The Definition of Points
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From: Mike Kelly on 31 Mar 2007 19:41 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? > >> Good enough? > > > You tell me. Did you mean to say "a sequence is a set"? If so, good > > enough. > > > -- > > mike. > > Not exactly, and no, what I said is not good enough. > > A set with an order where each element has a unique successor is a > forward-infinite sequence. Each can have a unique predecessor, and then > it's backward-infinite. And if every element has both a unique successor > and predecessor, then it's bi-infinite, like the integers, or within the > H-riffics, the reals. One can further impose that x<y ->~y<x, to > eliminate circularity. > > Good enough? Probably not yet. So when you say "sequence" you're refering to a set and an ordering on that set? There are some conditions on the properties of the ordering. You're not, as yet, able to coherently explain what those conditions are. 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.
From: Tony Orlow on 31 Mar 2007 19:44 Virgil wrote: > In article <460e8251(a)news2.lightlink.com>, > Tony Orlow <tony(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. >> >>>> Good enough? >>> You tell me. Did you mean to say "a sequence is a set"? If so, good >>> enough. >>> >>> -- >>> mike. >>> >> Not exactly, and no, what I said is not good enough. >> >> A set with an order where each element has a unique successor is a >> forward-infinite sequence. Each can have a unique predecessor, and then >> it's backward-infinite. And if every element has both a unique successor >> and predecessor, then it's bi-infinite, like the integers, or within the >> H-riffics, the reals. One can further impose that x<y ->~y<x, to >> eliminate circularity. >> >> Good enough? Probably not yet. > > > You are right, not yet. > I'm right!!! :D heh. > Every "sequence" must be a totally ordered set which is order isomorphic > either to the ordered set of naturals, if it has a first element, or to > the ordered set of integers, if it does not have a first element. Okay. How would you boil down that statement, and in which cases would you say it applies? > > Note that since the obvious mapping between the natural numbers and the > negative integers is an order isomorphism with order reversal, one need > not include that third case separately. Sure. One not even really distinguish between '<' and '>', but it helps. :) Tony
From: Tony Orlow on 31 Mar 2007 19:47 Lester Zick wrote: > On Fri, 30 Mar 2007 12:13:57 -0500, Tony Orlow <tony(a)lightlink.com> > wrote: > >> Lester Zick wrote: >>> On Thu, 29 Mar 2007 09:37:21 -0500, Tony Orlow <tony(a)lightlink.com> >>> wrote: >>> >>>>>> It's the same as Peano. >>>>> Not it isn't, Tony. Cumulative addition doesn't produce straight lines >>>>> or even colinear straight line segments. Some forty odd years ago at >>>>> the Academy one of my engineering professors pointed out that just >>>>> because there is a stasis across a boundary doesn't necessarily mean >>>>> that there is no flow across the boundary only that the net flow back >>>>> and forth is zero.I've always been impressed by the line of reasoning. >>>> The question is whether adding an infinite number of finite segments >>>> yields an infinite distance. >>> I have no idea what you mean by "infinite" Tony. An unlimited number >>> of line segments added together could just as easily produce a limited >>> distance. >>> >>> ~v~~ >> Not unless the vast majority are infinitesimal. > > No that isn't what I'm talking about. You seem to assume consecutive > segments would have to be colinear and lie along a straight line. I've > already tried to explain why this isn't so. They could all connect in > completely different directions even though mathematikers commonly > assume they somehow for some reason would very plolitely line up in > one direction alone. Line segments are only connected by points, Tony. > And their direction is not determined by those points because there is > no definable slope at point intersections. > I'm sorry Lester. Perhaps I misunderstood. When you used the word "added", I assumed you meant addition. That assumes a linear construction. But, perhaps, you meant some other form of addition. Addition is linear, as commonly understood... >> If there is a nonzero >> lower bound on the interval lengths, an unlimited number concatenated >> produces unlimited distance. > > And if segments were all of equal finite size we could make a finite > plane hexagon out them which would be quite limited in distance. > > ~v~~ Not exactly, but that's a complicated topic.... 01oo
From: Tony Orlow on 31 Mar 2007 19:48 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... :) Tony
From: Lester Zick on 31 Mar 2007 19:50
On Fri, 30 Mar 2007 11:39:53 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Thu, 29 Mar 2007 09:37:21 -0500, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>> Hi Lester - >>> >>> Glad you responded. I was afraid I put you off. This thread seems to >>> have petered unlike previous ones I've participated in with you. I hope >>> that's not entirely discouraging, as I think you have a "point" in >>> saying points don't have meaning without lines, and that the subsequent >>> definition of lines as such-and-such a set of points is somewhat >>> circular. Personally, I think you need to come to grips with the >>> universal circularity, including on the level of logic. Points and lines >>> can be defined with respect to each other, and not be mutually >>> contradictory. But, maybe I speak too soon, lemme see... >> >> Hey, Tony - >> >> Yeah I guess I'm a glutton for punishment with these turkeys. The >> trick is to get finite regressions instead of circular definitions. We >> just can't say something like lines are the set of all points on lines >> because that's logically ambiguous and doesn't define anything. I >> don't mind if we don't know exactly what points are in exhaustive >> terms just that we can't use them to define what defines them in the >> intersection of lines and in the first place. >> >> The problem isn't mathematical it's logical. In mathematics we try to >> ascertain truth in exhaustively demonstrable terms. That's what >> distinguishes mathematics from physics and mathematicians from >> mathematikers and empirics. >> >> (By the way, Tony, I'm chopping up these replies for easier access and >> better responsiveness.) >> >> ~v~~ > >Sounds like a good idea. Well sure. It fertilizes the ground. At 800+ lines I don't know too many mathematikers capable of that level of reading comprehension. >I've certainly taken my licks around here too, but that's a big part of >this process - debate. You win some and lose some. Or rather, some lose >and some win, when it comes to ideas. And some just spar forever before >they finally realize they're actually dancing together, like waves and >particles, in a fluid universe. > >When you say it's not necessary to know exactly what points "are", >that's somewhat true. We don't even know what mass "is", but as in >science, we define objects by their properties and the predictions we >can make. So, if we can characterize the relationship between points and >lines, then we can define them relative to each other, which may be the >best we can do. But, that is not what you desire. You want a "ground >zero" starting point upon which all else is built. This is akin to set >theorists' e operator: "is an element of". They start with that one >operator, then supposedly measure is built upon that. Well, they do the >same thing you are doing when you assume an implied OR in "not a not b", >and then derive OR from AND, which you define as not(not a not b). They >introduce the von Neumann ordinals defined solely by set inclusion, and >yet, surreptitiously introduce the notion of order by means of this set. >Order, '<', is another operator and should be recognized as such. One >should allow that there are always two first elements or operators, >whose interplay produces the whole we're considering. That's the Tao. >It's not wrong. There is no single perspective, and there is no straight >line. It's all circles. You want tao, Tony, talk to Brian. You want mechanics talk to me. ~v~~ |