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.

That assumes that time is continuous as well.

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

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.

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.

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 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.

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 (2007-06-13 09:08:17)

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 (2007-06-14 02:31:39)

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.

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).

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.

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.

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.

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

work that out!!!