A very large number that has all of these qualities?

[Mentallic]

No, I know. I know he was right. Like what would you say x should be in the power tower? Does 10^{10^{10^{10^7}}} seem too big, or about right?

Well, 10⁷ is 10 million, then 10^10,000,000 is the number 1 followed by 10 million zeroes. Then 10 to the power of that is again a number beginning with 1 and ending with (that number with a million zeroes) number of zeroes, then we take 10 to the power of that once more.

Clearly bigger than anything we can conceptualize.

 

[thefinalson]

Clearly bigger than anything we can conceptualize.

Yes, clearly bigger than anything we can conceptualize. But is it bigger than the number of universes there are probable to be, assuming that everything that can happen, will happen in at least one of those universes?

Also, was I correct or incorrect in my little theory about Graham's number, wikipedia pages, and universes? I'm just curious because I thought it was pretty clever, and I'd like to know whether I was right or wrong lol.

 

[mikeph]

how about g_(first g_64 digits of the golden ratio), using the iteration of Graham's number.

I think we can agree that is unimaginably larger than Graham's number, and uses the golden ratio

 

[GenePeer]

If everything that can happen will happen, then there are an infinite number of universes.

Assuming all universes were identical and the only difference is when someone was asked to think of an integer n, a positive real number less than 1, or one thing in an infinite set. If in all the universes, all the possible choices were taken, then there must be an infinite number of universes as well.

 

[agentredlum]

Using arrow notation GN is not that big because GN is in between 3-->3-->64-->2 and 3-->3-->65-->2. As a matter of fact, abstractly, almost ALL numbers are bigger than GN. To put it another way, the probability of picking a number less than GN is zero, the probability of picking a number greater than GN is one.

 

[mikeph]

If everything that can happen will happen, then there are an infinite number of universes.

Assuming all universes were identical and the only difference is when someone was asked to think of an integer n, a positive real number less than 1, or one thing in an infinite set. If in all the universes, all the possible choices were taken, then there must be an infinite number of universes as well.

It takes a finite amount of time to say/think a number, so the set of possible numbers any human (or anything) can randomly generate (can a human randomly generate a number) before the universe ends. I'd guess at 1 digit per second we are limited to maybe 10^20 digits?

 

[GenePeer]

It takes a finite amount of time to say/think a number, so the set of possible numbers any human (or anything) can randomly generate (can a human randomly generate a number) before the universe ends. I'd guess at 1 digit per second we are limited to maybe 10^20 digits?

Then I fail

 

[thefinalson]

It takes a finite amount of time to say/think a number, so the set of possible numbers any human (or anything) can randomly generate (can a human randomly generate a number) before the universe ends. I'd guess at 1 digit per second we are limited to maybe 10^20 digits?

So would that mean there's 10²⁰ universes in the multiverse? Yeah, that definitely sounds way too low. I am still curious if my wikipedia theory was correct or not.

To be perfectly honest, I really don't know exactly what I mean by this either, lol. I'm just spitballing ideas here and seeing if anything sticks. I think what I mean by the "outcome of Graham's number" is actually the possible outcomes of Graham's problem, of which Graham's number is one. "Consider an n-dimensional hypercube, and connect each pair of vertices to obtain a complete graph on 2ⁿ vertices. Then color each of the edges of this graph either red or blue. What is the smallest value of n for which every such coloring contains at least one single-colored 4-vertex planar complete subgraph?" That's Graham's problem.

The upper bound for the value of n, of course, we all know is Graham's number. The lower bound is currently said to be 13. So the range of all possible outcomes for what n equals, would be, I believe (and you can correct me if I'm wrong on this), GN minus 13. Which we can of course round off to GN, since obviously if multiplying it by the number of atoms in the entire universe doesn't noticeably affect it, subtracting 13 sure isn't going to, either.

So we have basically Graham's number of possible values for n. Now let's look at the wikipedia article for Graham's number. http://en.wikipedia.org/wiki/Graham's_number It contains a graphical representation of one of these possible solutions, a 2-colored 3-dimensional cube containing one single-colored 4-vertex planar complete subgraph. The subgraph is shown below the cube. Now, this is just one of an \approx Graham's number of examples they could show, is it not? So if the wikipedia page for GN is showing this exact example in our universe, and in another universe it shows another, and in another it shows another and so on, then doesn't there have to be at least Graham's number of universes, just for wikipedia pages about GN lol?

Please correct me if I'm missing something.

Yea or nay?

 

[mikeph]

So would that mean there's 10²⁰ universes in the multiverse? Yeah, that definitely sounds way too low. I am still curious if my wikipedia theory was correct or not.

No, that's just an example of finite possibilities in a finite universe. You could ask 6 billion people to think of random numbers and then you have (10^80)^(10^9) possible results. Asking someone to say a number is a very small sample of the overall amount of information in a universe, but in my opinion, the amount will still be finite (or at least finitely obtainable due to limitations on light travel speed / size of observable universe). But it's a bit of a digression.

Wikipedia- I can't see why any species in any time would attempt to demonstrate a 130923-dimensional illustration of this problem on their wikipedia page! Assuming the universes have the same dimension. But I do get your point in that there are perhaps a very (perhaphs unknowably) large number of possibilities. Or perhaps there is only one possibility.

