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midtskogen
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Largest number using 5 symbols

22 May 2022 13:27

Here's a challenge: What is the largest number possible to express using 5 symbols?  Only standard mathematical notation.

My suggestion
► Show Spoiler
But I'm curious whether it's possible to express an even (ridiculously) larger number than this using only 5 symbols.
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Re: Largest number using 5 symbols

22 May 2022 15:35

Is Knuth's up-arrow notation allowed? :D

Image

What is this abomination? We're putting a superscript 9! on the up arrow, which means we're applying 9 factorial or 362880 up arrows. According to the wiki this is a standard notation.

What does each up arrow do? The first means iterated multiplication. a↑b means to multiply a by itself b times, which is the same as exponentiation: a^b. A second arrow means iterated exponentiation, called tetration. a↑b is a raised to the power of a, b times.

A third arrow? Now we're iterating tetration. a↑↑↑b is a tower of (a↑↑b) a's.

Every additional arrow means to iterate the previous procedure, which quickly takes us to utterly ridiculous numbers. I don't want to think about what 9! up arrows do. I don't think I even can think about what 9! up arrows do.

Maybe it could be argued that a superscript should be considered an extra implied symbol (9^9 is three symbols for instance). In that case I'd drop the factorial and keep 9 up arrows. Even three up arrows, 9↑↑↑9, is already an ungodly big number. You can think of it as a tower of exponents, 9^9^9^9^9^9^9..., going onward and upward, but there are 9^9^9^9^9^9^9^9^9 nines in the tower, and then the number you get from that tower is fed into a new tower to tell you how many exponents to do, and that goes into yet another tower, and this is repeated 9 times. 

Each of those towers has more exponents in it than there are protons in the universe. Heck, more than 9!!!! of them, even. Which just leaves me thinking:

Image

(Actually this is a stepping stone towards trying to comprehend Graham's number, which is profoundly bigger still and did have a practical purpose, sort of...)
 
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Re: Largest number using 5 symbols

23 May 2022 01:45

Lovely.  To keep this thought experiment interesting, I think we can be fairly admissive about accepted notation as long as actual use can be documented.  Regarding this as a stepping stone towards Graham's number, how about utterly ruin that by allowing to replace 9 with G - Graham's number itself?
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Re: Largest number using 5 symbols

23 May 2022 03:00

Regarding this as a stepping stone towards Graham's number, how about utterly ruin that by allowing to replace 9 with G - Graham's number itself?
:???:  It would be so mind boggling that I struggle to think of useful examples that illustrate how insane it is (like odds of every person on Earth choosing one particle out of the universe at random and everyone happens to pick the same particle), that don't apply just as well to something like 9↑↑↑9, or even 9!!!! for that matter. 

I was thinking as well of other methods of generating stupidly large numbers, but the constraint of using just 5 symbols makes it interesting. TREE(3) came to mind for instance, but you would have to discount the parentheses. I don't know if TREE(3) is bigger than G↑(G!)G, though I would not be surprised if it is. (If it's not then just feed it a slightly bigger number, since TREE(n) grows faster than the tetration process behind Graham's number.) Rather, what I find totally surprising is that it manages to be so huge (especially given that TREE(2) is 3 and TREE(1) is 1), and yet it isn't infinite.
 
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Re: Largest number using 5 symbols

23 May 2022 04:48

The 5 symbol limitation is only a notational technicality in your suggestion, though.  Instead of TREE it could have been written 𝚻 or something and the concept would be the same.  In which case we could try 𝚻(G!).
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Re: Largest number using 5 symbols

23 May 2022 05:07

Yup! And 𝚻(G!) with G standing for Graham's number (g_64) is by far the better option than g(𝚻!), with 𝚻 standing for TREE(3). The sequence TREE(n) grows way faster. This link is one of my favorite numberphile videos, and they are basically going through the same process that we're doing here, defining ways of constructing larger and larger numbers, or rather sequences that grow faster and faster -- a fast-growing hierarchy. It gets absolutely bonkers. 
 
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Re: Largest number using 5 symbols

23 May 2022 14:27

Yup! And 𝚻(G!) with G standing for Graham's number (g_64) is by far the better option than g(𝚻!), with 𝚻 standing for TREE(3).
My intuition said so, but proving it does not sound intuitive.  The constraint "5 symbols" should probably allow any digit, constant, operator or function even if several signs are used to express the function.  After all, the factorial could have been written as a function !(x), but by convention we use postfix notation.  Conceptually, operators and functions are equal.  This means that 𝚻(G!) really is only three symbols, so we can have 𝚻(𝚻(𝚻(𝚻(G)))), which we could write as G𝚻𝚻𝚻𝚻 in postfix notation.  Digit also becomes superfluous since we include constants.  We still have to keep a somewhat loose requirement that the function or constant must have some kind of use in reasoning about something.  Otherwise there is nothing stopping us from defining endless new functions (consider TREEω using the reasoning in the video, and so on).

