A very large number that has all of these qualities?

[thefinalson]

Hello all! I'm new here; I found these forums by way of a google search after I was unable to get my question answered on yahoo answers. In any event, I don't see any place for new members to "introduce themselves" like a lot of forums have, so I guess I figured it would be alright if I just jump right into my question. Hope that's cool with everyone?

Just incidentally, I am working on a science fiction setting (well sort of science fantasy, it's got a mix of hard and soft sci-fi elements, but I want the hard sf parts to stick within real laws of physics and mathematics), so I will likely be asking more questions in the future as far as this type of thing, I hope you guys can help me out, and I'm certainly happy to credit you in the book acknowledgments as science/math consultants!

So, I'm trying to come up with a number that has both of these qualities and matches both of these criteria:

• It must be a very large number, so large that it must use Conway chained arrow notation. It must be larger than Graham's number; significantly so.

• It must be a mathematical anagram of the Golden Ratio (or Golden Mean, or whatever you want to call it.) By which I mean, it must be composed of only the numbers from that fraction (and zeroes). So it could be like 3 → 1 → 6 → 1 → 8 → 3 and so on…

I hope someone can help (and I hope this is categorized in the right forum, I always seem to be a little unsure about that when I first post in a new forum; I'm not sure if someone might consider this to be a "textbook-style problem" or not)!

 

[thefinalson]

:bump:

Can anyone please help me out here with this, I really need to know!

I know it's a complicated question…should I move this to the Number Theory section?

In case anyone needed to refer to any of the concepts I mentioned in my question (I'm sure you don't, because honestly I believe all of you are probably far better mathematicians than I, but I just thought I would put this, in case anyone does), here are wikipedia articles that describe them:

• http://en.wikipedia.org/wiki/Conway_chained_arrow_notation
• http://en.wikipedia.org/wiki/Graham's_number
• http://en.wikipedia.org/wiki/Golden_Ratio

Thank you. Please help if you know about this!

 

[chasrob]

First impressions-
You say you want to involve the digits of the Golden Ratio--the GR is an irrational number. Therefore its digits can be expanded to infinity. Therefore there's an excellent chance all single digits 0-9 can be found somewhere in its expansion, so...

[...]
So it could be like 3 → 1 → 6 → 1 → 8 → 3 and so on…

This appears to be in chained arrow form. However, according to rule 3 in the definition and overview section of the Wikipedia article you linked, the expression is truncated after the first "1". So your example shortens to 3 -> 1, which is equal to 3.

 

[thefinalson]

This appears to be in chained arrow form. However, according to rule 3 in the definition and overview section of the Wikipedia article you linked, the expression is truncated after the first "1". So your example shortens to 3 -> 1, which is equal to 3.

Ok, right, see, this is why I need help! :shy: Could you please help me get a similar number (involving the digits of the Golden Ratio in roughly the order that they appear within that irrational number's expansion) but one that contains no "1's" and is an ungodly huge number, like much bigger than Graham's number, and way too big to write within the observable universe?

 

[chasrob]

Ok, right, see, this is why I need help! :shy: Could you please help me get a similar number (involving the digits of the Golden Ratio in roughly the order that they appear within that irrational number's expansion) but one that contains no "1's" and is an ungodly huge number, like much bigger than Graham's number, and way too big to write within the observable universe?

How about the natural base, e, raised to the power 1618033988 (the first ten digits of the Golden Ratio)? That's equal to 3.3136932969011345 × 10^702703232. Or 1618033988! (same digits factorial ten times), equal to a power tower of 14,197,751,787 tens with a 5.211971113009518 on top. Both far too big to write within the observable universe. Though neither are not as big as Graham's number.

 

[Mentallic]

like much bigger than Graham's number, and way too big to write within the observable universe?

That first remark guarantees no digital representation of the number could be contained within our universe

I'm not quite sure of what you're asking. Of course you could create a number bigger than Graham's number, but what is the purpose? Does this number have to represent something?

 

[chasrob]

Yeah, your reasons for needing Graham's number are not clear. The golden ratio is fine. But GN is impossible to represent without special notation. The standard exponents and power towers, or log of the log etc., don't work. Chained arrow notation would approximate GN: its between 3→3→64→2 and 3→3→65→2. So 3→3→66→2 would be larger. But combining chained arrow and the digits of the golden ratio... you could drop all ones and zeros but then it wouldn't be the golden ratio anymore.

