CANTOR DISPROOF <<<<<<<<<<<<<<<<
in [Logic]
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From: George Greene on 20 Jun 2010 15:59 On Jun 20, 2:29 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: > I should add, the only reason anyone reads > anything you write is because you begin > every post with the recognized mantra CANTOR PROVED THAT.... This is NOT TRUE, dumbass. YOU are the one who re-invoked Cantor. YOU decided to call this an attack on Cantor. That forces us to remind you of the existence OF THE PROOF. Seriously, people should Google the contents of this newsgroup for the word "Cantor". They will discover that we basically went over 2 years WITHOUT IT UNTIL YOU RETURNED.
From: Graham Cooper on 20 Jun 2010 17:17 On Jun 21, 5:59 am, George Greene <gree...(a)email.unc.edu> wrote: > On Jun 20, 2:29 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: > > > I should add, the only reason anyone reads > > anything you write is because you begin > > every post with the recognized mantra CANTOR PROVED THAT.... > > This is NOT TRUE, dumbass. > YOU are the one who re-invoked Cantor. > YOU decided to call this an attack on Cantor. > That forces us to remind you of the existence OF THE PROOF. > Seriously, people should Google the contents of this newsgroup > for the word "Cantor". They will discover that we basically went over > 2 > years WITHOUT IT UNTIL YOU RETURNED. So you did learn something 2 Years ago! But you still can't answer how wide is a set? Herc
From: porky_pig_jr on 20 Jun 2010 19:11 On Jun 20, 7:07 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: > > I disproved Turing, Halt, Godel and Cantor Don't Halt here.
From: |-|ercules on 20 Jun 2010 19:15 <porky_pig_jr(a)my-deja.com> wrote > On Jun 20, 7:07 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: > >> >> I disproved Turing, Halt, Godel and Cantor > > Don't Halt here. > You on the other hand cannot answer how wide this set is! 3 31 314 .... You um and arr and cannot parse words like CONTAIN, BELOW, BIG, SET and accuse me of shifting language to make nonsense claims. How wide is the set porky, George is really stumped on this one! Herc
From: porky_pig_jr on 20 Jun 2010 20:05
On Jun 20, 7:15 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: > <porky_pig...(a)my-deja.com> wrote > > > On Jun 20, 7:07 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: > > >> I disproved Turing, Halt, Godel and Cantor > > > Don't Halt here. > > You on the other hand cannot answer how wide this set is! > > 3 > 31 > 314 > ... > > You um and arr and cannot parse words like CONTAIN, BELOW, BIG, SET > and accuse me of shifting language to make nonsense claims. > > How wide is the set porky, George is really stumped on this one! > > Herc This is the first time I see someone's using the adjective "wide" with the set. I assume you're asking about the cardinality of a set. Or may be you're asking what happens with the number of digits in each element of the set, as we generate the successive approximations of the pi? Well, pi is computable real. We're all agree on that. Generating successive approximations of pi is a counting process, so the number of such approximations is countably infinite. Each entry of such list contains some finite number of digits, we can call it a finite prefix of a pi. OK, I guess I see your point. By width you mean "the largest such prefix", right? And you want to imply that it is also countably infinite. So the "width" of this set is countably infinite. Well, there's a subtlety here. On one hand, certainly the width of this list (using your terminology) is not bounded by any natural n, right? We can always make one more iteration and create a prefix with the width n +1. So, since the width of this list is not bounded by any natural n, it must be infinite. Does it follow then that the list actually *contains* one entry which is the infinite string (and, of course, that infinite string *is* pi)? No, it does not follow at all, and this is where I believe you get off the track. I don't think I can help you with that. But now I'm thinking of the question someone asked me once. Suppose we prove something by induction. That is, we prove something for any natural n. Suppose we have some sequence with some limiting behavior. So we can think of some limit point. Does proof of induction include that limit point? The answer is categorical "No, it does not!" You can certainly think of a limit as a point, but it's a special point, never to be accessed in a finite number of steps. This is what Cantor (your hero, as I know), called "transfinite ordinals", the first such limit point is normally designated as little omega. Each predecessor of it is a finite number, yet it has no immediate predecessor and can't be accessed in a finite number of steps. Well, think of that little omega as a horizon. You sort of know it's there but you can never reach it in finite number of steps. One may say that that little omega-entry in our list is indeed our goal, our dream, the infinite string representing Pi. But it's not reachable. We can never generate such a string in a finite number of steps. In this respect, even if "the width of this set is infinite", the list does not contain that infinite string. Exactly for the same reason the list (1/2, 2/3, 3/4, 4/5, ...) will never contain 1. 1 is what happens at the limit, but we never reach that limit in a finite number of steps. We can stack that 1 on a top of the list, it will correspond to little- omega entry. But it's not reachable by our algorithm we use to generate the list (1/2, 2/3, 3/4 ...). So the infinite string representing pi won't be on the list (3, 31, 314, ...) for exactly the same reason. Because the limit is an imaginary never-reachable-in- finite-number-of-steps point. The fact that the width of such list is not bounded by any n, and hence infinite does not imply that the infinite string representing pi *is* on that list. Just we can stack 1 on a top of a sequence (1/2, 2/3, 3/4 ...), we can stack your infinite string on a top of (3, 31, 314 ...), but that's not a part of that sequence, it does not belong to it. That's as much as I can say. A while ago I wasn't clear on that, but after taking a few courses on real analysis, doing lots of exercises involving limits, and then a little bit involving transfinite induction vs the regular induction, I finally started getting the hold of that. Your mistake is thinking of a limiting behavior as an actual point, or actual location in that list. It isn't there. As I've said, I'm thinking of such entry as little omega-entry. You can mentally stack it on a top of your list, but it's *not* part of the list. It's never reachable. It's not there. The fact that the width of the list is not bounded by any real n *still* does not mean that it's there. Gee whiz, my answer is a bit long. And, as I said, at some point I wasn't that clear on that whole thing. The clarity came when I started reading about transfinite induction. PPJ. |