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 Re: hehe, Infinity Posted By: Forrest of B.org Date: 12/15/06 1:15 p.m. In Response To: Re: hehe, Infinity (treellama) : You can count rational numbers, the same as you can count natural numbers, : just have to do it in the right order: 1/1, 1/2, 2/1, 3/1, 2/2, 1/3, 1/4, : 2/3, 3/2, 4/1, 5/1, etc. Right, I got that, but that's the proof that I didn't find convincing. Why does both being countable mean that they have the same cardinal value? It seems like there should be exponentially more rational numbers than integers... that the cardinality of set of rationals should be the square of the cardinality of set of integers. Given there are n integers, there should be n rational numbers with any given integer as it's numerator or denominator; or in other words, there should be as many rationals on the number line between any two integers, as there are integers. The countability proof was shown to me graphically, as a grid of the natural numbers over the natural numbers, and they counted diagonally from 1/1 on down like you did. But it seems that, since a 3x3 grid has 3^2=9 boxes, and a 4x4 grid had 4^2=16 boxes, and so on, an aleph-null-by-aleph-null grid should have (aleph-null)^2 boxes. For that matter, shouldn't the set of natural numbers be smaller than the set set of integers? And shouldn't the set of prime numbers be smaller than that? I mean, they're all going to be "infinity" but we're already talking different values of "infinity" so what's the problem there? I'd think that aleph-null would be the cardinality of the set of natural numbers, or maybe the primes, and everything else would be based around that. I guess I just don't see why both being countable automatically makes two sets have the same cardinality. Is there only one countable cardinality? I gather the answer is supposedly "yes", but why? Is there only one uncountable cardinality? I gather the answer is supposedly "no", but why not? Maybe I'm just not understanding something about the concept of countability.
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 Replies:

 The Garden of Forking Paths Document 11/29/06 8:50 p.m. Re: The Garden of Forking Paths Forrest of B.org 11/29/06 10:11 p.m. *sniff* Duality *sniff *NM* treellama 11/30/06 2:51 a.m. That's not fair. RyokoTK 11/30/06 5:26 a.m. Re: That's not fair. McNutcase 11/30/06 5:40 a.m. Re: That's not fair. RyokoTK 11/30/06 8:51 a.m. Re: That's not fair. McNutcase 11/30/06 9:25 a.m. Re: That's not fair. RyokoTK 11/30/06 9:59 a.m. Re: That's not fair. D-M.A. 11/30/06 10:05 a.m. Re: That's not fair. RyokoTK 11/30/06 10:22 a.m. Ahaa, I see what you mean now, point taken. *NM* D-M.A. 11/30/06 10:33 a.m. define "well" MrHen 11/30/06 10:24 a.m. Re: define "well" RyokoTK 11/30/06 11:31 a.m. Re: define "well" Aaron Sikes 11/30/06 12:19 p.m. Re: define "well" Forrest of B.org 11/30/06 1:26 p.m. Re: define "well" Aaron Sikes 12/1/06 6:01 a.m. Mmm... House of Leaves MrHen 11/30/06 8:00 a.m. Re: The Garden of Forking Paths Vid Boi 11/30/06 8:13 a.m. Re: The Garden of Forking Paths sdwoodchuck 11/30/06 12:39 p.m. So, what was your conclusion? *NM* Frungi 11/30/06 3:57 p.m. Re: So, what was your conclusion? sdwoodchuck 11/30/06 6:18 p.m. in your theory, the dreams... MrHen 12/1/06 4:44 a.m. Re: in your theory, the dreams... thermoplyae 12/1/06 6:42 a.m. Re: So, what was your conclusion? Frungi 12/4/06 6:31 p.m. Re: So, what was your conclusion? Forrest of B.org 12/4/06 9:07 p.m. Time Travel and the Psychology of Gods Forrest of B.org 12/4/06 9:25 p.m. Re: Time Travel and the Psychology of Gods Frungi 12/5/06 8:46 a.m. Re: Time Travel and the Psychology of Gods Forrest of B.org 12/5/06 4:29 p.m. heck, I would buy 'em MrHen 12/5/06 6:39 p.m. Philosophy anyone? Icarus 12/6/06 8:29 a.m. Re: Philosophy anyone? Forrest of B.org 12/6/06 10:46 a.m. Re: The Garden of Forking Paths *LINK* Hamish Sinclair 12/2/06 5:06 a.m. Re: The Garden of Forking Paths Document 12/2/06 6:17 p.m. Re: The Garden of Forking Paths Document 12/4/06 7:03 p.m. Official Bungie Canon? Shoeless 12/7/06 7:00 a.m. Re: Official Bungie Canon? Bob-B-Q 12/7/06 10:14 a.m. Define "all" MrHen 12/7/06 10:20 a.m. Re: Define "all" Document 12/7/06 4:20 p.m. Re: Define "all" Document 12/7/06 4:21 p.m. Re: Official Bungie Canon? Chris Biberstein 12/11/06 10:28 a.m. Re: Official Bungie Canon? Shoeless 12/11/06 12:00 p.m. Re: Official Bungie Canon? Forrest of B.org 12/11/06 12:03 p.m. Re: Official Bungie Canon? MrHen 12/11/06 1:45 p.m. Re: Official Bungie Canon? *LINK* Frungi 12/11/06 3:22 p.m. uh, thanks... MrHen 12/11/06 6:01 p.m. Re: Official Bungie Canon? Forrest of B.org 12/11/06 9:44 p.m. I like being confused... MrHen 12/12/06 5:13 a.m. Re: I like being confused... Forrest of B.org 12/12/06 3:53 p.m. Re: I like being confused... Frungi 12/12/06 5:51 p.m. Re: I like being confused... Forrest of B.org 12/12/06 9:42 p.m. timelines and their glory MrHen 12/13/06 5:14 a.m. Re: timelines and their glory Forrest of B.org 12/13/06 8:05 a.m. Mmm... trippy... MrHen 12/13/06 8:20 a.m. Re: timelines and their glory Frungi 12/13/06 1:43 p.m. Re: Official Bungie Canon? Chris Biberstein 12/12/06 12:37 p.m. Re: Official Bungie Canon? Forrest of B.org 12/12/06 4:02 p.m. Re: Official Bungie Canon? Chris Biberstein 12/12/06 4:43 p.m. Re: Official Bungie Canon? Forrest of B.org 12/12/06 9:52 p.m. Re: Official Bungie Canon? Chris Biberstein 12/13/06 4:17 p.m. Re: Official Bungie Canon? Forrest of B.org 12/13/06 6:06 p.m. Re: Official Bungie Canon? Chris Biberstein 12/16/06 5:42 p.m. what? why? MrHen 12/16/06 5:47 p.m. Re: Official Bungie Canon? kyjel 12/16/06 6:40 p.m. Re: Official Bungie Canon? Shoeless 12/17/06 4:13 p.m. Re: Official Bungie Canon? Frungi 12/18/06 5:09 p.m. Re: Official Bungie Canon? Chris Biberstein 12/18/06 8:47 p.m. Re: Official Bungie Canon? Frungi 12/18/06 9:22 p.m. Re: Official Bungie Canon? Chris Biberstein 12/22/06 9:58 a.m. questions and answers MrHen 12/22/06 12:44 p.m. Re: Official Bungie Canon? Shoeless 12/22/06 1:24 p.m. Re: Official Bungie Canon? Shoeless 12/18/06 10:21 p.m. rabbit trail, sorry... MrHen 12/13/06 5:23 a.m. Re: rabbit trail, sorry... Chris Biberstein 12/13/06 4:29 p.m. and the problem was... where? MrHen 12/13/06 7:06 p.m. Re: and the problem was... where? Forrest of B.org 12/13/06 9:13 p.m. ah, my bad. I understand. :) *NM* MrHen 12/14/06 4:54 a.m. Re: ah, my bad. I understand. :) Forrest of B.org 12/14/06 1:22 p.m. I am the same way. ;) MrHen 12/15/06 5:18 a.m. Re: I am the same way. ;) Forrest of B.org 12/15/06 7:27 a.m. Branching MrHen 12/15/06 9:46 a.m. Re: Branching Forrest of B.org 12/15/06 11:04 a.m. Actually, I think I did understand. MrHen 12/15/06 4:32 p.m. Re: Actually, I think I did understand. Forrest of B.org 12/17/06 11:55 a.m. Oh, okay, then we do disagree. MrHen 12/17/06 6:50 p.m. Re: Oh, okay, then we do disagree. Forrest of B.org 12/17/06 10:08 p.m. so where do you get hyper-time? MrHen 12/18/06 5:13 a.m. Re: so where do you get hyper-time? Forrest of B.org 12/18/06 8:20 a.m. whoops... no... that is not what I meant. MrHen 12/18/06 9:49 a.m. Re: whoops... no... that is not what I meant. Forrest of B.org 12/18/06 1:58 p.m. right, yeah, that is the identity problem MrHen 12/18/06 4:41 p.m. Re: right, yeah, that is the identity problem Forrest of B.org 12/18/06 9:55 p.m. ohhh... MrHen 12/19/06 5:44 a.m. Re: ohhh... Forrest of B.org 12/19/06 9:20 a.m. back to the math ;) MrHen 12/19/06 11:07 a.m. Re: back to the math ;) Forrest of B.org 12/19/06 1:36 p.m. wait, so my model is too... real? ;) MrHen 12/22/06 1:57 p.m. Sort of. Forrest of B.org 12/22/06 3:22 p.m. and the light turns on... MrHen 1/7/07 4:41 p.m. Re: and the light turns on... Forrest of B.org 1/8/07 8:12 p.m. Re: Official Bungie Canon? Shoeless 12/13/06 8:11 a.m. Re: Official Bungie Canon? Chris Biberstein 12/13/06 4:34 p.m. Re: Official Bungie Canon? Shoeless 12/14/06 8:01 a.m. Re: Official Bungie Canon? Frungi 12/15/06 8:02 a.m. Re: Official Bungie Canon? Forrest of B.org 12/15/06 8:51 a.m. Re: Official Bungie Canon? Frungi 12/15/06 9:22 a.m. hehe, Infinity MrHen 12/15/06 9:51 a.m. Re: hehe, Infinity Forrest of B.org 12/15/06 11:16 a.m. Re: hehe, Infinity treellama 12/15/06 12:18 p.m. Re: hehe, Infinity Forrest of B.org 12/15/06 1:15 p.m. Re: hehe, Infinity treellama 12/15/06 