I'm writing a sci-fi story myself, and it involves large numbers also--why I'm interested.

 

[pmsrw3]

Using the digits of the decimal expansion of the golden ration seems kind of arbitrary to me. 1, 6, 1, 8, 0, 3, ... are consequences of arbitrary evolutionary accidents that resulted in us having 10 fingers. If we had 12, it would be 1, 7, 4, b, b, 7, ... If you want an approximation scheme (which is all that a decimal expansion is) for the golden ratio that has more universal validity, how about the Fibonacci numbers?

 

[thefinalson]

Yeah, your reasons for needing Graham's number are not clear. The golden ratio is fine. But GN is impossible to represent without special notation. The standard exponents and power towers, or log of the log etc., don't work. Chained arrow notation would approximate GN: its between 3→3→64→2 and 3→3→65→2. So 3→3→66→2 would be larger. But combining chained arrow and the digits of the golden ratio... you could drop all ones and zeros but then it wouldn't be the golden ratio anymore.

I'm writing a sci-fi story myself, and it involves large numbers also--why I'm interested.

That's very cool! PM me, we ought to compare notes .

I'll try to explain all my logic as best as I can. It's not that I need something that approximates GN, it's simply that I want this number I'm creating (or really defining; you can't actually create a number, they all exist already lol, but, you know what I mean) to be larger than Graham's number—I'm simply using it as a yardstick. The reason for this is pretty arbitrary really, just that I've seen it mentioned a couple of times as an awfully big number, and I want mine to be even bigger! :tongue2:

Dropping all the 1's and 0's out of the Golden Ratio, I think is what I'm going to have to do. It won't be the GR anymore, but oh well, it'll have to be close enough. So, doing this, we would get:

6 → 8 → 3 → 3 → 9 → 8 → 8 → 7

That number would be much larger than Graham's number, wouldn't it?

Oh, and you were curious what this number was for within my SF story, right? Well, basically, to make a long story (or, in this case, book lol :tongue:) short…this number is used by an immensely powerful, immensely ancient multiverse-spanning civilization for measuring horrendously huge numbers, such as all of the possible configurations of atoms in all of the universes that exist!

 

[thefinalson]

Using the digits of the decimal expansion of the golden ration seems kind of arbitrary to me. 1, 6, 1, 8, 0, 3, ... are consequences of arbitrary evolutionary accidents that resulted in us having 10 fingers. If we had 12, it would be 1, 7, 4, b, b, 7, ... If you want an approximation scheme (which is all that a decimal expansion is) for the golden ratio that has more universal validity, how about the Fibonacci numbers?

I could use the Fibonacci numbers instead, they seem to basically just be a different "version" (if you will) of the Golden Ratio; wikipedia describes them as, quote, "intimately connected". So, if I were to make my super-number with the Fibonacci numbers, it would be this:

2 → 3 → 5 → 8 → 13 → 21 → 34 → 55 → 89 → 144 → 233 → 377

Am I correct in assuming this would also be much larger than Graham's number?

 

[Hurkyl]

I strongly suspect you are running down a false path -- you may be better served by thinking more upon your actual goal, rather than spending all of your effort upon this idea you had to achieve the goal.


One of the features of a "horrendously large number" is that it's pretty much cannot every be used except for one thing (and even that use might not be very useful). No other number you ever encounter will ever be similar enough to it to make a comparison useful.

(Recall that Graham's number originally came about to prove a certain number N was actually finite -- they did so by proving N had to be less than Graham's number. At the time, I think, people thought that N=6. Today, I think people estimate it to be 13)



I also believe that any number you imagine as being horrendously large is actually far, far, far smaller than Graham's number. If the number in your example is finite, I imagine it's unlikely to be bigger than

$$10^{10^{10^{10^x}}}$$​

for some modest number x.


Have you considered suggesting and hinting at the number you're thinking of? From a story telling perspective, doing so might be far more compelling than trying to come up with an explicit number (which you're going to have to explain anyways, if you want people to feel the way you want them to).

 

[agentredlum]

Why don't you use googol-->GN-->googolplex.

This is certainly greater than Grahams Number, and makes a sandwich between googol-googolplex that Sci-Fi readers may find tasty, I would.

 

[thefinalson]

Thank you both very much for your advice, Hurkyl and Agentredlum! I appreciate the feedback.

