The Definition of Points
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From: Bob Kolker on 1 Apr 2007 07:27 Tony Orlow wrote: > > 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. Not a fact. Infinite sets have cardinality (a kind of "size") and do not necessarily have largest elements. You really want all sets to be finite, don't you? That will produce mathematics that is barely capable of keeping accounts at a bank. Bob Kolker
From: Tony Orlow on 1 Apr 2007 11:07 cbrown(a)cbrownsystems.com wrote: > 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". I defined it as a sequence where there exist elements infinitely beyond other elements. > > 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. > Straw man argument. >> >>>>>>>>>> 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. > Don't be a boor. "Quantitative results" are quantities indicated by the resulting symbolic expression. "Manipulate" means "produce a new string from an existing one according to rules". A quantity is a point on the real line. A number is a string that indicates a point. Arithmetic is the manipulation of strings. But, you know all that. >> 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 > That was tongue-in-cheek. I know there's no definition of "number",and mathematicians seem quite satisfied with that for themselves, but insist that I produce a definition of a word they use every day without knowing what they even mean themselves.
From: Tony Orlow on 1 Apr 2007 11:29 cbrown(a)cbrownsystems.com wrote: > On Mar 31, 8:38 pm, Tony Orlow <t...(a)lightlink.com> wrote: >> cbr...(a)cbrownsystems.com wrote: >>> On Mar 31, 5:33 pm, Tony Orlow <t...(a)lightlink.com> wrote: >>>> Mike Kelly wrote: > >>>>> 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? >> > > It's certainly /necessary/ for "<" to be a partial order; but it's not > sufficient. Just like it's necessary for my car to have gasoline to > run; but not sufficient. A partial order is transitive; but not every > transitive relation is a partial order. > > See: > > http://en.wikipedia.org/wiki/Partial_order > > It's just three simple rules, man. Sheesh! > So, add a<b v b<a v a=b, and make it a total order. If we're trying to address the real line and its subsets, we can assume total order. >>> 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. > > Right. That's why it's called a /partial/ order, and not a /total/ > order; there are elements which are incomparable - i.e., they cannot > be compared in the ordering. > Is the true for R or N? No. >> 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. >> > > And sometimes, depending on the ordering, there is no particularly > useful way to extend that ordering. > > Consider the subsets of {a, b, c}, ordered by inclusion. I can say > that every subset A <= {a, b, c} in this ordering; and I can say that > {} <= A for every subset A; but some subsets can;t be compared in this > ordering; for example, {a,b} and {b,c}. > > That's a perfectly reasonable state of affairs; not every partial > order is somehow "required" to be a particular canonical total order. > There is an obvious and useful way to provide total order on the power set, and that is to assign the value 2^n to the element n (starting at 0), and order according to the sum of those values corresponding to the elements of the subsets. For a set of size n, your subsets are numbered 0 through 2^n-1. Power set is the same as binary. >>> 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. > > Yup. If by "uh oh", you mean "he's actually asking me to /think/ about > what the heck I'm /saying/". Uh oh, indeed! > Total order is what I'm talking about. >>>> 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'm going to guess that what you wanted to say was "... the union of > two continuous sets, ...". > > But who the heck knows? > Discrete in the sense that they do not overlap. Internally, they are continuous, sure. >> I'd say if you break the real line into >> (x,x+1] for xeZ, those are discrete partitions of R. >> > > That's not a definition: it's an /example/. If I ask you to define > "Chassian number", the response "3 is an Chassian number" is not a > response which is a definition. > >> 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. :) >> > > Not so much wrong as inconsistent to the point of incomprehensibility. > There are /no/ two real numbers x, y in the set [0,1) union (1, 2] > with x < y such that there exists no element z in [0,1) union (1,2] > with x < z < y. 1 is /not/ an element of [0,1) union (1,2]. > Um, you're being inconsistent, or at least unclear. Alright, let's see. I defined a discrete order as one where there exist pairs of elements not separated by any intermediate elements. You offered [0,1)u(1,2] as a counterexample. Can you name two elements in the union which do not have an element between them? Try again. > So when you say "discrete partions of R", you either mean something / > different/ from saying "the ordering on that partition is discrete", > or else you don't have a good sense of what you really mean yourself > when you say "discrete". > I think you should reconsider my original definition, and see whether you can find anything else wrong with it. >>>> 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. >> > > Yes, a sentence is indeed a kind of sequence. Because otherwise: "a a > is of yes kind indeed sentence sequence" would mean the same thing. > > However, it is not a /definition/ of a sequence, nor of a sort of > sequence. > What does that have to do with anything? You asked which came first, and I answered in the context of the sentential sequence. >>>> 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.... >> > > Funny. But it also shows that you can't actually answer: what "time" > does 2^pi "happen" in the function f: R->R defined by f(x) = 2^x? > What is your point? It happens after 2^3 and before 2^4. >>>> 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. >> > > I can't give you proper directions to Springfield, until you tell me / > which/ Springfield you actually want to go to. > > Cheers - Chas > Er, okay. Tony
From: Tony Orlow on 1 Apr 2007 11:34 Virgil wrote: > In article <460f22e6(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >>> 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?" > > The set of real numbers, whether with or without infinitesimals is an > uncountable linearly ordered set, but of course not discretely ordered. > > And I cannot believe that TO, who is usually quite inventive, if not > always accurate, cannot create one. I did, the H-riffics. > >>>>>> However, where every element of a set has a well defined >>>>>> successor and predecessor, it's a sequence of some sort. > > Not necessarily. If a set is partitioned into two or more subsets each > with such an order on it, but with no order between partitions, then the > set itself is not even an ordered set even though every member has a > well defined predecessor and successor. It's certainly easy enough to order the partitions, though not without infinite descending sequences either within or between the partitions, in an uncountable set, as far as I can see.
From: Mike Kelly on 1 Apr 2007 11:37
On 1 Apr, 16:07, Tony Orlow <t...(a)lightlink.com> wrote: > cbr...(a)cbrownsystems.com wrote: > > On Mar 31, 7:27 pm, Tony Orlow <t...(a)lightlink.com> wrote: > >> step...(a)nomail.com wrote: > >>> 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". > > I defined it as a sequence where there exist elements infinitely beyond > other elements. As you're incapable of providing your definition of "sequence" that doesn't elucidate anything. > >>>>>>>>>> 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. > > Don't be a boor. "Quantitative results" are quantities indicated by the > resulting symbolic expression. "Manipulate" means "produce a new string > from an existing one according to rules". A quantity is a point on the > real line. A number is a string that indicates a point. Arithmetic is > the manipulation of strings. But, you know all that. Do you think people are *lying* when they say they can't understand you? > >> 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 > > That was tongue-in-cheek. I know there's no definition of "number",and > mathematicians seem quite satisfied with that for themselves, but insist > that I produce a definition of a word they use every day without knowing > what they even mean themselves. Mathematicians refer to "natural numbers" and "real numbers" and "ordinal numbers" and "cardinal numbers" and so on. They don't refer to plain old "numbers", because "number" isn't defined. You don't seem to comprehend this. -- mike. |