An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: Tony Orlow on 24 Oct 2006 13:16 David Marcus wrote: > Tony Orlow wrote: >> David Marcus wrote: >>> Tony Orlow wrote: >>>> David Marcus wrote: >>>>> Tony Orlow wrote: >>>>>> Your examples of the circle and rectangle are good. Neither has a height >>>>>> outside of its x range. The height of the circle is 0 at x=-1 and x=1, >>>>>> because the circle actually exists there. To ask about its height at x=9 >>>>>> is like asking how the air quality was on the 85th floor of the World >>>>>> Trade Center yesterday. Similarly, it makes little sense to ask what >>>>>> happens at noon. There is no vase at noon. >>>>> Do you really mean to say that there is no vase at noon or do you mean >>>>> to say that the vase is not empty at noon? >>>> If noon exists at all, the vase is not empty. All finite naturals will >>>> have been removed, but an infinite number of infinitely-numbered balls >>>> will remain. >>> "If noon exists at all"? How do we decide? >>> >> We decide on the basis of whether 1/n=0. Is that possible for n in N? >> Hmmmm......nope. > > So, noon doesn't exist. And, there is no vase at noon. I thought you > were saying the vase contains an infinite number of balls at noon. > If the vase exists at noon, then it has an uncountable number of balls labeled with infinite values. But, no infinite values are allowed i the experiment, so this cannot happen, and noon is excluded.
From: Tony Orlow on 24 Oct 2006 13:18 David Marcus wrote: > Tony Orlow wrote: >> David Marcus wrote: >>> Tony Orlow wrote: >>>> David Marcus wrote: >>>>> Tony Orlow wrote: >>>>>> David Marcus wrote: >>>>>>> Tony Orlow wrote: >>>>>>>> Virgil wrote: >>>>>>>>> In article <4533d315(a)news2.lightlink.com>, >>>>>>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>>>>>>>> Then let us put all the balls in at once before the first is removed and >>>>>>>>>>> then remove them according to the original time schedule. >>>>>>>>>> Great! You changed the problem and got a different conclusion. How >>>>>>>>>> very....like you. >>>>>>>>> Does TO claim that putting balls in earlier but taking them out as in >>>>>>>>> the original will result in fewer balls at the end? >>>>>>>> If the two are separate events, sure. >>>>>>> Not sure what you mean by "separate events". Suppose we put all the >>>>>>> balls in at one minute before noon and take them out according to the >>>>>>> original schedule. How many balls are in the vase at noon? >>>>>> empty. >>>>> Why? >>>> Because of the infinite rate of removal without insertions at noon. >>> OK. Just to recall, this vase has all the balls put in at one minute >>> before noon, then taken out on the usual schedule. How many balls are in >>> this vase at times before noon? >> Some supposedly infinite number, as only a finite number have been removed. > > But, for this vase, at all times before noon, there are an infinite > number of balls in the vase. So, how does this vase become empty at > noon? > Using the time singularity of the Zeno machine, where there is a condensation point in the sequence that allows an infinite number of iterations to occur in a moment. Luckily for the vase, no one is inserting extra balls on the same schedule.
From: Tony Orlow on 24 Oct 2006 13:23 David Marcus wrote: > Tony Orlow wrote: >> David R Tribble wrote: >>> Tony Orlow wrote: >>>> You have agreed with everything so far. At every point before noon balls >>>> remain. You claim nothing changes at noon. Is there something between >>>> noon and "before noon", when those balls disappeared? If not, then they >>>> must still be in there. >>> Of course there is a "something" between "before noon" and "noon" where >>> each ball disappears. At step n, time 2^-n min before noon, ball n is >>> removed. This happens for every ball, since there is a step n for >>> every ball. The balls are removed, one by one, one at each step, >>> before noon. >>> >> As each ball n is removed, how many remain? > > 9n. > >> Can any be removed and leave an empty vase? > > Not sure what you are asking. > If, for all n e N, n>0, the number of balls remaining after n's removal is 9n, does there exist any n e N which, after its removal, leaves 0? If not, then no matter how many n e N you remove from the vase, even if you remove all of them, every removal leaves balls in the vase. Paradoxical? Sure. But it's easily explainable and resolvable once a proper measure is applied to the situation. Omega doesn't lend itself to proper measure. Infinite series do. Bijection loses measure for infinite sets. N=S^L and IFR preserve measure. So, how do you empty the vase? Ball removal? Every removal leaves balls in the vase, as is obvious.
