The Definition of Points
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From: Tony Orlow on 31 Mar 2007 23:16 Virgil wrote: > In article <460f19f5(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <460ef650(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>> Bob Kolker wrote: >>>>> Tony Orlow wrote:>> >>>>>> Measure makes physics possible. >>>>> On compact sets which must have infinite cardinality. >>>>> >>>>> The measure of a dense countable set is zero. >>>>> >>>>> Bob Kolker >>>> Yes, some finite multiple of an infinitesimal. >>> In any consistent system in which there are infinitesimals, none of >>> those infinitesimals are zero. >> On the finite scale, any countable number of infinitesimals has zero >> measure. > > Again with the undefined terms. What does it mean to have zero measure > in a field having infinitesimals? On the finite scale, it takes an infinite number of infinitesimals to achieve measure. Each infinitesimal unit has measure 1 on the infinitesimal scale. On the infinite scale, it takes an infinite number of finites to achieve measure.
From: Tony Orlow on 31 Mar 2007 23:17 Virgil wrote: > In article <460f1a41(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <460ef372$1(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>> Virgil wrote: >>>>> In article <460e82b1(a)news2.lightlink.com>, >>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>> >>>>> >>>>>> As I said to Brian, it's provably the size of the set of finite natural >>>>>> numbers greater than or equal to 1. No, there is no last finite natural, >>>>>> and no, there is no "size" for N. Aleph_0 is a phantom. >>>>> All numbers are equally phantasmal in the physical world and equally >>>>> real in the mental world. >>>> Virgule, you don't really believe that, do you? You're way too smart for >>>> that... :) >>> While I have seen numerals in the physical world, I have never seen any >>> of the numbers of which they are only representatives. >>> >>> And I suspect that any who claim to have done so have chemically >>> augmented their vision. >> Is that wrong? haha. Anyway... >> >> You have seen two apples, and three? >> > Are apples numbers? Naturals apply to objects.
From: cbrown on 31 Mar 2007 23:19 On Mar 31, 5:45 pm, Tony Orlow <t...(a)lightlink.com> wrote: > Mike Kelly wrote: > > When we say that a set has cardinality Aleph_0 we are saying it is > > bijectible with N. Are you saying it's impossible for a set to be > > bijectible with N? Or are you saying N does not exist as a set? > > Something else? > > I have been saying that bijection alone is not sufficient for measuring > infinite sets relative to each other. > Since it is certainly sufficient for comparing sets by their cardinality, I can only ask: what do you mean by "measuring infinite sets relative to each other"? There are many, many ways of "measuring" one set against another; which do you have in mind? > 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
From: Tony Orlow on 31 Mar 2007 23:19 Virgil wrote: > In article <460f1b3e(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <460ef839(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>> Virgil wrote: >>>>> In article <460ee056(a)news2.lightlink.com>, >>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>>> Please do expliculate what the contradiction is in an uncountable >>>>>> sequence. What is true and false as a result of that concept? >>>>> A mathematical sequence is a function with the naturals as domain. >>>>> If TO wishes to refer to something which is not such a function, he >>>>> should not refer to it as a sequence if he wishes to be understood in >>>>> sci.math. >>>>> >>>>> >>>> Pray tell, what term shall I use???? >>> TO is so inventive in so many useless ways that I cannot believe that >>> his imagination will fail him in such a trivially useful way. >>>>>>> I know you are incapable of actually thinking about all the elements of >>>>>>> N, >>>>>>> but that is your problem. In any case, N is not an element of N. >>>>>>> Citing Ross as support is practically an admission that you are wrong. >>>>>>> >>>>>>> Stephen >>>>>>> >>>>>> Sure, of course, agreeing with someone who disagrees with you makes me >>>>>> wrong. I'll keep that in mind. Thanks.. >>>>> >>>>> It is not so much that Ross disagrees with one person, it is that he >>>>> disagrees with everyone, frequently including himself. >>>> Ross has a vision, even if not axiomatically expressed. In fact, he's >>>> entirely honest about that, expounding an axiom free system. I like >>>> Ross. So do you. Admit it. :) >>>> >>> >>> Like Russell? >>> >>> What is there about him to like? >> You don't like Russell? > > I don't know him well enough to like or dislike. I dislike his > anti-mathematical idiocies. Define "mathematics" before you accuse anyone of being "anti-mathematical". I doubt I agree with everything Russell said, but, whatever. Never mind. Be as crotchety as you like. :) Tony
