SCI.MATH POLL - uncountable infinity
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From: George Greene on 3 Jun 2010 16:33 On Jun 3, 10:50 am, "dannas" <inva...(a)invalid.com> wrote: > those are just assumptions that you have made, which change the problem. Liar. The fact that you cannot read does not mean that I am "making assumptions". In any case, the OP is NOT an authority on the problem. The problem long predates him. He has some serious misunderstandings about the problem and is groping. > the OP did NOT say natural numbers were written on them, Liar. He said, "and the boxes have a unique number written on them." Unique means one of a kind. It means that each box has a different number on it. Or, at least, that's how HE meant it. I concede that other interpretations may be possible, but not that you actually made one. You just fucked up. > and he did NOT say that the number inside the box was Unique. I didn't either, DUMBASS. There IS NO "the" number inside the box! THE number is ON THE OUTSIDE of the box, as a label, or, as the OP said, "written on them". What is INside each and every box is a ("unique", i.e., different for each one) SUBSET of the naturals! It is not so important that you admit that you were wrong in the past, as it is that you change to correctly understanding what is going on in the present.
From: George Greene on 3 Jun 2010 16:43 On Jun 3, 4:08 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: > You really are full of sh1t then! Of course he is, but look who's talking. You are the one who is too stupid to understand the proof of Cantor's theorem. The most important thing you need to understand about Cantor's theorem is that it has ABSOLUTELY NOTHING WHATSOEVER TO DO WITH INFINITY. The SAME proof works for a FINITE number of boxes. Obviously, if infinity isn't enough, then a finite number won't be enough other, so you will STILL get the same result. The infinity does NOT matter. You are the one trying to pretend that it does. In the infinite version, every number in a box (or on a box) has to be LESS THAN the NUMBER OF boxes (because every number in or on a box is finite, but the number of boxes is infinite). You can have a finite version of THE SAME problem simply by allowing/requiring the numbers in and on the boxes to start WITH ZERO instead of 1 -- just as the subsets must start with the empty set. If you do this then you can pose the problem (as you so ignorantly tried to do in your previous proof) for 5 boxes labeled 0,1,2,3 and 4, with a sublist of the 5 numbers 0,1,2,3 and 4 in each box. Obviously, since there are 32 subsets and only 5 boxes, the subset of numbers n for which the-box-with-number-n-does-NOT-contain--number-n-on-its- list will be one of the 27 lists-NOT-in-a-box AND NOT one of the 5 lists-IN-a-box. Going up to infinity (or up to the lowest infinity) SIMPLY MAKES NO DIFFERENCE. THE SAME argument goes through in EXACTLY the same way. In neither case does it prove the existence of "bigger" numbers (that has to be proven some OTHER way). It just proves that the set of subsets is too big for the-number-of- boxes to be its size. Because if it could be, you could fit EVERY subset into its own box, SOMEhow. This theorem proves that you can't do that, NOhow.
From: Daryl McCullough on 3 Jun 2010 17:15 George Greene says... > >On Jun 3, 4:08=A0pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: >> You really are full of sh1t then! > > Of course he is, but look who's talking. >You are the one who is too stupid to understand the proof of Cantor's >theorem. >The most important thing you need to understand about Cantor's theorem >is that it has >ABSOLUTELY NOTHING WHATSOEVER TO DO WITH INFINITY. In my opinion, what's counter-intuitive about Cantor's results is not that the reals have a different "size" than the naturals, it is that the naturals have the *SAME* size as the rationals. Generalizing from our experience with finite sets, I think most people would expect that if a set A is a proper subset of set B, then A is smaller than B. With Cantor's suggestion to use the existence of a bijection as the definition of two sets having the "same size", it's possible for a proper subset to be the same size as the full set. That's surprising, if you have never studied set theory. The fact that the reals have a larger size than the naturals is *not* surprising---that's what people would expect, if Cantor hadn't shown that a subset can have the same size as the full set. -- Daryl McCullough Ithaca, NY
