User blog:Plutfuse/You came to right post

There were formulas with one f at left and crazy tower of Greek letters in my Probabilistic The Box Supercluster page. Also, there was some up arrows. What does it mean?

⌬Up arrow notation
This notation is infamous notation that is capable of writing tetration, pentation, hexation, and those sort of things very efficiently with up arrows.

⌬Introduction to arrow notation
One of most famous BIG number notations is up arrow notation. It can express exponentiation, tetration, pentation, hexation, and so on. You probably heard or saw 3↑↑3, and its value is a number about 7.6 trillion.

This is famous example of up arrow notation… but how the hell does it give out result of surprising number bigger than 7.6 trillion? How to calculate is little complicated, but relatively simple compared to other big number notations involving Turing machine or fast-growing hierarchy:

3↑3 = 3^3 = 27

3↑↑3 = 3↑(3↑3) = 3^27 = 7,625,597,484,987

3↑↑↑3 = 3↑↑(3↑↑3) = 3↑↑(7,625,597,484,987) = 3^…(7,625,597,484,987-2 more 3^)…3

3↑↑↑↑3 = 3↑↑↑(3↑↑↑3) = BAAM

This allows many numbers that are hard to describe with standard squaring notations to be shortened drastically. For example, the megafuga-n numbers can be described with up arrow notation like this:

Megafuga-n = n↑↑n

The easier way to understand tetration (↑↑) is if the formula is n↑↑m, there are m-1 number of n^ and a single n at the end of formula. But sorry, there is no way to easily understand pentation or hexation than:

“build an insanely high power tower”

⌬The Graham's Number
Ok, the number grew indescribably quickly, but why three? Can’t it be 4, 1,000,000, or other number? The reason is, that 3 with up arrows are used in Graham’s number, the most famous number besides googol series.

3↑↑↑↑3, or 3 hexation 3 is often called G(1), which is the first step of making Graham’s number. It is the upper limit answer to higher dimension’s hypercube coloring problem. By the way, 3↑↑↑↑3 can be expressed as 3(↑^4)3. Description of graham’s number is…

“Suppose there is n dimensional hypercube, with 2n of corners (For example, number of corners for square (2D cube) and cube (literally cube, 3D) is 4(=22) and 8(=23)). Then connect every point with diagonal line, and color it with red and blue.

But you must avoid coloring one whole side of that hypercube with red or blue (for more understandable sentence, you must avoid coloring 4 edges of hypercube's square side and its 2 diagonal lines with same color).

However, if dimension is sufficiently large, no matter how hard you try, you will have to color the edges of one square side of that hypercube and its diagonal lines with same color”

The description is from my Emoji growth page and I already described: because experts suspect 13th dimension as the answer to hypercube coloring problem, the answer is somewhere between 13 and graham’s number.

(meaning that Graham's number might be horribly wrong upper bound for coloring problem)

But back to the topic. Below are steps required to make G(64), graham’s number.

3↑↑↑↑3 = G(1)

3(↑^G(1))3 = G(2)

3(↑^G(2))3 = G(3)

3(↑^G(3))3 = G(4)

…

3(↑^G(62))3 = G(63)

3(↑^G(63))3 = G(64)

The Graham’s number is one of most famous big number using up arrow notation. It is also by far the biggest calculatable (meaning Busy Beaver or Frantic frogs don’t count) number in mathematics used for real purpose (that is not drastically decreasing every day like Skewes’ number) (also not counting fast-growing hierarchy).

Unless another problem involving dimensional hypercube coloring appears, Graham’s number will remain as the biggest number in real mathematics (again, not counting Turing machine functions and fast-growing hierarchy. It is a SEQUENCE, not a NUMBER).

⌬Fast-growing hierarchy
"formulas with one f at left and crazy tower of Greek letters" is called the "fast-growing hierarchy", and is a powerful and efficient number expression system. Those Greek alphabets are called "ordinals", but let's not get into that scary thing right away.

⌬Non-crazy numbers level
These are normal, plain numbers with quite boring growth. First, f0(number) is number+1. So, f0(1) is 2, f0(4024398) is 4024399. NOT IMPRESSIVE. VERY NOT IMPRESSIVE. If that tiny number next to f becomes 1, so it is f1(number), things barely start to get interesting. Lengthened formula of f1(number) is f0(f0(...f0(f0(number))...)) where there are 'number' amounts of f0(, and equals multiplying number by 2. You know how to multiply, and it is SO, SO BORING. Let's skip.

Formula "number x 2number" is result of f2(number). f2(15) is "15 x 215", and its value is kinda big at 491520 (type the formula in calculator if you cannot trust this result). 10100, or number that is called "googol", can give number MUCH bigger than googolplex, or 1010^100 when plugged to f2(number). Wonder f3(number)? Well, it is f2(number) applied to that "number", "number" times. For example, f3(15) is staggering f2(f2(f2(f2(f2(f2(f2(f2(f2(f2(f2(f2(f2(f2(f2(15))))))))))))))). Literally explosive! f4(number) is f3(number) applied to "number", "number" times, and change all 2 in longer formula of f3(15) to 3 to see how crazy FGH (shorten word of fast-growing hierarchy) gets.

⌬Extremely crazy numbers level
Next level of growth that surpasses EVERYTHING listed above is fω(number). It is fnumber(number), and is literally indescribable. Example: fω^2ω x ω^ω^ω^ω(2) is (some extremely long function formula). You can keep build tiny formulas with omega to get truly incomprehensible number. But omega ordinal level FGH is no match for fε0(2), which is fω↑↑ω(2). And you know what is value of it?

I don't know either.

🌕 🔮 🌎

¯\_(ツ)_/¯

Btw, fε0+1(2) is fε0(fε0(2)).

These ordinals can evolve to fζ0(n), to fη0(n), ...... to incomprehensible oblivion like gamma level, or Bachmann's psi function (which it's number array kinda looks almost same as EUSI) levels, next level being a fractal (?) of former step.

Extra: infamous TREE(3) that is known well for its immense size can be wrote as this:

$$f_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_ {\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_ {\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_ {\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_ {\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi_{\Phi}}}}} } } } } }}}}}} } } } } }}}}}} } } } } }}}}}} } } } } }}}}}} } } } } }}}}}} } } } } }}}}}} } } } } }}}}}} } } } } }}}}}} } } } } }}}}}} } } } } }}}}}} } } } } }}}}}} } } } } }}}}}} } } } } }}}}}} } } } } }}}}}} } } } } }}}}}} } } } } }}}}}} } } } } }}}}}} } } } } }}}}}} } } } } }}}}}} } } } } }$$Oops the formula is out of screen even here. It is sad that there is only few tens of these greek letters fit in the screen and there has to be exactly 187196 of that phi's with (1) next to that extreme height formula to get most accurate approximation of TREE(3). The reason why I am not writing TREE(3)'s value in here is because our planet will collapse into black hole that is capable of destroying entire universe, and even xenoverse if it really exists.

⌬The emoji growth
There was :D(number) kind of function at the beginning. What is it? Well.... It is over 100000 bytes long, so how about checking it out (The page says that the author of that page is... Plutfuse!!! :OOOO).