be. You argue endlessly about cardinality not matching your intuition for "size". But set theory doesn't claim cardinality is "size". > > 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. Yeah you are a mathematical visionary, right? I'm probably not good enough to lick your boots. > > 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. I'm not worthy to tell anything to you. [Side note : you haven't come up with a reason why your ideas are interesting. Or acknowledged that cardinality doesn't preclude other "measures".] > >>>>> 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? Breathtaking strawman. Are you serious here? I've never claimed cardinality to be the "correct version of infinite size" (whatever that would mean). I've spent thousands of words trying to explain to you that there is no "correct version of infinite size". Hello? Are you even reading this? > >>> 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. Fairly easy to define an "asymptotic density in the naturals", I think. Not very interesting though. > >>> 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". But set theory doesn't say "a set plus additional elements is not "a greater set". It says something about bijectibility. You are arguing against something that is just in your head. > So, blah, blah, blah to you. Yes, that's all you read of what people say. Then you respond to something else. I think it's quite rude, actually. > >> 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? :) No, I don't really care what you consider a reasonable definition of "size". Your "considering" does not change the theorems of utility of set theory. > 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. Another cogent argument from you. > 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. Hmm, what does this have to do with cardinality? Or matter even. Banach-Tarski says that IF we can divide up matter infinitely finely, even into non-measurable bits and have a choice function on arbitrary sets then we can double the ball. I find that quite interesting theorem actually. It's NOTHING to do with physical balls though. Apparently all you can do is complain that it's not with your intuition about real balls. Well, why should it be? > that, even though only half the integers are even, there are as many > even integers as integers. Laugh. Again, you are hallucinating. Cardinality says the evens and the naturals are bijectible. What you are disagreeing with is in your own head. > 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... Well if you LEARNED some logic maybe you could understand what a "theorem" is and see that it is not exactly the same thing as the informal natural language people use to describe it. > >>>>> 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? Oh, dodge the question :) Do you have any ACTUAL objections to cardinality, rather than your hallucinations of what you think it says? In answer : I don't see the point in Bigulosity. Your informal descriptions sound uninteresting (actually, quite incoherent). > >>>>>> 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! >From the man who said "The problem with cardinality is that it sucks". Righhhht. > >> 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, as I explain below : > 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. No response from you though :-{ Maybe you can't read what doesn't match your personal view? > >> 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... :) Zzz. > >> 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. Well, according to you there are more naturals than evens. Even though there is a bijection! Seems weird to me. That's just intuition though, taint nothing of consequence. > I know you'ld never > want to learn anything from some old loner thinker like me, but you > might as well taste the soup. But you can't even coherently explain to anyone what an "uncountable sequence" might be. Problem is on your end. > >>>> 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. > > > 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? I read a few of their posts. It's very obvious to me. I'm not really interested in articulating why cause Scrubs is on now and I just got to the end here :{ -- mike.
From: stephen on 17 Apr 2007 16:23 In sci.math Alan Smaill <smaill(a)spaminf.ed.ac.uk> wrote: > Tony Orlow <tony(a)lightlink.com> writes: >> Alan Smaill wrote: >>> Lester Zick <dontbother(a)nowhere.net> writes: >>> >>>> On Sun, 15 Apr 2007 21:34:09 +0100, Alan Smaill >>>> <smaill(a)SPAMinf.ed.ac.uk> wrote: >>>> >>>>>>>>>>>>> Dear me ... L'Hospital's rule is invalid. >>>>>> So returning to the original point, would you care to explain your >>>>>> claim that L'Hospital's rule is invalid? >>>>> haha! >>>> Why am I not surprized? Remarkable how many mathematiker opinions on >>>> the subject of mathematics don't quite hold up to critical scrutiny. >>> knew you wouldn't get it, >>> irony is not a strong point with Ziko. >>> nor indeed do you bother defending your own view that you can use >>> Hospital to work out the value for 0/0. >>> well, there you go. >>> >>>> ~v~~ >>> >> >> In all fairness to Lester, I am the one who said 0 for 0 is >> 100%. T'was a joke, > of course! >> ala L'Hospital's theft from the Bernoullis, and >> the division by 0 proscription. >> >> :) > and Zick was the one who claimed that he would use l'Hospital to work > out the right answer for 0/0. > such a japester, eh? I wonder what Lester thinks 0/0 is? All he needs to do is to demonstrate how one uses l'Hospital to work out the value for 0/0. Lester is always going on about demonstrations, maybe just once he will actually demonstrate something. Stephen
From: Mike Kelly on 17 Apr 2007 17:12 On 17 Apr, 18:09, Tony Orlow <t...(a)lightlink.com> wrote: > 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? You're supposed to be tellling us what "cardinality pretends it can tell". "AC" does not seem like much of an answer to this. > Extensionality. But, cardinality is a Galilean, read "primitive", > extension of Extensionality. I don't know what you are trying to say here. > >>>>> 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'"! > > You may laugh as you like, but numbers represent measure, and measure is > built on "size" or "count". This seems unconnected to the previous posts. I'm not sure what you're trying to say. Are you, perhaps, complaining about how aleph_0 is called a cardinal number, even though you don't think it "represents measure"? If so, you are still missing the point quite spectacularly. > 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... You've lost me again. A bad analogy is like a diagonal frog. -- mike.
From: stephen on 17 Apr 2007 17:19
In sci.math Tony Orlow <tony(a)lightlink.com> wrote: > Mike Kelly wrote: >> >> 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". What "measure", "size" or "count" does the imaginary number i represent? Is i a number? The word "number" is used to describe things that do not represent any sort of "size". Stephen |