The Definition of Points
in [Math]
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From: Tony Orlow on 31 Mar 2007 20:25 stephen(a)nomail.com wrote: > In sci.math Tony Orlow <tony(a)lightlink.com> wrote: >> stephen(a)nomail.com wrote: >>> In sci.math Tony Orlow <tony(a)lightlink.com> wrote: >>>> stephen(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? >>>>>> 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? There is no standard definition > of "size", as you have been told countless times for a couple > of years now. Size is an ambiguous word in any situation, and > there is no argument in set theory that depends on the word "size". > > Oh ambiguous... >>>>> It should be obvious that the number of elements between 1 and w is >>>>> larger than any finite natural number. Let X be a finite >>>>> natural number > 1. Then {2, 3, .. X, X+1, .. 2X } is a subset >>>>> of the elements between 1 and w that has more more than X elements. >>>>> >>>>> As I said, even you do not accept your own definition of "actually >>>>> infinite". >>>>> >>>>> Stephen >>>>> >>>> If you paid attention, the apparent contradiction would evaporate. The >>>> number of elements up to and including any finite element of N is >>>> finite, and equal to that element in magnitude. If the number is n, then >>>> there's an nth, and its value is n. As Ross like to say, NeN. We are not >>>> alone. :D >>>> Tony >>> But the question is not about the number of elements up and including >>> any finite element of N. I asked how many elements are between 1 and w >>> in the set {1, 2, 3, ..., w }. > >> w-2 are between w and 1. x-2 are between 1 and x. > > What is w-2? Remember, I am talking about the standard definition > of w. The set I am talking about does not contain a w-2. It > contains all the finite elements of N, and the element w. > How convenient. You can't move left from w. Well, that simplifies your dance, now, doesn't it? >> w is not an element of N, nor is it finite. > >> Oh, then why mention it? > > Is there some rule saying that we can only mention finite elements, > or elements of N? I can describe all sorts of sets such as > N U { 1/2 }, or N U { w } or N U { {1, 2}, {2, 3}, {3, 4} ... }. > Describe away - just don't expect it to prove anything if it's not pertinent. > The reason I mentioned it is because the set {1, 2, 3, ... w } > has the property that there exist two elements between which > there is an infinite number of elements, namely 1 and w. I know > that you do not consider {1, 2, 3, ... , w} an actually > infinite set, so I brought this up as an example of the fact > that even you do not agree with your own statement, which was: > >>> An actually infinite sequence is one where there exist two elements, one >>> of which is an infinite number of elements beyond the other. Prove to me, logically, that there exist more than any finite number of elements between 1 and w. > > And of course that was my whole point. Despite the fact that > you posted that as a definition of an actually infinite sequence, > even you do not think it is the definition of an actually infinite > sequence. > I do not think your example qualifies, logically. Sorry. >>> I know you are incapable of actually thinking about all the elements of N, >>> but that is your problem. In any case, N is not an element of N. >>> Citing Ross as support is practically an admission that you are wrong. >>> >>> Stephen >>> > >> Sure, of course, agreeing with someone who disagrees with you makes me >> wrong. I'll keep that in mind. Thanks.. > >> Tony > > No, agreeing with someone who makes absolutely no sense, such as > Ross, is tantamount to admitting you are wrong. Whether Ross makes any sense or not is a personal judgment, based on whether what he says jibes with anything one may or may not think. Some of what he says jibes for me. So Ross doesn't make no sense, from where I sit, even if he doesn't have a system that I completely grok. His is not incompatible with mine. Of course you > do seem to have caught on to the fact that Lester is full of nothing > but nonsense, so maybe there is hope for you yet. I see that Lester doesn't get the "establishment" position on logic, and I'd like to help. I don't think all his points are simply lines... ;) > > If you think Ross makes sense, explain his null axiom theory. > > Stephen > > > > I don't understand a theory without axioms, but I do understand the sentiment, and it's not dissimilar to Lester's. It's all about getting to the roots of the Tree of Knowledge, without undue assumptions. It's a worthy endeavor, even if fraught with entanglement and personal woe. The problem is, there's always two roots to every sprout...so let's all get used to it. Tony
From: Lester Zick on 31 Mar 2007 20:30 On Fri, 30 Mar 2007 12:49:54 -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: >> >>>> This is why science is so useful because you stop arguing isolated >>>> problems to argue demonstrations instead which subsume those isolated >>>> problems. There's simply no point to arguing such problems >>>> individually as to whether "not" is universally true of everything or >>>> whether there are such things as conjunctions not reducible to "not" >>>> in mechanically exhaustive terms unless the demonstration itself is >>>> defective and not true. And just claiming so per say won't cut it. >>>> >>> Your "not a not b" has an assumed OR in it. >> >> The problem is not whether it has or doesn't, Tony, but how do you >> know and how can you demonstrate the truth of that claim. I mean there >> is no visible indication what the relation between A and B is. You >> might consider the relation between them is "or" but we have no >> evidence that this conjecture is right and not just rank speculation. >> I mean there are plenty of people out there who insist that relations >> between any two items like A and B are theistic, deistic, or even the >> product of aliens and UFO's. > > >Please choose true or false, if you didn't do it last time: > >a b not a not b > >true true true or false? >true false true or false? >false true true or false? >false false true or false? Kinda hard to tell what these terms mean, Tony? Are there clues or do we just wing it? >> Consequently it's not my assumption of any relation between A and B >> but my demonstrations of relations between them that matters. Sure I >> can assume anything I want. And on previous occasions I certainly have >> assumed the relation between them was a functional if not explicit or >> because it seems to me the most plausible mechanical relation likely. >> But that doesn't mean it's necessarily true. > >You need to define what relation your grammar denotes, or there is no >understanding when you write things like "not a not b". Of course not. I didn't intend for my grammar to denote anything in particular much as Brian and mathematikers don't intend to do much more than speak in tongues while they're awaiting the second coming. >> However the fact is that given two different things A and B we can >> combine them with compoundings of "not" and when we do certain >> conjunctive relations between them fall out the first of which is >> "and" and the next of which is "or". That's how we can tell what the >> originary implications between two distinct items is and has to be. > >Not if you assumed OR to begin with. In that case, you're as circular as >anyone else, and more. Better to build up from true() and false() as >0-place predicates. Of Christ. Don't you and the zero place predicates wait up for us. >> But that doesn't mean there is any assumption of "or" between them >> only that given two distinct things like A and B we can determine any >> conjunctive relations between them without the implicit assumption of >> or explicit use of conjuctions. And that means conjunctions and so on >> are "in here" and not "out there" among distinct things themselves. > >Choose true or false above, and I guarantee you'll see it's the relation OR. Of course it is, Tony. I just tried to slip one over on you. OR Brian OR Virgil OR Stephen OR PD OR David Or Mikey Or someone else without saying so. ~v~~
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. |