The Definition of Points
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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
From: cbrown on 1 Apr 2007 00:22 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! > > 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. > 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. > > 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! > > >> 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? > 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]. 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". > >> 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. > >> 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? > >> 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
From: Virgil on 1 Apr 2007 00:40
In article <460f1e93(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <460f00a0(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Look back. The nth is equal to n. Inductive proof holds for equality in > >> the infinite case > > > > Not in vN. > > I know that statement is not generally acceptable. I don't care. It's > true. Not in ZF or NBG, so just where does TO claim it is true? > Infinite-case induction has not been disproved It has not been proved, which is more to the point. And without a clear statement of what it says, it cannot be. > > And inductive proofs do not work that way. One can prove by > > induction that something is true for each natural, but that does not > > create any infinite naturals for which it is true. > > You can agree with your buddies these are the rules of your club, and > you can hang around the pickup drinking cans of beer and talking about > trucks and ladies, but somes of us gots interstates to travel, and, you > know, big cities to deliver to. The keep on truckin', since you don't seem to be good for anything else. > > Where an equality is proven true for all x>k, then any positive infinite > value is greater than any finite k, and the proof holds. Simple clear > logic. Refute, please. Refute what? So far you have produced nothing worth refuting. |