Hurkyl, I understand what you're saying, and you're probably right. In this case, the number actually is useful for more than one thing, because it's sort of the "base number" for a new measurement system used to deal with things that exist on a nearly-infinite scale, which this particular fictional civilization encounters on a regular basis! It's probably not that realistic to have any number larger than $$10^{10^{10^{10^x}}}$$ but this book is meant to be over-the-top and "beyond the impossible", so unrealistically huge is how the number should be! :tongue:

Agentredlum, googol → GN → googolplex indeed would make a cool "number sandwich", and one that I think would be more immediately recognizable to a lot of SF readers, so I'm definitely going to consider using that. On the other hand, though, my fictional civilization's whole numerical system is based around natural units, physical constants, and numbers inherent to the universe itself. For example, their units of distance are based around the Planck length. So that kind of makes me lean towards the Fibonacci numbers instead.

Just to check/confirm, 2 → 3 → 5 → 8 → 13 → 21 → 34 → 55 → 89 → 144 → 233 → 377

This would be much larger than Graham's number, right? I'd just like to be clear on that.

 

[Mentallic]

I've yet to even be able to concoct a problem where I need to input fathomably large numbers (powers) to yield an unfathomably large number such as Graham's number. I'm not even remotely close!

for measuring horrendously huge numbers, such as all of the possible configurations of atoms in all of the universes that exist!

For example, these numbers wouldn't even need to be expressed in up-arrow notation.
All you have there is xyz where x is the number of configurations of an atom, y is the number of atoms in the each universe, and z is the number of universes. If we take each as being 10¹⁰⁰ which is already more than the estimated number of atoms in the universe by a factor of 10²⁰, but nonetheless, this would produce an answer of (10¹⁰⁰)³=10³⁰⁰ possible configurations of all the atoms in all the universes. This is nothing compared to Graham's number, which is why I agree with Hurkyl that such a number he posed would be more than enough for anything you would need.

 

[thefinalson]

Ok, well, still, first off I'd like to just get a straight, yes or no answer to my initial question, if I could please:

Just to check/confirm, 2 → 3 → 5 → 8 → 13 → 21 → 34 → 55 → 89 → 144 → 233 → 377

This would be much larger than Graham's number, right? I'd just like to be clear on that.

And secondly,

For example, these numbers wouldn't even need to be expressed in up-arrow notation. All you have there is xyz where x is the number of configurations of an atom, y is the number of atoms in the each universe, and z is the number of universes. If we take each as being 10¹⁰⁰ which is already more than the estimated number of atoms in the universe by a factor of 10²⁰, but nonetheless, this would produce an answer of (10¹⁰⁰)³=10³⁰⁰ possible configurations of all the atoms in all the universes. This is nothing compared to Graham's number, which is why I agree with Hurkyl that such a number he posed would be more than enough for anything you would need.

Alright well, hold your horses there for one minute, Mentallic. $$10^{100}$$ may be a lot more than the number of atoms in one universe, but how many possible positions and quantum states and combinations of possible positions and quantum states are there for all of those atoms? And then how many universes are there? It could be infinite. Plus you're only counting the observable universe, as the entire universe. There could be a lot more. And remember, this is science fantasy here!

 

[GenePeer]

The problem with what you're aiming for is the number will be so huge, you won't even be able to describe how huge it is. Just from reading the wikipedia page, $g_{1}$, which is the first step for a sequence of 64 terms that are growing unimaginably faster than exponentiation, is already too big to be written in the observable universe. It is simply impossible to fathom how large Graham's number, $g_{64}$, actually is. No sort of comparison system you can ever come up with will help. It will take the fun out of the book, if all the reader knows about that measure is it's "unimaginable size". What's fiction if you can't imagine it?

 

[chasrob]

[...]
Just to check/confirm, 2 → 3 → 5 → 8 → 13 → 21 → 34 → 55 → 89 → 144 → 233 → 377

This would be much larger than Graham's number, right? I'd just like to be clear on that.

Yep, that would be true. But its not the Fibonacci sequence, you left off the first three digits- 0,1,1. Which would truncate the notation and it would be equal to zero.

 

[Mentallic]

Alright well, hold your horses there for one minute, Mentallic. $$10^{100}$$ may be a lot more than the number of atoms in one universe, but how many possible positions and quantum states and combinations of possible positions and quantum states are there for all of those atoms?