 

[GenePeer]

No, that's just an example of finite possibilities in a finite universe. You could ask 6 billion people to think of random numbers and then you have (10^80)^(10^9) possible results. Asking someone to say a number is a very small sample of the overall amount of information in a universe, but in my opinion, the amount will still be finite (or at least finitely obtainable due to limitations on light travel speed / size of observable universe). But it's a bit of a digression.

So my first example wasn't perfect, but are you implying that time and distances are also discrete as well?

If in one universe an atom decays at 00:00 and in another universe the atom decays at 00:01. Shouldn't there be infinite universes where it decays somewhere in between? We probably can't measure the time accurately enough to always note the difference, but there should be a difference, right?

 

[thefinalson]

No, that's just an example of finite possibilities in a finite universe. You could ask 6 billion people to think of random numbers and then you have (10^80)^(10^9) possible results. Asking someone to say a number is a very small sample of the overall amount of information in a universe, but in my opinion, the amount will still be finite (or at least finitely obtainable due to limitations on light travel speed / size of observable universe). But it's a bit of a digression.

Wikipedia- I can't see why any species in any time would attempt to demonstrate a 130923-dimensional illustration of this problem on their wikipedia page! Assuming the universes have the same dimension. But I do get your point in that there are perhaps a very (perhaphs unknowably) large number of possibilities. Or perhaps there is only one possibility.

Right. There could be just the one universe, or an infinite number of universes, or any finite amount in-between. My book is, in large part, about characters who travel around between universes, so it would be hard to tell that story if there was only one universe. While it might seem unlikely, it's possible (and therefore, in my hypothetical multiverse, inevitable) that some civilization in some universe will attempt to demonstrate a 130923-dimensional illustration of this problem on their wikipedia page (remembering also that different universes have different numbers of spatial dimensions in my multiverse).

I'm just going to arbitrarily say that my multiverse has 2 → 3 → 5 → 8 → 13 → 21 → 34 → 55 → 89 → 144 → 233 → 377 universes in it. Maybe in real life it has a lot less, or a lot more, but no one can really know for sure either way, and anyway my book isn't real life.

 

[thefinalson]

So my first example wasn't perfect, but are you implying that time and distances are also discrete as well?

If in one universe an atom decays at 00:00 and in another universe the atom decays at 00:01. Shouldn't there be infinite universes where it decays somewhere in between? We probably can't measure the time accurately enough to always note the difference, but there should be a difference, right?

Well, the difference can't possibly be less than one Planck time, so, there will be an extremely large but not infinite number…

 

[GenePeer]

Well, the difference can't possibly be less than one Planck time, so, there will be an extremely large but not infinite number…

Why not? From what I just read, Planck time is, theoretically, the smallest time-difference measurable. If things are only defined by what's measurable, then wouldn't it be pointless to bring in the concept of other universes when we can only observe one?

Edit:

I'm just going to arbitrarily say that my multiverse has 2 → 3 → 5 → 8 → 13 → 21 → 34 → 55 → 89 → 144 → 233 → 377 universes in it. Maybe in real life it has a lot less, or a lot more, but no one can really know for sure either way, and anyway my book isn't real life.

Do you have a way of explaining that notation in layman's terms, so that it won't make the novel less enjoyable? You'll spend a couple of paragraphs just explaining what the arrows mean yet other (smaller) big numbers like googolplex only take a sentence or two. Is it worth the time and effort?

 

[chasrob]

[...]

I'm actually curious now as to how Graham's number was developed...

As it turns out, you're not alone. Graham has never published that proof.

http://groups.google.com/group/sci.math.research/msg/0d6de64133da7e1c?hl=en

 

[thefinalson]

Why not? From what I just read, Planck time is, theoretically, the smallest time-difference measurable.

Well, Planck time is not just the smallest time difference measurable, it's also the shortest time difference possible. At periods of time shorter than one Planck time (just as with lengths shorter than one Planck length) time breaks down into a weird kind of quantum foam and stops acting like time as we understand it. This is the time scale at which the structure of spacetime becomes dominated by quantum effects, and it would become impossible to determine the difference between two times less than one Planck time apart. Not just impossible to determine the difference because we lack the technology to probe these scales (though that is also true), but impossible to determine the difference because there physically is no difference anymore.

You'll spend a couple of paragraphs just explaining what the arrows mean yet other (smaller) big numbers like googolplex only take a sentence or two. Is it worth the time and effort?

Sure, why not?

 

[chasrob]

[...]


I'm just going to arbitrarily say that my multiverse has 2 → 3 → 5 → 8 → 13 → 21 → 34 → 55 → 89 → 144 → 233 → 377 universes in it. Maybe in real life it has a lot less, or a lot more, but no one can really know for sure either way, and anyway my book isn't real life.

You don't even have to go that far. Drop the ones in the Golden Ratio(just as "natural" as Fibonacci numbers), like you said in an earlier post, and you get 3 → 6 → 8 → 3. That short expression is inconceivably larger than Graham's #.

 

[thefinalson]

You don't even have to go that far. Drop the ones in the Golden Ratio(just as "natural" as Fibonacci numbers), like you said in an earlier post, and you get 3 → 6 → 8 → 3. That short expression is inconceivably larger than Graham's #.

Yeah. Either way works. Anyway, thank you very much, Chasrob, and everyone else who helped; not just for helping me find my über-number, but for an extremely engaging, lively, and fascinating discussion of the practice and theory of ridiculously gargantuan numbers as well. I learned a lot (and, I bet, so will people who might stumble across this thread in the future by way of a google search or what-have-you). Thanks again, everyone!