The Definition of Points
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From: Lester Zick on 31 Mar 2007 20:33 On 30 Mar 2007 09:58:43 -0700, "JAK" <jak(a)theoryofmind.org> wrote: >I believe a fine answer was posted earlier (by Eric, as I recall) >noting that points are relative. Well they're not relatives of mine. Of yours perhaps. Of the top of your head most likely of all. > And the posting of "not a not >b" (Tony?) is also excellent. Either response was great. Combined, >they are superb. Can we take your word for it per say? ~v~~
From: Tony Orlow on 31 Mar 2007 20:33 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 Continuous means x<z -> Ey: x<y ^ y<z Discrete means not continuous, that is, given x and z, y might not exist. For something to "occur", it must happen "at some time". 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. >>>> 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. Explain away. > > 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... tony.
From: Tony Orlow on 31 Mar 2007 20:45 Mike Kelly wrote: > On 31 Mar, 16:47, Tony Orlow <t...(a)lightlink.com> wrote: >> Mike Kelly wrote: >>> On 31 Mar, 13:48, Tony Orlow <t...(a)lightlink.com> wrote: >>>> step...(a)nomail.com wrote: >>>>> In sci.math Virgil <vir...(a)comcast.net> wrote: >>>>>> In article <460d4...(a)news2.lightlink.com>, >>>>>> Tony Orlow <t...(a)lightlink.com> wrote: >>>>>>> An actually infinite sequence is one where there exist two elements, one >>>>>>> of which is an infinite number of elements beyond the other. >>>>>> Not in any standard mathematics. >>>>> It is not even true in Tony's mathematics, at least it was not true >>>>> the last time he brought it up. According to this >>>>> definition {1, 2, 3, ... } is not actually infinite, but >>>>> {1, 2, 3, ..., w} is actually infinite. However, the last time this >>>>> was pointed out, Tony claimed that {1, 2, 3, ..., w} was not >>>>> actually infinite. >>>>> Stephen >>>> No, adding one extra element to a countable set doesn't make it >>>> uncountable. 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 (countable) sequences have a last element? What's the last finite >>> natural number? >>> -- >>> mike. >> As I said to Brian, it's provably the size of the set of finite natural >> numbers greater than or equal to 1. > > Provable how? > Look back. The nth is equal to n. Inductive proof holds for equality in the infinite case >> No, there is no last finite natural, > > You keep changing your position on this. > Nope, I don't. >> and no, there is no "size" for N. Aleph_0 is a phantom. > > 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. > I find it very hard to understand what you are even trying to say when > you say "Aleph_0 is a phantom". It seems a bit like Ross' meaningless > mantras he likes to sprinkle his posts with. > > -- > mike. > Yes, NeN, as Ross says. I understand what he means, but you don't. Where taking away makes something less, aleph_0-1<aleph_0, and there is no smallest infinity, except in the nonlogical imagination. Chase that tail! tony.
From: Tony Orlow on 31 Mar 2007 20:47 Lester Zick wrote: > On Fri, 30 Mar 2007 12:04:33 -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: >>> >>>>>>> Okay, Tony. You've made it clear you don't care what anyone thinks as >>>>>>> long as it suits your druthers and philosophical perspective on math. >>>>>>> >>>>>> Which is so completely different from you, of course... >>>>> Difference is that I demonstrate the truth of what I'm talking about >>>>> in mechanically reduced exhaustive terms whereas what you talk about >>>>> is just speculative. >>>> You speculate that it's agreed that not is the universal truth. It's not. >>> No I don't, Tony. It really is irritating that despite having read >>> E201 and E401 you call what I've done in those root threads >>> "speculation". What makes you think it's speculation? I mean if you >>> didn't understand what I wrote or how it demonstrates what I say then >>> I'd be happy to revisit the issue. However not questioning the >>> demonstration and still insisting it's speculation and no different >>> from what you say is just not okay. >> I've questioned that assumption all along. We've spoken about it plenty. > > What assumption, Tony?You talk as if there is some kind of assumption. > That "not not" is self-contradictory, as if "not" is a statement.... >>> I don't speculate "it's agreed" or not. I don't really care whether >>> it's agreed or not and as a practical matter at this juncture I'd have >>> to say it's much more likely not agreed than agreed. What matters is >>> whether it's demonstrated and if not why not and not whether it's >>> agreed or not since agreements and demonstrations of truth are not the >>> same at all. Agreements require comprehension and comprehension >>> requires study and time whereas demonstrations of truth only require >>> logic whether or not there is comprehension. >>> >>> ~v~~ >> Demonstrate what the rules are for producing a valid one of your logical >> statements from one or more other valid ones of your logical statements, >> because "not not" and "not a not b" are not standard valid logic >> statements with known rules of manipulation. What are the mechanics? As >> far as I can tell, the first is not(not(true))=true and the second is >> or(not(a),not(b)), or, not(and(a,b)). > > Or you could demonstrate why the standard valid logic you cite is > standard and valid. > > ~v~~ Okay, I'll take that as a disinclination and failure to comply. You have the right to remain silent... ;) 01oo
From: Tony Orlow on 31 Mar 2007 20:50
Lester Zick wrote: > On Fri, 30 Mar 2007 12:22:30 -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: >>> >>>>> Finite addition never produces infinites in magnitude any more than >>>>> bisection produces infinitesimals in magnitude. It's the process which >>>>> is infinite or infinitesimal and not the magnitude of results. Results >>>>> of infinite addition or infinite bisection are always finite. >>>>> >>>>>> Wrong. >>>>> Sure I'm wrong, Tony. Because you say so? >>>>> >>>> Because the results you toe up to only hold in the finite case. >>> So what's the non finite case? And don't tell me that the non finite >>> case is infinite because that's redundant and just tells us you claim >>> there is a non finite case, Tony, and not what it is. >>> >> If you define the infinite as any number greater than any finite number, >> and you derive an inductive result that, say, f(x)=g(x) for all x >> greater than some finite k, well, any infinite x is greater than k, and >> so the proof should hold in that infinite case. Where the proof is that >> f(x)>g(x), there needs to be further stipulation that lim(x->oo: >> f(x)-g(x))>0, otherwise the proof is only valid for the finite case. >> That's my rules for infinite-case inductive proof. It's post-Cantorian, >> the foundation for IFR and N=S^L. :) >> >>>> You can >>>> start with 0, or anything in the "finite" arena, the countable >>>> neighborhood around 0, and if you add some infinite value a finite >>>> number of times, or a finite value some infinite number of times, you're >>>> going to get an infinite product. If your set is one of cumulative sets >>>> of increments, like the naturals, then any infinite set is going to >>>> count its way up to infinite values. >>> Sure. If you have infinites to begin with you'll have infinites to >>> talk about without having to talk about how the infinites you >>> have to talk about got to be that way in the first place. >>> >>> ~v~~ >> Well sure, that's science. Declare a unit, then measure with it and >> figure out the rules or measurement, right? > > I have no idea what you think science is, Tony. Declare what and then > measure what and figure out the rules of what, right, when you've got > nothing better to do of an afternoon? > > ~v~~ I've been dropping feathers and bowling balls out my window all morning.... What do YOU think science is? 01oo |