From: cbrown on 31 Mar 2007 23:38
On Mar 31, 7:27 pm, Tony Orlow <t...(a)lightlink.com> wrote: > step...(a)nomail.com wrote: > > In sci.math Tony Orlow <t...(a)lightlink.com> wrote: > >> step...(a)nomail.com wrote: > >>> In sci.math Tony Orlow <t...(a)lightlink.com> wrote: > >>>> step...(a)nomail.com wrote: > >>>>> In sci.math Tony Orlow <t...(a)lightlink.com> wrote: > >>>>>> step...(a)nomail.com wrote: > >>>>> So in other words > >>>>>>>>>>> An actually infinite sequence is one where there exist two elements, one > >>>>>>>>>>> of which is an infinite number of elements beyond the other. > >>>>> is not your "correct" definition of an "actually infinite sequence", > >>>>> which was my point. You are so sloppy in your word usage that you > >>>>> constantly contradict yourself. > > >>>>> If all you mean by "actually infinite" is "uncountable", then > >>>>> just say "uncountable". Of course an "uncountable sequence" > >>>>> is a contradiction, so you still have to define what you mean > >>>>> by a "sequence". > > >>>> Please do expliculate what the contradiction is in an uncountable > >>>> sequence. What is true and false as a result of that concept? > >>> A infinite sequence containing elements from some set S is a function > >>> f: N->S. There are only countably infinite many elements in N, > >>> so there can be only countably infinite many elements in a sequence. > >>> If you want to have an uncountable sequence, you need to provide > >>> a definition of sequence that allows for such a thing, and until > >>> you do, your use of the word "sequence" is meaningless, as nobody > >>> will know what you are talking about. > > >> Oh. What word shall I use? Supersequence? Is that related to a > >> subsequence or consequence? > > > As long as you define your terms it does not matter to much what you > > call it. You could just call it an uncountably infinite sequence, but you > > need to define what that is if you want anyone to know what you are > > talking about. Why are you so reluctant to define your terms? > > I did that, and was told no such thing exists. Gee, then, don't talk > about unicorns or alephs. > No, you were told that to define "an uncountable sequence" as "a sequence which is uncountable" makes about as much sense as defining "a quadriangle" as a "a triangle that has four sides". That is /not/ the same as being told "there is no such thing as a polygon with four sides". If you'd pull your head out of, err, the sand, it's quite possible that your ideas can be formalized; but no one is going to accept that there exists a triangle with four sides. Not for "political" or "religious" reasons; but because it simply makes no sense - it's either false or gibberish. > > > >>>>>>>> If all other elements in the sequence are a finite number > >>>>>>>> of steps from the start, and w occurs directly after those, then it is > >>>>>>>> one step beyond some step which is finite, and so is at a finite step. > >>>>>>> So you think there are only a finite number of elements between 1 and > >>>>>>> w? What is that finite number? 100? 100000? 100000000000000000? > >>>>>>> 98042934810235712394872394712349123749123471923479? Which one? > > >>>>>> Aleph_0, which is provably a member of the set, if it's the size of the > >>>>>> set. Of course, then, adding w to the set's a little redundant, eh? > >>>>> Aleph_0 is not a finite number. Care to try again? > > >>>> It's also not the size of the set. Wake up. > >>> It is the cardinality of a set. > > >> Is that a number? > > > What is your definition of "number"? aleph_0 is called a transfinite > > number, but definitions, not names, are the important thing. > > A number is a symbolic representation of quantity which can be > manipulated to produce quantitative results in the form of symbols. Great! All that's left then is for you to define "quantity", "manipulated", and "quantitative results" without using the words number, quantity, manipulated, and quantitative results. > I > might be wrong, but I'm sure you can apprise me of the official meaning > of "number", mathematically. ;) > There really isn't one. Honest. Sure, there's a definition of natural number, rational number, algebraic number, adic number, complex number, Stirling number, number field, and so on. But by itself, the word "number" is too vague to have a useful mathematical definition; just like the word "size". Cheers - Chas |