The Definition of Points
in [Logic]
|
Prev: On Ultrafinitism
Next: Modal logic example
From: Virgil on 31 Mar 2007 19:25 In article <460ee48e(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Why? What have I defined, if not a sequence? Is there a word for it? It > must "exist", if I assert so. Does TO now claim the right of God to make things exist by His command? I understand that God is a jealous God, and takes such usurpations in very bad part.
From: stephen on 31 Mar 2007 19:29 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. >>>>> 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. 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". >>>> 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. > 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} ... }. 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. 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 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. 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. If you think Ross makes sense, explain his null axiom theory. Stephen
From: Tony Orlow on 31 Mar 2007 19:30 Virgil wrote: > In article <460e812f(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Surely, you don't think me fool enough to think that Virgil would >> actually give me a sincere compliment, or acknowledge that any of my >> nonstandard points actually has any merit, do you? Still, it was nice of >> Virgil to say I'm not worst ignoramus he knows. That warmed my heart. >> >> Still, I don't know what Virgil's comment about me says about my future >> responses to you. See above for a characterization of Q. > > > I have, upon occasion, found, and stated, that TO was correct on some > point or another. Yes, you have conceded on several occasions that my basic statement was correct, but not that my point in general had any merit, that I've noticed. That's okay. You do very well as a corrections officer, and I appreciate your role, and try not to take it too personally. I like you, Virgule. :) (I just googled Virgule, for the first time - whaddya know? How very fitting!) > > I have never found Zick to be correct on any point. But then I have long > since stopped looking at Zick's posts. I suppose that it is marginally > possible that Zick may have been right about something since then. Lester has a vision, but his formalization is flawed, as I see it. I think he intuits some valid issues, but as smart as I think he is to intuit and see what he sees, I don't think he's analyzed the situation properly. While that's perhaps disappointing, the very will to address fundamental issues is telling, and such devotion should not go unappreciated. <3 ToeKnee 01oo tony. :)
From: Tony Orlow on 31 Mar 2007 19:36 Lester Zick wrote: > On Fri, 30 Mar 2007 12:10:12 -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: >>> >>>>>> Their size is finite for any finite number of subdivisions. > >>>>> And it continues to be finite for any infinite number of subdivisions >>>>> as well.The finitude of subdivisions isn't related to their number but >>>>> to the mechanical nature of bisective subdivision. >>>>> >>>> Only to a Zenoite. Once you have unmeasurable subintervals, you have >>>> bisected a finite segment an unmeasurable number of times. >>> Unmeasurable subintervals? Unmeasured subintervals perhaps. But not >>> unmeasurable subintervals. >>> >>> ~v~~ >> Unmeasurable in the sense that they are nonzero but less than finite. > > Then you'll have to explain how the trick is done unless what you're > really trying to say is dr instead of points resulting from bisection. > I still don't see any explanation for something "nonzero but less than > finite". What is it you imagine lies between bisection and zero and > how is it supposed to happen? So far you've only said 1/00 but that's > just another way of making the same assertion in circular terms since > you don't explain what 00 is except through reference to 00*0=1. > > ~v~~ But, I do. I provide proof that there exists a count, a number, which is greater than any finite "countable" number, for between any x and y, such that x<y, exists a z such that x<z and z<y. No finite number of intermediate points exhausts the points within [x,z], no finite number of subdivisions. So, in that interval lie a number of points greater than any finite number. Call |R in (0,1]| "Big'Un" or oo., and move on to the next conclusion....each occupies how m,uch of that interval? 01oo
From: Tony Orlow on 31 Mar 2007 19:40
Lester Zick wrote: > On Fri, 30 Mar 2007 12:11:23 -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: >>> >>>>> Equal subdivisions. That's what gets us cardinal numbers. >>>>> >>>> Sure, n iterations of subdivision yield 2^n equal and generally mutually >>>> exclusive subintervals. >>> I don't know what you mean by mutually exclusive subintervals. They're >>> equal in size. Only their position differs in relation to one another. >>> >>> ~v~~ >> Mutually exclusive intervals : intervals which do not share any points. > > What points? We don't have any points not defined through bisection > and those intervals do share the endpoints with consecutive segments. > > ~v~~ Okay, lay off the coffee. Sure. Now subdivide the line so that the left endpoint is always included and the right never. [x,y). Then each is mutually exclusive of all others. 01oo |