It still wouldn't be anywhere close. These problems can be solved with manageable numbers (although huge that we can't give a digital representation of the figure).
For example, if we take the approximate number of atoms in the universe as 10¹⁰⁰ and assume each atom is unique, and then to find each and every possible position that the atoms can arrange themselves in, it will be 10¹⁰⁰! But n!<nⁿ so 10¹⁰⁰!<(10¹⁰⁰)¹⁰¹⁰⁰=10¹⁰¹⁰² which definitely doesn't require up arrow notation to be represented. You can go on and find all the quantum states, all the possibilities, but if you try to describe anything physical in our universe (or all the finite number of universes), it won't come close to Graham's number.

And then how many universes are there? It could be infinite.

Well I assumed that if they're going to be counting all the universes combined, then it wouldn't be infinite. If it is, then there's no point in talking about large numbers because infinite is just that, infinite.

And remember, this is science fantasy here!

Which is what we've been trying to do here! We can't even concoct a fictitious scenario that would require us to express the figure in terms of up arrow notation.

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

 

[thefinalson]

Yep, that would be true. But its not the Fibonacci sequence, you left off the first three digits- 0,1,1. Which would truncate the notation and it would be equal to zero.

Ok, well firstly, thank you very much, Chasrob, for being the only one to actually answer my question; that was the only thing I really wanted to know! Yeah, I know it's not exactly the Fibonacci sequence without the 0, 1, and 1, but obviously I can't have my super-number be equal to zero, so it'll have to be close enough.

It still wouldn't be anywhere close. These problems can be solved with manageable numbers (although huge that we can't give a digital representation of the figure).
For example, if we take the approximate number of atoms in the universe as 10¹⁰⁰ and assume each atom is unique, and then to find each and every possible position that the atoms can arrange themselves in, it will be 10¹⁰⁰! But n!<nⁿ so 10¹⁰⁰!<(10¹⁰⁰)¹⁰¹⁰⁰=10¹⁰¹⁰² which definitely doesn't require up arrow notation to be represented. You can go on and find all the quantum states, all the possibilities, but if you try to describe anything physical in our universe (or all the finite number of universes), it won't come close to Graham's number.Well I assumed that if they're going to be counting all the universes combined, then it wouldn't be infinite. If it is, then there's no point in talking about large numbers because infinite is just that, infinite.Which is what we've been trying to do here! We can't even concoct a fictitious scenario that would require us to express the figure in terms of up arrow notation.

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

Ahh, well, as I've said previously, I'm not really as good a mathematician as most of you probably are, (I'm more of a writer and a storyteller and world-builder with just a layman's interest in, and knowledge of, math and science) but still, that being said, even with my dilettante level of expertise, I'm still really sure that you have a way low estimate there. If we are to assume that there are 10¹⁰⁰ atoms in an "average" universe, then all of the possible arrangements and configurations (10¹⁰⁰!) has to be far higher than 10¹⁰⁰. I say this with confidence because the number of possible configurations of a set of things must always be exponentially greater than the set of things itself. For example, look at our Latin alphabet. It contains only 26 letters, yet the number of possible words that could be formed is enormous. Just try to imagine how many words we could make if our alphabet had 10¹⁰⁰ letters in it! Of course I realize atoms aren't words, but the analogy is illustrative nonetheless.

Graham's number was developed to answer the question: "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?

Graham & Rothschild (1971) proved that this problem has a solution, N*, and gave as a bounding estimate 6 ≤ N* ≤ N, with the upper bound N a particular, explicitly defined, very large number. (In terms of Knuth up-arrow notation, N=F⁷(12), where F(n)=2 up-arrow ⁿ3.) The lower bound of 6 was later improved to 11 by Geoff Exoo of Indiana State University (2003) and even further to 13 by Jerome Barkley in 2008. Thus, the best known explicit bounding estimate for the solution N* is now 13 ≤ N* ≤ N."

Which brings me to my next point, by the way. What if I propose, in my sf setting, that each possible thing (including different laws of physics and logic, different spatial dimensions, and even different laws of math than we enjoy in our universe) must exist in some universe out there? Including every possible solution N* for Graham's number, as well as lots of other such problems as the one GN was produced to solve—then I am quite certain there would be far more than 10¹⁰⁰ things for my civilization to measure!