Invoking recursion is powerful.  If we can't think of anything larger than G𝚻𝚻𝚻𝚻, the next challenge could be to reason about the largest number that can be expressed by G𝚻𝚻𝚻𝚻 symbols.  For this we could define a new operator ↟ writing G𝚻𝚻𝚻𝚻 as G↟4. And we could define a new number like Graham's number using ↟ instead of ↑.  Why not G↟G?
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Re: Largest number using 5 symbols

23 May 2022 18:12

G↟G!!
[dah<500,26>dah<180,14>dah<180,21>dah<500,19>dah<180,26>dah<500,21>]
 
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Re: Largest number using 5 symbols

24 May 2022 00:41

G↟G!!
G↟[sup]G![/sup]G
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Re: Largest number using 5 symbols

28 May 2022 05:57

Heh, due to my recent youtube history, the algorithm decided I should watch this video: Ultimate List of Large Numbers. It is exactly what it says it is: an ordered sequence of growing numbers (some boring, others interesting or famous), ending with Graham's number. More than 3 hours long though, so recommend skipping around through it. But for fun, we can relate the sizes of some of our suggested numbers by where in the timeline of the video something comparable or larger than them appears. 

Number         alternate representations                                         When in video?
------------------------------------------------------------------------------------------------------------
9!!!!               ~10^1202                                                               25:25
9↑↑9              {9,9,2} = 9^9^9^9^9^9^9^9^9 ~ f3(9)                        1:14:28
9↑↑↑9            {9,9,3} ~ 9↑↑(9↑↑...) (9 times) ~ f4(9)                        1:34:23
9(↑9)9           {9,9,9} ~ f10(9) ~ fω(9)                                            2:19:02
9(↑9!)9          {9,9,9!} ~ f9!(9) ~ f362880(9) ~ fω(3.6*10^5)             3:01:31
G                  g(64) ~ {64,64,1,2} ~ fω+1(64) ~ fω^64(64)               3:11:52

-----------numbers below here were too big for the video, and even all of its follow-up videos!------------------------
number            alternative representations and comments

TREE(3)            T(3)                This number is so much bigger than G that it puts the difference between G and 3 to shame.
TREE(G)            T(G)               This number way bigger than TREE(3), or even g(TREE(64)).
T(T(T(T(G))))     GTTTT = G↟4.  NOW we're talking. Iterating on TREE itself gets to a new level of crazy.
G↟G                                         Oh God...          

G↟G!!                Comment: You may be surprised how little of a difference the factorials make here now. This is because n! is less than n^n (for n>1), and n^n is equivalent to just one up arrow. But we have already long passed the point of iteratively growing the number of arrows! This is sort of the same notion as taking 10^100 and adding 1 to it. 10^100 just devours it and stays practically the same.

G(↟G!)G            Putting the factorial here is much better (now we're hitting the number of applications of ↟ in a big way). 

And of course we can go further.  We can keep increasing the number of ↟'s... even to infinity. But then we don't have to stop there! How about diagonalize again? Turn two indices into one. Generalize the expression G(↟G!)G as G(↟m)n, but then make m=n (which brings us up to a new ordinal much like how ω was introduced), and maybe give it a new symbol like ֍↟(n). Why not. This will grow faster with increasing n than the former does by increasing either m or n.  And we can keep going.  

Now it's getting hard even just to describe what our definition is doing, let alone comprehend the numbers it spits out! We've literally gone beyond infinity so many times (by introducing new procedures to iterate) that it becomes difficult to hold in the mind what the nested set of procedures even is. 

But wait. How many times could we diagonalize in order to define a new procedure? Infinitely many times. But that infinity in itself is just another limit, and we don't have to stop there! That's what separates the ordinals like ω from the higher ones like ε and η and so forth. What totally blows my mind is that sequences like TREE(n) are still yet more powerful than any of them.
 
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Re: Largest number using 5 symbols

30 May 2022 08:19

G↟G                                         Oh God...          
I think I made a pretty lame definition of ↟ in saying that that a↟b is simply TREE(TREE(...TREE(a)...) with depth b.  How about something like this?:
RTREE(a, b) = TREE(RTREE(a, b - 1)) for b > 0, otherwise TREE(a).
where RTREE(a, b) is a↟b.  That will make G↟G interesting.

