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 Re: hehe, Infinity Posted By: Forrest of B.org Date: 12/15/06 3:15 p.m. In Response To: Re: hehe, Infinity (treellama) : You would think that, but we're mapping one natural number to one rational : number here; there's a bijection. For each rational number in that list, : there's exactly one natural number that goes with it, and the reverse is : true. And as many as you can ever count of one, there's that many you can : count of the other. So, they're the same cardinality. : Because if you can order and count the members of an infinite set, you can : show there's a bijection between natural numbers, and the members of that : set. I guess my problem with accepting this is that it seems like, for any amount of rational numbers, by the time you've counted them all, you've counted to a natural number higher than any thus far used in any of those rationals. E.g. if you count all rationals constructed of only 1 and 2: 1: 1/1 2: 1/2 3: 2/1 4: 2/2 You now have two new natural numbers, 3 and 4, that could be used to construct a bunch more rationals (namely 1/3, 2/3, 3/3, 4/3, 3/1, 3/2, 3/4, 1/4, 2/4, 4/4, 4/1, and 4/2, in no particular order). So then by the time you've counted all them, you've got natural numbers up to 16, so that's 12 more natural numbers to construct rational numbers out of, after which point you'll have counted up to 256, and so on. So by the time you've finished counting the rational numbers that can be constructed out of the complete set of natural numbers, you must have counted to a number higher than any natural number. Which, now that I say it, sounds like a contradiction, which should mean that you can't ever count that high. Which of course you practically can't since it would take forever to do so no matter how fast you could count, but then that makes me ponder, what exactly is meant by "countable" if not in the hypothetical "...given that we had infinite time to work with" sense, in which sense nothing but the natural numbers themselves seem countable? Just that they can be arranged into an ordered sequence, so that you would know how to even proceed in the incompletable task of trying to count them? But then I don't see how that's supposed to prove that all such countable sets have the same cardinality. Mind you, I'm not asking you, Mr. Music Major, to answer all these things. Just kind of mumbling my confusions out loud. Though if you (or any one else) have answers, please, let me know. Postscript: a picture just popped into my head of two curves, both of which hit an asymptote at the same x-value, but one of which increases much more quickly; so that at x=1, they have only slightly different y values, and x=2, they have bigger differences in y values, and x=3 they have vastly different y values, and so on... but both reach infinity at, say, x=7. Is that even possible? Is that something like what's happening here; even though the cardinality of a subset of rationals is exponentially greater than the cardinality of the subset of naturals from which they're constructed, somehow both sets reach infinity "at the same time"?
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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: : I guess my problem with accepting this is that it seems like, for any amount : of rational numbers, by the time you've counted them all, you've counted : to a natural number higher than any thus far used in any of those : rationals. E.g. if you count all rationals constructed of only 1 and 2: 1: : 1/1 : 2: 1/2 : 3: 2/1 : 4: 2/2 : You now have two new natural numbers, 3 and 4, that could be used to : construct a bunch more rationals (namely 1/3, 2/3, 3/3, 4/3, 3/1, 3/2, : 3/4, 1/4, 2/4, 4/4, 4/1, and 4/2, in no particular order). So then by the : time you've counted all them, you've got natural numbers up to 16, so : that's 12 more natural numbers to construct rational numbers out of, after : which point you'll have counted up to 256, and so on. So by the time : you've finished counting the rational numbers that can be constructed out : of the complete set of natural numbers, you must have counted to a number : higher than any natural number. : Which, now that I say it, sounds like a contradiction, which should mean that : you can't ever count that high. Which of course you practically can't : since it would take forever to do so no matter how fast you could count, : but then that makes me ponder, what exactly is meant by : "countable" if not in the hypothetical "...given that we : had infinite time to work with" sense, in which sense nothing but the : natural numbers themselves seem countable? Just that they can be arranged : into an ordered sequence, so that you would know how to even proceed in : the incompletable task of trying to count them? But then I don't see how : that's supposed to prove that all such countable sets have the same : cardinality. : Mind you, I'm not asking you, Mr. Music Major, to answer all these things. : Just kind of mumbling my confusions out loud. Though if you (or any one : else) have answers, please, let me know. : Postscript: a picture just popped into my head of two curves, both of which : hit an asymptote at the same x-value, but one of which increases much more : quickly; so that at x=1, they have only slightly different y values, and : x=2, they have bigger differences in y values, and x=3 they have vastly : different y values, and so on... but both reach infinity at, say, x=7. Is : that even possible? Is that something like what's happening here; even : though the cardinality of a subset of rationals is exponentially greater : than the cardinality of the subset of naturals from which they're : constructed, somehow both sets reach infinity "at the same : time"? 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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