The Definition of Points
in [Math]
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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
From: Tony Orlow on 31 Mar 2007 23:38 cbrown(a)cbrownsystems.com wrote: > On Mar 31, 5:33 pm, Tony Orlow <t...(a)lightlink.com> wrote: >> Mike Kelly wrote: >>> On 31 Mar, 16:46, Tony Orlow <t...(a)lightlink.com> wrote: >>>> Mike Kelly wrote: >>>>> On 31 Mar, 13:41, Tony Orlow <t...(a)lightlink.com> wrote: >>>>>> Mike Kelly wrote: >>>>>>> On 30 Mar, 18:25, Tony Orlow <t...(a)lightlink.com> wrote: >>>>>>>> Lester Zick wrote: >>>>>>>>> On Thu, 29 Mar 2007 09:37:21 -0500, Tony Orlow <t...(a)lightlink.com> >>>>>>>>> wrote: >>>>>>>>>>>> If n is >>>>>>>>>>>> infinite, so is 2^n. If you actually perform an infinite number of >>>>>>>>>>>> subdivisions, then you get actually infinitesimal subintervals. >>>>>>>>>>> And if the process is infinitesimal subdivision every interval you get >>>>>>>>>>> is infinitesimal per se because it's the result of a process of >>>>>>>>>>> infinitesimal subdivision and not because its magnitude is >>>>>>>>>>> infinitesimal as distinct from the process itself. >>>>>>>>>> It's because it's the result of an actually infinite sequence of finite >>>>>>>>>> subdivisions. >>>>>>>>> And what pray tell is an "actually infinite sequence"? >>>>>>>>>> One can also perform some infinite subdivision in some >>>>>>>>>> finite step or so, but that's a little too hocus-pocus to prove. In the >>>>>>>>>> meantime, we have at least potentially infinite sequences of >>>>>>>>>> subdivisions, increments, hyperdimensionalities, or whatever... >>>>>>>>> Sounds like you're guessing again, Tony. >>>>>>>>> ~v~~ >>>>>>>> An actually infinite sequence is one where there exist two elements, one >>>>>>>> of which is an infinite number of elements beyond the other. >>>>>>>> 01oo >>>>>>> Under what definition of sequence? >>>>>>> -- >>>>>>> mike. >>>>>> A set where each element has a well defined unique successor within the >>>>>> set. >>>>> So any set is a sequence? For any set, take the successor of each >>>>> element as itself. >>>> There is no successor in a pure set. That only occurs in a discrete >>>> linear order. >>> What does it mean for an ordering to be "discrete" or "linear"? What >>> does it mean for something to "occur in" an ordering? >> Linear means x<y ^ y<z ->x<z > > Funny; everyone else calls that "a transitive relation". Yes, is that unrelated? > > Let S = {a,b,c,f,g, h} > > Impose the following ordering: > b < a > c < b > c < a > f < a > g < f > g < a > if x <> h, then h < x. > > The equivalent Hasse diagram: > > a > / \ > b f > / \ > c g > \ / > \ / > \ / > h > > This ordering satisfies, for all x,y,z in S: if x<y and y<z, then x<z. > > This is not what most people mean when they say "a linear ordering". > Instead, it's an example of what people usually call a partial order. Okay, but that's an ordering that is based on some finite set of rules regaring some finite set of points, which doesn't suffice to specify the relationships between every pair of points. We can't say, from the specified relationships, whether f<b or f<c or b<g or c<g. That's why there's a parallel route, and so the diagram is "nonlinear". It could be all on a line, but there would be several possible ordering given the stated relationships. > See: > > http://en.wikipedia.org/wiki/Partial_order > >> Continuous means x<z -> Ey: x<y ^ y<z > > Is "<" a partial order? a pre-order? a total order? Unless you > specify, I might say that in a triangle, the third vertex is "between" > any two given distinct vertices. > Uh oh. >> Discrete means not continuous, that is, given x and z, y might not exist. > > So [0,1) u (1,2], with the usual ordering of the reals, is a > "discrete" ordering? > I appears to be the union of two discrete sets, mutually exclusive, and without mutual continuity. I'd say if you break the real line into (x,x+1] for xeZ, those are discrete partitions of R. What I said was that a discrete order will have pairs of elements which have no elements between them, whereas a continuous order will not. But, I'm sure I'm wrong. :) >> For something to "occur", it must happen "at some time". > > Does "1 + 5" "occur", i.e., happen, "at some time" different than when > "2 + 4" "occurs"? > "1 + 5" occurred earlier in that sentence than "2 + 4" did. A sentence is a kind of sequence. Thanks for the example. >> In a sequence, >> this is defined as after some set of events and before some other >> mutually