#fgh

The fast growing hierarchy (or fgh) is used to determine the growth rate of a Computable function. The rules are as follows (more detail of these rules later on):

f₀(x) = x+1

fₐ₊₁(x) = fₐⁿ(n)

fₐ⁰(n) = n

fₐⁱ⁺¹(n) = fₐⁱ(fₐ(n))

fₐ(n) = fₐ₍ₙ₎(n) for limit ordinal ‘a’ (ₐ₍ₙ₎ means a[n] in subscript)

Chapter 1, level 0

A hierarchy in math is an archief of functions, these functions are recalled by the name of the hierarchy followed by the index in subscript, like g₁, the hierarchy is called ‘g’ and the index is one. The first rule of the fgh says ‘f₀(x) = x+1’, this means the function f₀ (fgh level zero) takes a number and returns the successor, it plus one. This function grows liniarly and is comparable to a function that adds or subtracts a fixed number to the input, and everything beneath.

Chapter 2, level 1 to ω

Rule two, three and four imply ‘fₐ₊₁(x) = fₐⁿ(n)’, ‘fₐ⁰(n) = n’ and ‘fₐⁱ⁺¹(n) = fₐⁱ(fₐ(n))’, what means if the index is a successor of an other number, repeat the function in wich the index of the hierarchy is that other number the same amount of time as the input of the function. This means the function ‘f₁’ repeats the function ‘f₀’, the successor function, the same amount of time as the input, it doubles the number. An example for this:

f₁(4) = f₀⁴(4) = f₀³(f₀(4)) = f₀³(4+1) = f₀³(5) = f₀²(f₀(5)) = f₀²(5+1) = f₀²(6) = f₀¹(f₀(6)) = f₀¹(6+1) = f₀¹(7) = f₀⁰(f₀(7)) = f₀⁰(7+1) = f₀⁰(8) = 8 = 4*2

And so is the function ‘f₂’ the function ‘f₁’ iterated on the input the input amount of times (f₂(x) = x*2^x). f₁ is comparable to multiplying by a fixed number or a number lower then the input, f₂ is comparable to exponentiation and the triangle function and f₃ is comparable to tetration. You can already see that just adding one to the index of this hierarchy makes the function much more powerful each time. For the people who can understand it, a simple function to approximately compare the fgh to Knuth's up arrow notation is this: fₐ₊₁(n) ≃ 2^ᵃn.

Chapter 3, ω to ω2

Up to this point, it looks like none of the fgh's functions are comparable to Graham's function (G(0) = 4, G(n+1) = 3{G(n)}3). But this isn't true, because we haven't looked yet at rule number five: ‘fₐ(n) = fₐ₍ₙ₎(n) for limit ordinal “a” ’. This means when the index isn't a successor of another ordinal or the number zero (it is a limit ordinal), take the nth index of the index and put it in the index. ω is a limit ordinal and ω[n] = n, this means f_{ω}(n) = f_{ω[n]}(n) = f_{n}(n), (_{...} means the same as subscript), and f_{ω+1}(n) = f_{ω}ⁿ(n). fgh level ω+1 is comparable to Graham's function and ω is comparable to the Ackermann function. Some examples of this:

f_{ω+1}(2) = f_{ω}²(2) = f_{ω}¹(f_{ω}(2)) = f_{ω}¹(f₂(2)) = f_{ω}¹(2*2^2) = f_{ω}¹(8) = f_{ω}⁰(f_{ω}(8)) = f_{ω}(8) = f₈(8) ≃ 10^^^^^^^8 > Trisept

And that was only two!

f_{ω+1}(3) = f_{ω}³(3) = f_{ω}²(f_{ω}(3)) = f_{ω}²(f₃(3)) ≃ f_{ω}²(10^10^8) = f_{ω}¹(f_{ω}(10^10^8)) ≃ f_{ω}(10{10^10^8}10^10^8) ≃ 10{10{10^10^8}10^10^8}10^10^8 > Boogolplex

The fast growing hierarchy realy is fast growing.

