The Definition of Points
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From: cbrown on 31 Mar 2007 23:11 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". 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. 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. > 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? > 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"? > 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"? > > 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
From: Tony Orlow on 31 Mar 2007 23:11 cbrown(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?" >> people have examined all sorts of orderings, partial, >> >>> total, and other. The fact that you prefer to remain ignorant of this >>> does not mean the issue has been skirted by anyone other than >>> yourself. >> 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. >>>> However, where every element of a set has a well defined >>>> successor and predecessor, it's a sequence of some sort. >>> Let S = {0, a, 1, b, 2, c}. >>> Let succ() be defined on S as: >>> succ(0) = 1 >>> succ(1) = 2 >>> succ(2) = 0 >>> succ(a) = b >>> succ(b) = c >>> succ(c) = a >> Okay you have two sequences. >> > > Why two? Why not one, or three, or six? Your definition fails to say. > > Is S = {0,1}, succ(0)=1, succ(1)=0 a "sort of sequence"? It has a well- > defined successor and predeccessor for each element. How about S = > {0}, and succ(0) = 0? > >> >>> Every element of S has a well-defined successor and predecessor. What >>> "sort of sequence" have I defined? Or have you left out some parts of >>> the /explicit/ definition of whatever you were trying to say? >>> Cheers - Chas >> Yes, I left out some details. > > Given that you are claiming that your definition is somehow being > surpressed by religious or political forces, why not take the > opportunity to provide these details (in which we all know the devil > resides)? > > As it stands, I have no real idea what you're talking about; and quite > frankly, I doubt you yourself have a clear idea of what you are > talking about. > > Other people have thought through similar ideas and presented a > comprehensive structure for understanding. See, e.g., > > http://en.wikipedia.org/wiki/Order_theory > > in case you'd actually like to try to /learn/ something about the > subject. > > Cheers - Chas > Gee, thanx.
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 |