exclusive set, in whatever order is under consideration. >> > > Is "1+1" an "event" which "occurs" or "happens" at some "time"? When > is that time? Has it already "occured"? > I think I just saw it. Look! Up there.... >>> So when you say "sequence" you're using an undefined term. As such, >>> it's rather hard to your evaluate claims such as "There are actually >>> infinite sequences". I have literally no idea what you are even trying >>> to say. >>> -- >>> mike. >> Oh gee, there has to be some word for it... >> > > There almost certainly is; but as usual, it depends on what the /heck/ > you're talking about. Perhaps the words "well-order" or "total order" > actually already satisfy your requirements; or some particular proper > subset of all non-isomorphic total orders satisfy your requirements. > Or not. But how will you ever know if you refuse to /learn/ what these > words refer to? > > Cheers - Chas > I guess not by asking around here. Geeze. Tally Ho! Tony
From: Tony Orlow on 31 Mar 2007 23:44 cbrown(a)cbrownsystems.com wrote: > 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? > Let's examine what '<' means. x<y ^ y<z -> x<z. True for real quantities, and true if '<' is taken to mean "is a proper subset of". The proper subset is less than the whole, and the evens are half the naturals. That's a very primitive result. >> 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, given the fact that its size must be equal to the largest element, which doesn't exist. The whole concept is a phantom. Toodles - Tony
From: cbrown on 31 Mar 2007 23:48
On Mar 31, 8:11 pm, Tony Orlow <t...(a)lightlink.com> wrote: > cbr...(a)cbrownsystems.com wrote: > > On Mar 31, 3:30 pm, Tony Orlow <t...(a)lightlink.com> wrote: > >> cbr...(a)cbrownsystems.com wrote: > >>> On Mar 31, 5:30 am, Tony Orlow <t...(a)lightlink.com> wrote: > >>>> Virgil wrote: > >>>>> In standard mathematics, an infinite sequence is o more than a function > >>>>> whose domain is the set of naturals, no two of which are more that > >>>>> finitely different. The codmain of such a function need not have any > >>>>> particular structure at all. > >>>> That's a countably infinite sequence. Standard mathematics doesn't allow > >>>> for uncountable sequences like the adics or T-riffics, because it's been > >>>> politically agreed upon that we skirt that issue and leave it to the > >>>> clerics. > >>> That's false; > >> Please elucidate on the untruth of the statement. It should be easy to > >> disprove an untrue statement. > > > I did in the continuation of that sentence; but I'll repeat myself. > > > You claimed that mathematics doesn't "allow for uncountable > > sequences" (for which you agree you've given no real useful > > definition). But on the contrary, "people" (aka, mathematicicians) > > have studied all sorts of ordered sets, finite, countable, and > > uncountable; and functions from them (whether they use the term > > "uncountable sequence" or not). > > > So your claim that ordered sets which are not countable have not been > > studied is false; and therefore your comments that the reasons /why/ > > they have not been studied (political or religious) are non-sequiturs. > > > The obvious question is why haven't /you/ studied them; instead of > > making vague and uninformed statements about them (regardless of what > > you choose to call these ordered sets). > > The question is, "is there an acceptable term with which to refer to > such uncountable linearly ordered sets?" > Sure. "S is an uncountable linearly ordered set" has a perfectly common generally meaning: S is a set with uncountable cardinality, upon which we in addition have a total (aka, linear) order. For any given such uncountable set S, there are uncountably many non- isomoprhic total orderings; so in general we naturally like to know more about exactly what type of linear order is being referred to. > >> There have always been religious and political pressures on this area of > >> exploration. > > > How would you know what has "always" been the case in this area? Is > > there an example please of a /religious/ or /political/ pressure that > > you can cite? Besides the quite reasonable recommendation to educate > > yourself regarding the subject matter, using the freely available > > material relating to the subject, e.g., > > >http://en.wikipedia.org/wiki/Order_theory > > > ? > > Google it up. Why should I? /You're/ making the claim, not me; /you/ google it up, and post your links. Cheers - Chas |