And we can go on:
RRTREE(a, b) = RTREE(RRTREE(a, b - 1)) for b > 0, otherwise RTREE(a), and we can generalise the number of iterations like this by a superscript to ↟.  Then we can try to reason about G↟[sup]G[/sup]G.

Ok, so this is turning into finding a function blowing up in the most spectacular fashion.  TREE is impressive, and invoking recursion on TREE itself isn't equally clever, I suppose.
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Re: Largest number using 5 symbols

16 Jul 2022 05:20

Regarding this as a stepping stone towards Graham's number, how about utterly ruin that by allowing to replace 9 with G - Graham's number itself?
:???:  It would be so mind boggling that I struggle to think of useful examples that illustrate how insane it is (like odds of every person on Earth choosing one particle out of the universe at random and everyone happens to pick the same particle), that don't apply just as well to something like 9↑↑↑9, or even 9!!!! for that matter. 

I was thinking as well of other methods of generating stupidly large numbers, but the constraint of using just 5 symbols makes it interesting. TREE(3) came to mind for instance, but you would have to discount the parentheses. I don't know if TREE(3) is bigger than G↑(G!)G, though I would not be surprised if it is. (If it's not then just feed it a slightly bigger number, since TREE(n) grows faster than the tetration process behind Graham's number.) Rather, what I find totally surprising is that it manages to be so huge (especially given that TREE(2) is 3 and TREE(1) is 1), and yet it isn't infinite.
Would Graham's number even exceed the number of all subatomic particles in the multiverse if we consider our universe to be average and string theory landscape's prediction of 10^500 universes to be correct, Wats?
By the way, now that I have you here, I wanted to ask what you thought of Anna Iljas and Paul Steinhardt's new paper on their modified cyclic model of the universe that suggests the universe will stop expanding in 65 million years and will then start to contract?  It's a modified version of the original cyclic model because it irons out the problems with the original.  I was at UCSD recently and there was physics convention going on there where this was being discussed.
 
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Re: Largest number using 5 symbols

24 Jul 2022 02:37

Would Graham's number even exceed the number of all subatomic particles in the multiverse if we consider our universe to be average and string theory landscape's prediction of 10^500 universes to be correct, Wats?
You have no idea how small that number of subatomic particles is compared to Graman's number.  Not even the number of permutation's of Planck volumes over the number of Planck times in the universe since the Big Bang will even begin to describe Graham's number.  There simply isn't a direct way to relate Graham's number with anything physical.  To give you an idea: The number of subatomic particles in the universe is much smaller than even the number of digits in Graham's number.  Much smaller than the number of digits of the number of digits.  And so on for a number of times greater than the number of subatomic digits.  Only now are we approaching the order (or shall I say order-order or order-order-order) of magnitude of Graham's number.  And we were just getting started with Graham's number using it as seed for the most rapidly growing function imagined with a simple description, and then applying recursion on that.
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Re: Largest number using 5 symbols

24 Jul 2022 03:32

Would Graham's number even exceed the number of all subatomic particles in the multiverse if we consider our universe to be average and string theory landscape's prediction of 10^500 universes to be correct, Wats?
You have no idea how small that number of subatomic particles is compared to Graman's number.  Not even the number of permutation's of Planck volumes over the number of Planck times in the universe since the Big Bang will even begin to describe Graham's number.  There simply isn't a direct way to relate Graham's number with anything physical.  To give you an idea: The number of subatomic particles in the universe is much smaller than even the number of digits in Graham's number.  Much smaller than the number of digits of the number of digits.  And so on for a number of times greater than the number of subatomic digits.  Only now are we approaching the order (or shall I say order-order or order-order-order) of magnitude of Graham's number.  And we were just getting started with Graham's number using it as seed for the most rapidly growing function imagined with a simple description, and then applying recursion on that.
The argument can be made that if the universe/multiverse is mathematical than rather than the Big Bang, there must be a Big Bounce because if the universe/multiverse is mathematical than all numbers must be able to be described within the entirety of its existence.  So would say Max Tegmark.  We could apply a recursive function to the multiverse also, as you can envision universes existing inside black holes inside universes, and so on (and on and on lol)
 
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Re: Largest number using 5 symbols

25 Jul 2022 07:51

The argument can be made that if the universe/multiverse is mathematical than rather than the Big Bang, there must be a Big Bounce because if the universe/multiverse is mathematical than all numbers must be able to be described within the entirety of its existence.  So would say Max Tegmark.  We could apply a recursive function to the multiverse also, as you can envision universes existing inside black holes inside universes, and so on (and on and on lol)
That doesn't make sense to me.  I don't see why a "mathematical" universe couldn't be a subset of mathematics. If "mathematical" universe means that the universe can be fully described by mathematics, why couldn't mathematics also describe abstract, non-physical things?
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