Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ ¹ ² ³ °
 

You are not logged in. #26 20070607 07:35:20
Re: Number Giants
If the universe is continuous then there are an infinite amount of states. If it is discrete, then it's still just about impossible to calculate since there are known knowns, known unknowns, and unknown unknowns. We may be able to account for everything but the last, but that still leaves quite a lot. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #27 20070611 15:39:13
Re: Number GiantsBasically, that number is pretty much infinite, being that time can be divided into infinitely small units. "Knowledge is directly proportional to the amount of equipment ruined." "This woman painted a picture of me; she was clearly a psychopath" #28 20070611 15:49:29
Re: Number GiantsThat assumes that time is continuous as well. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #29 20070612 01:53:52
Re: Number GiantsActually, I gave that statement some thought today, and, well, it seems that there may be a point at which the division of time is so short that nothing actually happens during it  like the time it take for one electron to do a complete revolution around its nucleus, though that's still something happening. "Knowledge is directly proportional to the amount of equipment ruined." "This woman painted a picture of me; she was clearly a psychopath" #30 20070612 23:41:55
Re: Number Giantspicoseconds????????????????????? Character is who you are when no one is looking. #31 20070613 02:51:18
Re: Number Giants
Well, well, I see you guys are still tying yourselves in knots. What is "pretty much infinite"? Like 0.999... is "pretty much" 1? Oh dear, oh dear. And...
Think on Cantor's theorem, Ricky. If what you're all calling the "universe" is discrete, then it is countable. If it's not discrete, then it's not countable, by Cantor's thm. Surely you don't need me to show you, do you? #32 20070613 02:57:14
Re: Number Giants
I'm pretty much sure his use of "pretty much" was just stating "it's infinite" with an amount of uncertainty. That is, he did not mean "almost infinity". But this is just interpretation.
No, but you do need to take a step down from your perch and realize you're confusing two very different concepts. Continuous and discrete are not the equivalent of uncountable and countable. Discrete means there is a smallest step. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #33 20070613 03:32:06
Re: Number GiantsHere's what you could say about the number in question, in order to calculate it: Last edited by Laterally Speaking (20070613 20:16:55) "Knowledge is directly proportional to the amount of equipment ruined." "This woman painted a picture of me; she was clearly a psychopath" #34 20070613 03:45:50
Re: Number GiantsThe thing that allows us to say that the number of combined circumstances leading up to any point in time is the following: "Knowledge is directly proportional to the amount of equipment ruined." "This woman painted a picture of me; she was clearly a psychopath" #35 20070613 06:30:36
Re: Number Giants
Countably finite. Countable implies a bijection to the natural numbers, which in turn implies infinite. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #36 20070613 08:35:36
Re: Number Giants
This is weird, Ricky. surely you're not suggesting the possible existence of an "uncountable finite" set?
No, this is wrong. Why do you think this? All finite sets are countable, almost by definition. It's true that a set is said to be countable if there is a bijection to a subset of N, and as N is "countably infinite" again by definition, and as N is always a subset of N, the bijection you refer to may or may not imply a set is countably infinite, it could easily be finite (we don't need the countably bit for finite sets). But it is most certainly not the case that countable implies infinite. Last edited by ben (20070613 09:08:17) #37 20070613 09:07:07
Re: Number Giants
You know you were projecting when you said, "Surely you don't need me to show you, do you?", and you know it.
There are two different ways to define countable. I choose the way I've used because it explicitly identifies that finite sets are of a different cardinality than the natural numbers, which I like. After doing some searching, it does seem though that your definition is vastly more common. The book I first got my version from came out of India. Not sure if that has something to do with it. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #38 20070613 15:53:07
Re: Number GiantsPlease, stop arguing. The point is that the number in question is very likely to be infinite
No, something more like an attosecond which is 10^18 seconds. To give you an idea, an attometer is roughly the size of a quark... Last edited by Laterally Speaking (20070614 02:31:39) "Knowledge is directly proportional to the amount of equipment ruined." "This woman painted a picture of me; she was clearly a psychopath" #39 20070614 01:58:49
Re: Number Giants
I might if I knew what "projecting" means. Obviously, by your tone, it's a negative thing, but in fact, my comment was intended as flattery  as in, of course you know Cantor's thm. #40 20070614 02:45:18
Re: Number GiantsOk, in such a case I'm sorry, I jumped the gun. What I meant by projecting was that you were acting more so as "instructor" rather than having a discussion. But perhaps I misread.
Infinite has a whole bunch of different meanings. For example, all elements of A are infinite sets (every real number is). But their magnitude is of course finite.
Completeness gives R is uncountableness, and it can be proven using the construction of R from the rationals. I looked up continuity axiom and it seems to do with Euclidean space and circles, or Archimedes and the rationals. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #41 20070614 04:00:48
Re: Number GiantsI think we're all straying from the subject at hand: Number Giants. What we're discussing here would be better described as "Problems with the communication of people". "Knowledge is directly proportional to the amount of equipment ruined." "This woman painted a picture of me; she was clearly a psychopath" #42 20070614 23:58:41
Re: Number GiantsI doubt that! Character is who you are when no one is looking. #43 20070615 03:33:22
Re: Number GiantsI dunno... 10^(10!!!!!!!!!!) iterations loosely similar to the ones used to obtain G has gotta yield a HUGE number. "Knowledge is directly proportional to the amount of equipment ruined." "This woman painted a picture of me; she was clearly a psychopath" #44 20070616 00:08:56
Re: Number GiantsThis time you got it right! Certainly, the number you got using John Conway's chained arrow notation is greater than G. I think G is expressed as between 3>3>64>2 and 3>3>65>2. Character is who you are when no one is looking. #45 20070616 02:04:37
Re: Number GiantsHere's another big number, obviously larger than G, because of the number of iterations and the sheer size of the first step alone: "Knowledge is directly proportional to the amount of equipment ruined." "This woman painted a picture of me; she was clearly a psychopath" #46 20070617 05:51:46
Re: Number GiantsHere is (again) G: "Knowledge is directly proportional to the amount of equipment ruined." "This woman painted a picture of me; she was clearly a psychopath" #47 20070620 07:35:14
Re: Number GiantsHere's another: "Knowledge is directly proportional to the amount of equipment ruined." "This woman painted a picture of me; she was clearly a psychopath" #48 20070622 02:19:06
Re: Number GiantsI think that it's pretty obvious that if you use both iterations, factorials, Knuth's uparrow notation, and Conway's chainedarrow notation, you can get absolutely mindboggling numbers. "Knowledge is directly proportional to the amount of equipment ruined." "This woman painted a picture of me; she was clearly a psychopath" #49 20070623 04:42:25
Re: Number GiantsOn a loosely related issue, a very small (positive) number can be obtained in the following way: 10^(any number that has been posted here). "Knowledge is directly proportional to the amount of equipment ruined." "This woman painted a picture of me; she was clearly a psychopath" #50 20070623 04:49:21
Re: Number Giantsmy number 10!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 