And now for some larger ordinals:

fgh level ω+2 is comparable to Multiexpasion

fgh level ω+3 is comparable to Powerexplosion

fgh level ω+4 is comparable to Explodotetration

Chapter 4, ω2 to ω²

To calculate numbers in the fgh with an ordinal index larger than ω2 (ω times 2), use this rule:

(a+ω)[n] = a+(ω[n])

(There's actually an other rule but I think most people already know this: (a*b)+b = a*(b+1))

fgh level ω3 is comparable to copy notation, an example number in fgh level ω3 looks like (in Sbiis Sabians Hyper E notation):

f_{ω3}(4) = f_{ω2+ω}(4) = f_{ω2+4}(4) = f_{ω2+3}³(f_{ω2+3}(4)) = f_{ω2+3}³(f_{ω2+2}⁴(4)) = f_{ω2+3}³(f_{ω2+2}³(f_{ω2+2}(4))) = f_{ω2+3}³(f_{ω2+2}³(f_{ω2+1}³(f_{ω2}³(f_{ω2}(4))))) = f_{ω2+3}³(f_{ω2+2}³(f_{ω2+1}³(f_{ω2}³(f_{ω+4}(4))))) = f_{ω2+3}³(f_{ω2+2}³(f_{ω2+1}³(f_{ω2}³(f_{ω+3}³(f_{ω+2}³(f_{ω+1}³(f_{ω}³(f_{ω}(4))))))))) = f_{ω2+3}³(f_{ω2+2}³(f_{ω2+1}³(f_{ω2}³(f_{ω+3}³(f_{ω+2}³(f_{ω+1}³(f_{ω}³(f_{4}(4))))))))) = f_{ω2+3}³(f_{ω2+2}³(f_{ω2+1}³(f_{ω2}³(f_{ω+3}³(f_{ω+2}³(f_{ω+1}³(f_{ω}³(f_{3}³(f_{2}⁴(4)))))))))) = f_{ω2+3}³(f_{ω2+2}³(f_{ω2+1}³(f_{ω2}³(f_{ω+3}³(f_{ω+2}³(f_{ω+1}³(f_{ω}³(E2#3#5)))))))) ≃ E4###3

If you understand the extended hyper E function, you know how big of a number that is. For generalization: f_{ωa}(n) ≃ En###a.

Chapter 5, ω² to ζ₀

Here are the rest of the rules to go up to ε₀, I will not go higher and I will not go into detail in how every rule exactly works because this post is getting very long, also, a*b is not always equal to b*a:

(a*ω)[n] = a*n

ω^(a+1) = (ω^a)*ω

(ω^a)[n] = ω^(a[n]) for limit ordinal a

ε₀[0] = 1

ε_{a+1}[0] = ε_{a}+1

ε_{a+1}[n+1] = ω^(ε_{a+1}[n])

ε_{a}[n] = ε_{a[n]} for limit ordinal a

Conclusion

@mxrrymxdmxgcian​

The other gently, somewhat awkwardly, extending one hand. Glancing up at long last then, realizing just how long he must have been idly gazing at those slender digits. Yet the man stead of pulling away in ire or annoyance had seemed to only merely humor Rui, allowing him to take inordinate amounts of time peering at the hand offered to intensely study it.

Azure hues finally flickering up. To meet the other’s puzzled, curious, gaze. .

A heavy beat of silence then.

The weight of the air suddenly akin to a tomb, suffocating. Taken aback by the unexpected question. That struck right down to his very core, so very easily pushing past all of his defenses. Unable to help the soft flinch upon his face, yet still not drawing away. Merely inching closer to that strange human still...

Wavering hues tumbling down, quietly peering at the gloved digits once more.

Quaking digits, porcelain pale, ghosting upon that offered palm, surprisingly warm even through the thin fabric. Free hand clutching heavily the front of his thin yukata, just above his heart.

How had it come to this?

It had been such a simple wish to start. Kind, unfettered. Not wanting to see the gentle pity, the worry, that echoed within his parent’s warm eyes every. Single. Time. Every time he collapsed, wheezing softly upon the cold floor. Every time delicate hands brushed at damp bangs from a fevered brow...

Just wanting to be....stronger...

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