From: Tony Orlow on 24 Oct 2006 13:30 Lester Zick wrote: > On Mon, 23 Oct 2006 15:00:58 -0400, Tony Orlow <tony(a)lightlink.com> > wrote: > >> Lester Zick wrote: > > [. . .] > >>>> Unfortunately, transfinitology exists, despite the fact that it makes no >>>> sense underneath the hood. When it comes to arithmetic on them, it's one >>>> big kludge. But, there are forms of infinite numbers upon which one can >>>> define arithmetic. They just have nothing whatsoever to do with omega or >>>> the alephs. >>> Not sure what you're talking about here, Tony. Lots of things exist in >>> the sense of having been defined. That doesn't make them true and >>> doesn't mean they form any basis for the truth of other things defined >>> on them. There's no shortage of things other than infinity on which to >>> define arithmetic. >>> >> What is log2(0)? > > -00? Not sure what this is in aid of, Tony. What is log0(0)? For that > matter what is log0(00) or log-00(00) or x!=0? There are all kinds of > restrictions around 0 and 00 precisely because 0 is not a natural > number. > It was just a little example of where an infinity arises naturally. >>>>> Yet I've also been considering what it looks like you're trying to do >>>>> with trans finite arithmetic.In particular it occurs to me that if one >>>>> takes +00 to be larger than any positive finite -00 correspondingly >>>>> must be smaller than any negative finite such that your concept of >>>>> circularity among arithmetic numbers might be combined in the >>>>> following way: [-00, . . . 3, 2, 1, 0, 1, 2, 3 . . . +00]. The only >>>>> difference would be that whereas +00 represents the number of >>>>> infinitesimals, -00 would represent the size of infinitesimals. Thus >>>>> we'd have a positive axis with the number of infinitesimals and a >>>>> negative axis with the size of infinitesimals. At least that's the >>>>> best I can make of the situation. > >>>> Well, I rather think of 1/oo as the size of infinitesimals, or more >>>> precisely, for any specific infinite n, 1/n is a specific infinitesimal >>>> value. When it comes to the number circle, in some ways oo and -oo can >>>> be considered the same so the number line forms an infinite circle, but >>>> in others, such as lim(n->oo) as opposed to lim(n->-oo), there is a very >>>> clear difference between the two. I think it's a bit like the >>>> wave-particle dualism for physical objects, and may actually be directly >>>> connected. >>> Well now you're just back to the idea of arithmetic as some kind of a >>> TOE, Tony. It's very simple. The only mechanical definition for 00 is >>> any finite/0. And if that product can't be defined than neither can >>> 00. There is no specific size to infinitesimals because they're an >>> process not a static thing. Any series of infinitesimals varies in >>> size continuously. There's a reciprocity between number and size for >>> any infinite series but no "circle" between them. >> Technically, the number of reals in the unit interval (0,1] is Big'un. > > But the point is that they're within the interval. There is no > infinite set 1, 2, 3 . . . 00 outside of some interval. Sure there is, in an infinite interval. Of course, that requires the existence of infinite numeric values, but that well within our capabilities. Ross's EF is a special case of my IFR, where the mapping function is f(x)=1/Big'un. That maps the set of Big'un hypernaturals over the real line to the set of Big'un infinitesimals, natural multiples of Lil'un, within the first unit interval. > >> That's also the infinite length of the real number line, in unit >> intervals. The unit infinitesimal is Lil'un, or 1/Big'un. Now they're >> all specific and related to spatial measure and quantity. :) >> >>> On the other hand if you want to do transfinite arithmetic you might >>> ask yourself what the results of 00-00 or 00/00 are. The latter can be >>> addressed through application of L'Hospital's rule but I don't know >>> any way to address the former. >>> >>> ~v~~ >> The formulas that lend themselves to L'Hospital's Rule usually cannot be >> simplified any further to resolve that problem. Subtracting one simple >> formula from another is just a matter of combining like terms and >> finding the most significant to see if you get a finite result through >> mutual cancellations. > > But L'Hospital's rule applies to ratios, Tony. It only gives the > finite ratio between infinities. If you subtract 1/0 from 2/0 what do > you get? They both already have common denominators so the answer > would seem to be 1/0 which still remains infinite. Kluge is the right > word for transfinite arithmetic. > > ~v~~ You had oo-oo. I assume that's something like, say, the derivative of 1/x - 3/x^2 at x=0. Clearly, the second term dominates, and this tends to -oo. I don't see the difficulty there, but it's probably not important.
