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

You are not logged in.

- Topics: Active | Unanswered

**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

In one of the other threads, it was said that the number of possible states of the Universe was found by the amount of particles in the Universe multiplied by the amount of time that the Universe has existed for.

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..."

Offline

**Laterally Speaking****Real Member**- Registered: 2007-05-21
- Posts: 356

Basically, 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"

Offline

**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

That 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..."

Offline

**Laterally Speaking****Real Member**- Registered: 2007-05-21
- Posts: 356

Actually, 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"

Offline

**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 13,154

pico-seconds?????????????????????

Character is who you are when no one is looking.

Offline

**ben****Member**- Registered: 2006-07-12
- Posts: 106

Laterally Speaking wrote:

Basically, that number is pretty much infinite, being that time can be divided into infinitely small units.

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...

Ricky wrote:

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

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?

Offline

**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

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...

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.

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?

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.

So for example, if I had a box 5 inches to a side which I could only place 1 inch cubes in an exact number of inches from a side, there would be 125! different states of that box. If the universe is discrete like my box, then there are a finite number of possible states.

This of course is assuming the universe is not infinite, in which case it is obvious there is an infinite amount of states.

"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..."

Offline

**Laterally Speaking****Real Member**- Registered: 2007-05-21
- Posts: 356

Here's what you could say about the number in question, in order to calculate it:

If time is discrete, rather than continuous, which, as far as I know it isn't, then your fist factor is countable. If not, it isn't.

If the universe is infinite, or is continuous, or both, then your second factor is infinite. If the universe is both finite AND discrete, then that second factor is not infinite.

It's pretty obvious from this that the number in question is infinite.

*Last edited by Laterally Speaking (2007-06-12 22:16:55)*

"Knowledge is directly proportional to the amount of equipment ruined."

"This woman painted a picture of me; she was clearly a psychopath"

Offline

**Laterally Speaking****Real Member**- Registered: 2007-05-21
- Posts: 356

The thing that allows us to say that the number of combined circumstances leading up to any point in time is the following:

CHAOS THEORY

Chaos theory states that a small change will, over time, create a very large one in any iterative system. It can be said that evolution, breeding, and quite a few other things that directly or indirectly affect us are iterative systems. Also, you can see that very distant things can have big impacts on these systems.

The observation of some particular phenomenon on the other side of the galaxy which saves the life of the person who observed it, thus creating a slight change in the gene pool. This alteration might seem very minor, but, after a few generations, there might be a few traits that appear that the astronomer, long since dead, had. These traits could have some repercussions, which would have more, and so on.

This whole discussion was probably a bit astray from the main subject.

"This woman painted a picture of me; she was clearly a psychopath"

Offline

**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

If the universe is both finite AND discrete, then that second factor is countable.

Countably finite. Countable implies a bijection to the natural numbers, which in turn implies infinite.

Offline

**ben****Member**- Registered: 2006-07-12
- Posts: 106

Ricky wrote:

Countably finite.

This is weird, Ricky. surely you're not suggesting the possible existence of an "uncountable finite" set?

Countable implies a bijection to the natural numbers, which in turn implies infinite.

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 *sub*set 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.

Oh yes, and leave off the "your perch" bit if you don't mind - I am tolerant, but not infinitely so

*Last edited by ben (2007-06-12 11:08:17)*

Offline

**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

Oh yes, and leave of the "your perch" bit if you don't mind - I am tolerant, but not infinitely so

You know you were projecting when you said, "Surely you don't need me to show you, do you?", and you know it.

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.

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.

Offline

**Laterally Speaking****Real Member**- Registered: 2007-05-21
- Posts: 356

Please, stop arguing. The point is that the number in question is very likely to be infinite

picoseconds??

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...

Here are the prefixes for (relatively small) numbers from 10^18 to 10^-18:

Exa, Peta, Tera, Giga, Mega, Kilo.

Milli, Micro, Nano, Pico, Femto, Atto.

Here are some other ways to compare these:

Comparing a meter to a nanometer is like comparing the circumference of the Earth to that of a marble.

An atom is roughly 10 to 80 nanometers in diameter.

A picosecond is roughly the time between you hitting a key on your keyboard (contact being made) and the computer receiving the signal.

An exameter is over 300000000 times the distance between the Earth and the moon.

