Re: hehe, Infinity


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Re: hehe, Infinity

Posted By: treellama Date: 12/15/06 2:09 p.m.
In Response To: Re: hehe, Infinity (Forrest of B.org)

: 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.
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.
: 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.
That holds true for any finite number n you can conceive, but we're talking about the cardinality of the entire infinite set. There's still exactly one natural number for each rational and vice versa, no matter how high you count, using that mapping.
: 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?
It's trivial to order and count the primes:
2, 3, 5, 7
1, 2, 3, 4
So, you can set it up so that for each prime number, there's one and only one natural number that corresponds. Similarly, for integers
0, 1, -1, 2, -2
1, 2, 3, 4, 5
So, all three sets have the same cardinality
: 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.
Aleph null is the cardinality of the sets of all naturals, integers, rationals, and primes, since they all have the same cardinality.
: 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?
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.
: Maybe I'm just not understanding something about the concept of countability.
Maybe not :)

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Pre-2004 Posts

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

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: 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. : That holds true for any finite number n you can conceive, but we're talking : about the cardinality of the entire infinite set. There's still exactly : one natural number for each rational and vice versa, no matter how high : you count, using that mapping. : It's trivial to order and count the primes: 2, 3, 5, 7 : 1, 2, 3, 4 : So, you can set it up so that for each prime number, there's one and only one : natural number that corresponds. Similarly, for integers : 0, 1, -1, 2, -2 : 1, 2, 3, 4, 5 : So, all three sets have the same cardinality : Aleph null is the cardinality of the sets of all naturals, integers, : rationals, and primes, since they all have 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. : Maybe not :)

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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.



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	Message: : 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. : That holds true for any finite number n you can conceive, but we're talking : about the cardinality of the entire infinite set. There's still exactly : one natural number for each rational and vice versa, no matter how high : you count, using that mapping. : It's trivial to order and count the primes: 2, 3, 5, 7 : 1, 2, 3, 4 : So, you can set it up so that for each prime number, there's one and only one : natural number that corresponds. Similarly, for integers : 0, 1, -1, 2, -2 : 1, 2, 3, 4, 5 : So, all three sets have the same cardinality : Aleph null is the cardinality of the sets of all naturals, integers, : rationals, and primes, since they all have 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. : Maybe not :)
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