But as I stated before, I'm really not enough of an expert on these sorts of matters to truly know, and as such, I bow out of the debate graciously. If you say there's less, then there's probably less.

Finally, though,

The problem with what you're aiming for is the number will be so huge, you won't even be able to describe how huge it is. Just from reading the wikipedia page, $g_{1}$, which is the first step for a sequence of 64 terms that are growing unimaginably faster than exponentiation, is already too big to be written in the observable universe. It is simply impossible to fathom how large Graham's number, $g_{64}$, actually is. No sort of comparison system you can ever come up with will help. It will take the fun out of the book, if all the reader knows about that measure is it's "unimaginable size". What's fiction if you can't imagine it?

Well, this setting is meant to be "unimaginable"—but you're right, it's no good if my readers don't know what this number really means! What would you suggest would be the best way to succinctly explain how big a Conway chained arrow notation number is in a little story footnote or sidebar to readers who may or may not have any great background in the mathematics of ridiculously large numbers, but assuming at least that they understand how normal exponentiation, like 10²⁰, works? The wikipedia example is a little too long-winded and a little too dependent on links to other articles to try to summarize in my book! :tongue2:

 

[Mentallic]

I'm still really sure that you have a way low estimate there. If we are to assume that there are 10¹⁰⁰ atoms in an "average" universe, then all of the possible arrangements and configurations (10¹⁰⁰!) has to be far higher than 10¹⁰⁰. I say this with confidence because the number of possible configurations of a set of things must always be exponentially greater than the set of things itself.

You misread my post! :tongue:

What would you suggest would be the best way to succinctly explain how big a Conway chained arrow notation number is in a little story footnote or sidebar to readers who may or may not have any great background in the mathematics of ridiculously large numbers, but assuming at least that they understand how normal exponentiation, like 10²⁰, works? The wikipedia example is a little too long-winded and a little too dependent on links to other articles to try to summarize in my book! :tongue2:

You may find this person's blog interesting: http://frothygirlz.com/2010/01/14/big-numbers-part-2/"

And take particular note of the part where it actually starts to delve into the shear magnitude of Graham's number:

G1 = 3 ↑↑↑↑ 3 (4 up arrows)

G2 = 3 ↑↑↑↑…G1 up arrows…↑↑↑↑ 3

 

[thefinalson]

You misread my post! :tongue:

I apologize. I went back and re-read it and now I understand that you were saying this. However, I'm still not sure…10¹⁰⁰—which is also called a googol—yes, it's more atoms than there are in our universe (ours has about 10⁸⁰, but let's be generous and assume that others might be a little larger). However, firstly, are you sure that n!<nⁿ? That sounds wrong to me. If I had to guess, I would guess that n!>nⁿ. And secondly, even if you are right about that (I assume you are, I'm just trying to understand why), would 10⁸⁰! be the number of all the possible things (planets, stars, black holes, objects, organisms, etc.) that could exist in our universe or any other alternate version of our universe with an equal number of atoms? Because in our 3-dimensional universe, there may be more possible configurations of things than there would be if we lived in some kind of 1-dimensional universe made only of numbers. I would almost feel like it would be 10⁸⁰³!. I really have no idea though, and I guess that number still doesn't require even Knuth's up-arrow notation, no less Conway chained arrow notation, to write.

Still, if we presuppose other types of universes beyond just reconfigurations of our own universe, such as universes where each possible solution N* for Graham's number is there, than we might be able to require a much bigger number. Maybe…

You may find this person's blog interesting: http://frothygirlz.com/2010/01/14/big-numbers-part-2/"

And take particular note of the part where it actually starts to delve into the shear magnitude of Graham's number:

I do find it extremely interesting (and useful for my book) thank you so much for pointing me towards it!

 

[Mentallic]

I apologize. I went back and re-read it and now I understand that you were saying this. However, I'm still not sure…10¹⁰⁰—which is also called a googol—yes, it's more atoms than there are in our universe (ours has about 10⁸⁰, but let's be generous and assume that others might be a little larger). However, firstly, are you sure that n!<nⁿ? That sounds wrong to me. If I had to guess, I would guess that n!>nⁿ.

Nope. Note that
n!=n*(n-1)*(n-2)...2*1 which has n terms while,
nⁿ=n*n*n...n*n which also has n terms. Obviously nⁿ>n!