*Last edited by Laterally Speaking (2007-06-13 04:31:39)*

"This woman painted a picture of me; she was clearly a psychopath"

Offline

**ben****Member**- Registered: 2006-07-12
- Posts: 106

Ricky wrote:

You know you were projecting when you said, "Surely you don't need me to show you, do you?", and you know it.

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.

Anyway, you got me thinking. Ricky, I know you don't need this little tutorial, but here goes, for general consumption.

There is a load of laxity in language regarding "infinity". Consider the subset A = [a, b] of R. By the definition of [ , ], no element of A is infinite. But by the definition of R (well, it's not really a definition, it's a continuity axiom), the *cardinality* of [a, b] is infinite and uncountably infinite to boot.

See what I mean about laxity?

Offline

**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

Ok, 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.

no element of A is infinite.

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.

But I got to say I'm not quite sure where you're going with this.

But by the definition of R (well, it's not really a definition, it's a continuity axiom), the cardinality of [a, b] is infinite and uncountably infinite to boot.

Completeness gives R is uncountable-ness, 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.

Offline

**Laterally Speaking****Real Member**- Registered: 2007-05-21
- Posts: 356

I 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".

Here is a big number:

x1=10!!!!!!!!!!! (10 factorials)

x2=10!!!!!!!!!!! (x1! factorials)

x3=10!!!!!!!!!!! (x2! factorials)

...

x(10^x1)=10!!!!!!!!!! (x((10^x1)-1)! factorials)

The last step is probably larger than graham's number, being the number of iterations.

"This woman painted a picture of me; she was clearly a psychopath"

Offline

**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 13,154

I doubt that!

Remember, Graham's number grows very rapidly as we move from the first step onwards.

There is reason for me stating this. I had once come up with a number I thought was probably greater than Graham's Number, but the number was more comparable to Moser's. In fact, the number might be lesser than Moser's.

The number I had in my mind was something like

(10^10^10)^^^^^(10^10^10).

Character is who you are when no one is looking.

Offline

**Laterally Speaking****Real Member**- Registered: 2007-05-21
- Posts: 356

I dunno... 10^(10!!!!!!!!!!) iterations loosely similar to the ones used to obtain G has gotta yield a HUGE number.

Here's a number definitely larger than G: 100->100->100->100->100->100->100.

"This woman painted a picture of me; she was clearly a psychopath"

Offline

**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 13,154

This 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.

Offline

**Laterally Speaking****Real Member**- Registered: 2007-05-21
- Posts: 356

Here's another big number, obviously larger than G, because of the number of iterations and the sheer size of the first step alone:

x1=10^^^^^^^^^^10

x2=10^^^...^^^^^^x1 (x1 up-arrows)

...

x(x1)=10^^^...^^^^^^x(x1-1) (x1-1 up-arrows)

"This woman painted a picture of me; she was clearly a psychopath"

Offline

**Laterally Speaking****Real Member**- Registered: 2007-05-21
- Posts: 356

Here is (again) G:

G1=3^^^^3

G2=3^^...^^3 (G1 up-arrows)

...

G64=3^^...^^3 (G63 up-arrows)

Knowing this, you can say for certain that the number I posted just above is bigger than G.

"This woman painted a picture of me; she was clearly a psychopath"

Offline

**Laterally Speaking****Real Member**- Registered: 2007-05-21
- Posts: 356

Here's another:

10!!!...!!! ((10^^^^^^^^^^1000000)!!!...!!! (10! factorials) factorials)

Actually, this might be smaller than G, but it's still enormous.

"This woman painted a picture of me; she was clearly a psychopath"

Offline

**Laterally Speaking****Real Member**- Registered: 2007-05-21
- Posts: 356

I think that it's pretty obvious that if you use both iterations, factorials, Knuth's up-arrow notation, and Conway's chained-arrow notation, you can get absolutely mind-boggling numbers.

"This woman painted a picture of me; she was clearly a psychopath"

Offline

**Laterally Speaking****Real Member**- Registered: 2007-05-21
- Posts: 356

On a loosely related issue, a very small (positive) number can be obtained in the following way: 10^-(any number that has been posted here).

"This woman painted a picture of me; she was clearly a psychopath"

Offline

**henryzz****Member**- Registered: 2007-06-03
- Posts: 14

my number 10!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

work that out!!!

Offline