From: Lester Zick on 24 Oct 2006 13:46
On 23 Oct 2006 10:51:19 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >Lester Zick wrote: >> >I don't say that. And the definition of 'cardinality of' does use >> >'equinumerosity', but the definition of 'equinumerosity' does not use >> >'cardinality of', so there is not the circluarity you just arbitarily >> >claim there to be. >> >> Arbitrarily? And maybe you'd like to apprise us of a definition for >> equinumerosity which doesn't already assume cardinality for the >> elements which are "equally numerous"? > >This conversation with you is ridiculous. I agree but I doubt we'd agree on the reason why. At least they're short and pretty much to the point. > Of course I can keep defining >until I hit the primitives. You can keep asking me to do that, in a >backwards motion, from the complex to the primitives, for a long time, >and I finally will hit the primitives. No that's not at all what I'm asking of you. The problem is that your regressions seem to be unlimited. In the preceding message you say "By a 'theory' I mean a set of sentences closed under entailment" which just regresses one ambiguous term "theory" to another equally ambiguous term "closed under entailment" which is meaningless except presumably within the context of standard set analytical techniques. (Here I'm making the general assumption that you do know what you describe of standard set analysis in detail.) And if I ask for a definition of "equinumerosity" which doesn't rely on cardinality to begin with you just go on to another potpourri of standard set analytical terms and concepts which don't acturally say "cardinality". So all you're really doing is regressing your appraisal of standard set analysis and techniques to a group of terms which only appear to have meaning within standard set analysis. In other words you're arguing terminology from the perspective of standard set analysis and techniques presumably under the assumption that they're true whereas what I'm asking for is a reduction of mathematical terms and techniques to primitives which are actually true and exhaustive instead of just being assumed true for the purposes of analysis. > But it would be much more >efficient for me just to state the primitives and then demonstrate and >define in a forward direction. Which doesn't even address the problem of whether the primitives you argue from are actually true. They're just assumptions. Of course it would be more efficient; it just wouldn't tell us how or explain why they represented the correct perspective on mathematical issues such as set analysis. I don't say the terms and concepts involved are false just that they're not exhaustive and I'm trying to argue others are. >But I'l indulge you just one more time: > >x is equinumerous with y <-> Ef(f is a bijection from x onto y). > >There is no assumption of having defined 'cardinality' prior to the >above definition. So if you don't actually say the word "cardinality" it isn't there? So if you say the word "equinumerous" instead we can all go home and rest easy that there is no "primitive" implication than things which are equinumerous don't bear a cardinal relation to one another to begin with? >Now you can say, "define 'bijection'". And then I do that. Then you >say, "define 'function'". Then I do that. Finally, we reach the >primitives. Lord knows I have no intention of doing that and I could care less what your primitives are and certainly have no intention of suffering that kind of nonsense. The point is whether those primitives are true or not and whether they're exhaustive. And so far I see no evidence whatsoever that by saying "cardinality" instead of "number of" or "bijection" instead of "matching" you've improved the quality of mathematical life, reasoning, and set analysis one iotum. All you've done in practical terms is substitute one incomprehensible phrase for another in what I describe as a terminological regression masquerading as a mathematical reduction. > > I've already been over the subject of >mathematical >> >definitions with you in other threads. >> >> I've been over the issue with Randy and a few others but I can't >> specifically recall visiting the issue with you. > >You don't recall. I do. In particular, you brought the conversation to >a nadir with your mere harrumping about the criteria of non-creativity >and eliminability that I mentioned. More buzzwords. I do remember the "non creativity" and "eliminability" which I frankly thought were hilarious given the context. But I really don't remember specifically discussing the nature of definition with you in general. Maybe the conversation never got that far. Oh well. ~v~~ |