And secondly, even if you are right about that (I assume you are, I'm just trying to understand why), would 10⁸⁰! be the number of all the possible things (planets, stars, black holes, objects, organisms, etc.) that could exist in our universe or any other alternate version of our universe with an equal number of atoms? Because in our 3-dimensional universe, there may be more possible configurations of things than there would be if we lived in some kind of 1-dimensional universe made only of numbers. I would almost feel like it would be 10⁸⁰³!. I really have no idea though, and I guess that number still doesn't require even Knuth's up-arrow notation, no less Conway chained arrow notation, to write.

I'm not quite sure exactly what you wanted to calculate when you came up with that figure, although it looks wrong because what that says is first find 80³ (which couldn't be right, because you haven't dealt with 80*80*80 in any way that I can see) and then take 10 to the power of that number, then the factorial of it. Hmm... Well I'll give you an idea of some crazy combinations of atoms and whatnot which is what you're trying to get at:

Let's assume we have 10¹⁰⁰ atoms in the universe, 10¹⁰⁰ universes and each atom has 10¹⁰⁰ different states.
We want to find all the combinations of all the atoms in the all the universes.

For example, if an atom has 2 states (1 or 0), then 2 atoms have a total of 4 combinations (00,10,01,11) and 3 atoms have 8 combinations, n atoms have 2ⁿ combinations etc.

On top of this, we will assume that the position in space that each atom occupies is unique, so we want to find all the positions that the atoms can arrange themselves in.

So how many atoms? That's just 10¹⁰⁰.10¹⁰⁰=10²⁰⁰
How many different combinations of states? (10¹⁰⁰)¹⁰²⁰⁰ Since it's (number of states)^{(number of atoms)}
How many different orders in their positions? 10²⁰⁰! which as I've shown is less than (10²⁰⁰)¹⁰²⁰⁰$\approx$10¹⁰²⁰⁰

And now to answer our question, all we need to do is multiply these terms together, which gives us (remember this is larger than our answer, since we've approximated it)

10²⁰⁰.(10¹⁰⁰)¹⁰²⁰⁰.10¹⁰²⁰⁰=10¹⁰⁴⁰⁰

Still nothing...

Still, if we presuppose other types of universes beyond just reconfigurations of our own universe, such as universes where each possible solution N* for Graham's number is there, than we might be able to require a much bigger number. Maybe…

Of course we can take a simple approach and say, well, if there are Graham's number of universes then all the atoms in the universes will be approximately Graham's number, but that's cheating

I do find it extremely interesting (and useful for my book) thank you so much for pointing me towards it!

No worries!

 

[GenePeer]

"However, firstly, are you sure that $n!<n^n$?"

$n! = n \cdot (n-1) \cdot (n-2) \ldots 2 \cdot 1$

$n^n = n \cdot n \cdot n \ldots n \cdot n$

They both have the same number of terms in the product but after the first term, subsequent terms in $n!$ are always less than the corresponding terms in $n^n$.
As for a suggestion, I honestly don't know how you'd convey the size of Graham's number. I don't even understand how huge it is. I've accepted that it's beyond my comprehension. I guess that link posted by Mentallic would be the best explanation.

 

[thefinalson]

Of course we can take a simple approach and say, well, if there are Graham's number of universes then all the atoms in the universes will be approximately Graham's number, but that's cheating

Wait, I'm still confused, wouldn't that only be true if each universe contained only one atom? I don't know, I kind of get it but…so, are you saying that ^1010400 is more than all of the possible things there could be? And by "things", I realize I'm being extremely vague—I simply mean anything made out of atoms, like a book, an apple, a star, a person, or an alien spaceship.

I don't know, thanks everyone for trying to help me understand! I guess I just can't really envision how fantastically enormous a number ^1010400 really is, which is, paradoxically, probably why it doesn't seem big enough to me!

 

[pmsrw3]

Wait, I'm still confused, wouldn't that only be true if each universe contained only one atom?

GN is so much more enormously big than the number of atoms in a (finite) universe, that it doesn't really matter if each universe contains 1 atom or 10¹⁰⁰ or even 10¹⁰¹⁰⁰. At the level of approximation we're dealing with, that's still basically just a tiny bit more than GN of atoms.

 

[thefinalson]

GN is so much more enormously big than the number of atoms in a (finite) universe, that it doesn't really matter if each universe contains 1 atom or 10¹⁰⁰ or even 10¹⁰¹⁰⁰. At the level of approximation we're dealing with, that's still basically just a tiny bit more than GN of atoms.