2:09 p.m. Re: hehe, Infinity Forrest of B.org 12/15/06 3:15 p.m. Re: hehe, Infinity treellama 12/15/06 4:12 p.m. Re: hehe, Infinity Frungi 12/15/06 6:34 p.m. Re: hehe, Infinity treellama 12/16/06 3:38 a.m. So... do I have this right? MrHen 12/16/06 6:35 a.m. Re: So... do I have this right? treellama 12/16/06 11:25 a.m. Re: hehe, Infinity treellama 12/15/06 2:21 p.m. Re: Official Bungie Canon? treellama 12/15/06 9:55 a.m. Re: Official Bungie Canon? McNutcase 12/15/06 11:12 a.m. Re: Official Bungie Canon? Forrest of B.org 12/15/06 1:20 p.m. Re: Official Bungie Canon? McNutcase 12/15/06 9:32 a.m. Re: Official Bungie Canon? MrHen 12/15/06 9:49 a.m. Re: Official Bungie Canon? treellama 12/15/06 9:56 a.m. Re: Official Bungie Canon? McNutcase 12/15/06 11:09 a.m. Re: Official Bungie Canon? Forrest of B.org 12/15/06 1:34 p.m. Re: Official Bungie Canon? Forrest of B.org 12/15/06 1:39 p.m. Re: Official Bungie Canon? ukimalefu 12/15/06 6:29 p.m. Re: Official Bungie Canon? McNutcase 12/15/06 10:23 p.m. awesome, thanks! *NM* MrHen 12/16/06 6:49 a.m. Re: Official Bungie Canon? Chris Biberstein 12/16/06 5:51 p.m. Re: Official Bungie Canon? kyjel 12/16/06 7:08 p.m. Re: Official Bungie Canon? Chris Biberstein 12/18/06 8:36 p.m. Re: Official Bungie Canon? Frungi 12/18/06 9:48 p.m. Re: Official Bungie Canon? Chris Biberstein 12/22/06 10:00 a.m. Re: Official Bungie Canon? Forrest of B.org 12/22/06 10:19 a.m. what he said MrHen 12/22/06 12:38 p.m. Re: Official Bungie Canon? Shoeless 12/18/06 10:23 p.m. *sigh* MrHen 12/17/06 5:08 a.m. Re: Official Bungie Canon? Frungi 12/11/06 3:35 p.m. Re: Official Bungie Canon? Chris Biberstein 12/12/06 12:23 p.m. Re: Official Bungie Canon? Shoeless 12/13/06 7:56 a.m. Re: Official Bungie Canon? Forrest of B.org 12/13/06 8:32 a.m. another example (albeit overused) MrHen 12/13/06 8:57 a.m. Re: The Garden of Forking Paths *LINK* irons 2/23/18 1:11 a.m. LOKE *NM* W'rkncacnter 2/23/18 4:06 p.m. Re: LOKE *NM* *LINK* irons 2/23/18 4:14 p.m. LOKE *NM* *NM* *NM* *NM* *NM* *LINK* W'rkncacnter 2/23/18 11:09 p.m. Re: LOKE *NM* *LINK* irons 2/24/18 3:31 a.m.

 [ Post a Reply | Message Index | Read Prev Msg | Read Next Msg ]Pre-2004 Posts

 Message: : Right, I got that, but that's the proof that I didn't find convincing. Why : does both being countable mean that they have the same cardinal value? It : seems like there should be exponentially more rational numbers than : integers... that the cardinality of set of rationals should be the square : of the cardinality of set of integers. Given there are n integers, there : should be n rational numbers with any given integer as it's numerator or : denominator; or in other words, there should be as many rationals on the : number line between any two integers, as there are integers. The : countability proof was shown to me graphically, as a grid of the natural : numbers over the natural numbers, and they counted diagonally from 1/1 on : down like you did. But it seems that, since a 3x3 grid has 3^2=9 boxes, : and a 4x4 grid had 4^2=16 boxes, and so on, an aleph-null-by-aleph-null : grid should have (aleph-null)^2 boxes. : For that matter, shouldn't the set of natural numbers be smaller than the set : set of integers? And shouldn't the set of prime numbers be smaller than : that? I mean, they're all going to be "infinity" but we're : already talking different values of "infinity" so what's the : problem there? I'd think that aleph-null would be the cardinality of the : set of natural numbers, or maybe the primes, and everything else would be : based around that. : I guess I just don't see why both being countable automatically makes two : sets have the same cardinality. Is there only one countable cardinality? I : gather the answer is supposedly "yes", but why? Is there only : one uncountable cardinality? I gather the answer is supposedly : "no", but why not? : Maybe I'm just not understanding something about the concept of countability. If you'd like to include a link to another page with your message,please provide both the URL address and the title of the page: If necessary, enter your password below:

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