From: George Greene on 3 Jun 2010 19:19 On Jun 3, 5:15 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Generalizing from our experience with finite sets, I think most people > would expect that if a set A is a proper subset of set B, then A is > smaller than B. With Cantor's suggestion to use the existence of a > bijection as the definition of two sets having the "same size", it's > possible for a proper subset to be the same size as the full set. That's > surprising, if you have never studied set theory. Well, of course, but you don't have to invoke rationals or reals to show that; you just either 1) take 0 out of the set of naturals, or 2) look at the image of the naturals under successor (which is injective). 1) gets you the proper subset and 2) gets you the bijection between it and the originals. THAT surprise is best assimilated via a collection of tales about "the hotel infinity". > The fact that the reals have a larger size than the naturals is *not* > surprising---that's what people would expect, if Cantor hadn't shown > that a subset can have the same size as the full set. So NOW you are blaming Cantor BOTH for having a confusing proof AND for having sown the prior seeds of people's confusion about his own proof?!?? HERETIC! (Leper, outcast, unclean, one is tempted to further lament). I say we gotta keep'em separated. The point I am still trying to re-stress is that Cantor's theorem Simply Has NOTHING to do with infinity. The proof just doesn't care. Cantor's theorem holds FOR ANY set, PERIOD. EVERY set is SMALLER than its powerset. Whether it is or is not also "paradoxically not bigger" than a proper subset of itself (because it can be bijected with one) IS NOT relevant. Those are NOT ANY of the bijections that ARE being contemplated in the proof. They're just irrelevant. The proper-subset bijections (and infinity) are just ANOTHER topic. Right now, in this thread, we really are dealing with Herc's objections specifically, which have to do with (regardless of what has happened in the first n-1 lines of the proof) the jump to hyperspace, namely, the conclusion at the end of the proof that a higher infinity must exist. This is actually NOT proven by the theorem. To get the existence of an actual larger NUMBER for the size of the set of subsets, you have to invoke something else. In the case of the finite sets it would be something like pairing, which you would use, essentially, to prove that every set has a successor that is also a set (you would then define some subclass of sets-in-general as ordinal numbers, which, being sets, ALSO always have successors, and those successors are always ordinals too); meanwhile, in the case of an infinite set, you have to invoke the axiom of infinity (to get that N and the collection of its subsets are both sets) and the axiom of choice (to get that p(N) has a cardinality AT ALL -- and even then you still have no idea, thanks to the independence of the continuum hypothesis, WHAT that cardinality might BE -- you just know that it's "bigger"). So it's sort of no wonder that Herc doesn't think that the leap to "therefore hyper-infinity exists" is justified. Unfortunately, his permanent habit of jumping to strawman conclusions is continuing to prevent him from taking the relevant baby steps along the way.
From: dannas on 3 Jun 2010 19:22
"George Greene" <greeneg(a)email.unc.edu> wrote in message news:86a1f680-00d8-41da-9c7c-e14965762706(a)k39g2000yqd.googlegroups.com... On Jun 3, 10:50 am, "dannas" <inva...(a)invalid.com> wrote: > those are just assumptions that you have made, which change the problem. >Liar. you lie! > The fact that you cannot read does not mean that I am "making >assumptions". ditto, (WIKI "DITTO") >In any case, the OP is NOT an authority on the problem. The problem >long predates him. says you. He stated his problem clearly, YOU modified it into your problem. WRONG!! >He has some serious misunderstandings about the problem and is >groping. NSS. > the OP did NOT say natural numbers were written on them, >Liar. you lie again!! shame on you! >He said, >"and the boxes have a unique number written on them." which includes all other numbers as he did not specify "Natural" Numbers, by this he also included Complex numbers. >Unique means one of a kind. It means that each box has a different >number you use wiki well. but you have a weak brain. >on it. Or, at least, that's how HE meant it. I concede that other >interpretations You don't know how HE meant it. It is above your pay grade. >may be possible, but not that you actually made one. > and he did NOT say that the number inside the box was Unique. >I didn't either, DUMBASS. >There IS NO "the" number inside the box! He says there is, can't you read? >THE number is ON THE OUTSIDE of the box, as a label, or, as the OP >said, >"written on them". >What is INside each and every box is a ("unique", i.e., different for >each one) SUBSET >of the naturals! That is not what he said at all, he said, "with fridge magnets in the boxes that are any natural number" Try reading each word next time, it may sink in better that way. A Good Learning Lesson for you! |