Wow. Yeah, see, that's a perfect way of describing Graham's number for my book right there: it's a number so enormous that adding to it the number of atoms in the entire universe doesn't noticeably increase it.

But let's look at this problem from a different angle, and see if we can come up with a good estimation number. Clearly, it will be smaller than GN.

Let's say we're trying to approximately calculate how many universes there might be in my setting's multiverse. We'll start by assuming each universe has $\approx$ 10¹⁰⁰ atoms. Next we'll assume that the position in space that each atom occupies is unique, so we want to find all the positions that the atoms can arrange themselves in. We're further going to assume that every possible variation of the laws of physics, logic, mathematics etc. exists in some universe, and that each one of these variant types of universe also has all possible arrangements of its atoms.

We're going to further assume that each universe is only different from its neighboring universe in one way. So Universe A might be our universe that we're in right now, Universe B is exactly like ours in every respect, except that the position of one atom somewhere in the universe is different to one degree (for example, is one Planck length to the left or has a spin quantum number which is one different). Universe C, again, has only one infinitesimal difference from Universe B of the same type, and two such infinitesimal differences from Universe A, and so on.

Let's further assume that every possible outcome of every possible event, including the big bang, exists in one universe, and all those possible outcomes of all things (including Graham's number) do happen in my multiverse. Looking at this old post I dug up might help to visualize what I mean: https://www.physicsforums.com/showthread.php?t=238668

But also remember that there will be universes very unlike ours, such as universes where the laws of physics allow magick to be possible, others are universes of pure energy, or nearly so; composed of a single form of reality such as pure light, absolute darkness, deadly radiation, endless sound, pure thought, and so on. Still other universes are even stranger, harder to categorize: they may contain little or no solid ground, or energy vortices might twist space back upon itself, even concepts like "up" and "down" may be meaningless.

So, considering all that, is there any way to really estimate $\approx$ how many universes we're talking?

 

[pmsrw3]

Wow. Yeah, see, that's a perfect way of describing Graham's number for my book right there: it's a number so enormous that adding to it the number of atoms in the entire universe doesn't noticeably increase it.

That's true, but it's not what I said. Even multiplying by the number of atoms in the entire universe doesn't noticeably increase it.

Let's further assume that every possible outcome of every possible event, including the big bang, exists in one universe, and all those possible outcomes of all things (including Graham's number) do happen in my multiverse.

What do you mean by the outcome of GN? That doesn't make any sense to me.

So, considering all that, is there any way to really estimate $\approx$ how many universes we're talking?

What's confusing about this? The answer is clearly "No".

 

[GenePeer]

Wow. Yeah, see, that's a perfect way of describing Graham's number for my book right there: it's a number so enormous that adding to it the number of atoms in the entire universe doesn't noticeably increase it.

The same can be said for googolplex.

I think you've absolutely underestimated the size of GN by the way you keep trying to compare it to something we can conceptualize. No sort of simple layman's comparison is possible.

 

[Mentallic]

I think you've absolutely underestimated the size of GN by the way you keep trying to compare it to something we can conceptualize. No sort of simple layman's comparison is possible.

Exactly this. I still stand firm by Hurkyl's earlier response that GN is unnecessary and will only cause confusion. It would be best if you stick to moderately sized power towers:

If the number in your example is finite, I imagine it's unlikely to be bigger than

$$10^{10^{10^{10^x}}}$$​

for some modest number x.

 

[thefinalson]

What do you mean by the outcome of GN? That doesn't make any sense to me.

What's confusing about this? The answer is clearly "No".

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? :tongue2:

Please correct me if I'm missing something.

The same can be said for googleplex.

I think you've absolutely underestimated the size of GN by the way you keep trying to compare it to something we can conceptualize. No sort of simple layman's comparison is possible.

Yes, I know I'm vastly underestimating the size of GN. It's obviously really difficult to try to wrap one's mind around such an inconceivable number, I'm trying the best I can lol . As I said before, I have trouble even conceptualizing ^1010400, and that's so much smaller than Graham's number it's ridiculous. So…yeah.

Exactly this. I still stand firm by Hurkyl's earlier response that GN is unnecessary and will only cause confusion. It would be best if you stick to moderately sized power towers:

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?

 

[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. :tongue:

 

[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.