1. Introduction Factorial1.1. Simple FactorialDefinition and notation: For all n∈N,
n!
=n∙(n-1)!
=n(n-1)(n-2)... 3∙2∙1
Example:
1!
=1
2!
=2∙1=2
3!
=3∙2∙1=6
4!
=4∙3∙2∙1=24
5!
=5∙4∙3∙2∙1=120
10!
=362880≈3.62×106
100!
≈9.33×10157
1000!
≈4.62×102567
Trivia:100! already bigger than the number of all particles (all quarks in neutron and proton, all electrons, in all of atoms) in the whole observable universe, which is ±3.28×108011.2. Ultra-Factorial Notation1.3. First Ultra-Factorial.This is the factorial version of Knuths's-up arrow notation.Let us define the Knuths's-up arrow notation.
a↑b
=ab=a∙a∙a…a∙a∙a⏠⏣⏣⏣⏣⏡⏣⏣⏣⏣⏢b times
a↑↑b
=a↑(a↑(…(a↑a)))⏠⏣⏣⏣⏣⏣⏡⏣⏣⏣⏣⏣⏢b times=aaa⋰a⏠⏣⏣⏡⏣⏣⏢b times
The first ultra-factorial (written as n!1), thus defined as exponentiation with factorial-like notation at the level of double up-arrows.
n!1
=n↑(n-1)↑(n-2)↑…↑3↑2↑1
=n(n-1)(n-2)⋰21
Tower factorial defined recursively as following
n!1
=n↑(n-1)!1
Example
3!1
=3↑2↑1
=321=32
=9
4!1
=4↑3↑2↑1
=4321=49
=262144
5!1
=5↑4↑3↑2↑1
=54321=5262144
≈6.21×1018323
≈10105.26
Trivia:5!1 is much much larger than approximate number of Planck volumes composing the volume of the observable universe, which is 8.5×10184.21.4. Second Ultra-FactorialDefinition and notation
n!2
=n!1↑↑(n-1)!1↑↑(n-2)!1↑↑…↑↑3!1↑↑2!1↑↑1!1
=n!1↑↑(n-1)!2
Example
3!2
=3↑↑2↑↑1
First, we calculate 2↑↑1, which is only 2, since n raised to 1 is always 2.
3!2
=3↑↑2
=3↑3
=27
4!2
=4↑↑3↑↑2↑↑1
=4↑↑3!2
=4↑↑27
=4↑(4↑(…↑(4↑4)…))⏠⏣⏣⏣⏣⏣⏣⏡⏣⏣⏣⏣⏣⏣⏢27
=444⋰4⏠⏣⏣⏡⏣⏣⏢27
>f𝜔(3)(lower bound in fast-growing hierarchy)
<f𝜔(4)(upper bound in fast-growing hierarchy)
Trivia:4!2 already much much bigger (not even on the same ordinal) with Poincare recurrence time, the length of time elapsed until the recurrence of universe, which is 10101010101.1years.3 Even if you convert that time into second, no ... even if you convert that time into planck time, 4!2 is much larger than that.1.5. Third Ultra-FactorialDefinition and notation:
n!3
=n!2↑↑↑(n-1)!2↑↑↑(n-2)!2↑↑↑…↑↑↑2!2↑↑↑1!2
=n!3↑↑↑(n-1)!3
Example
3!3
=3↑↑↑2↑↑↑1
=3↑↑↑2
=3↑↑3
=3↑(3↑3)
=3↑27
=7 625 597 484 987
≈7.63 × 1012
4!3
=4↑↑↑3↑↑↑2↑↑↑1
=4↑↑↑3!3
=4↑↑↑(3↑27)
=4↑↑(4↑↑(…(4↑↑4)…))⏠⏣⏣⏣⏣⏣⏣⏡⏣⏣⏣⏣⏣⏣⏢3↑27
>f𝜔(5)
<f𝜔(6)
Trivia:1.6. wth Ultra-FactorialDefinition and notation: For all w∈N,
n!w
=n!w-1↑w(n-1)!w-1↑w(n-2)!w-1↑w…↑w 2!w-1↑w 1!w-1
=n!w-1↑w(n-1)!w
Here, n is called the argument, and w is called the iteration of ultra-factorial.Example:
n!69
=n!68↑69(n-1)!69
n!n!50
=n!n!50-1↑n!50(n-1)!n!50
In (1), the iteration (w) itself is in the form of ultra-factorial, with 50 iteration.2. Ultra2-Factorial2.1. Introduction to Ultram-FactorialWe can generalized Ultra-Factorial as Ultra1-Factorial, adding number 1 as subscript in symbol !n!w=n!w1Generalized notation of Ultram-Factorialn!wmHere, subscript m (m∈N+) is called the level of ultrafactorial, n is called the argument, and w is called the iteration of ultra-factorial.
Level
Name
Notation
1
Ultra1-Factorial
n!w=n!w1
2
Ultra2-Factorial
n!w2
3
Ultra3-Factorial
n!w3
⋮
⋮
⋮
m
Ultram-Factorial
n!wm
2.2. First Iteration of Ultra2-FactorialDefinition and notation
n!12
=n!a1
Where a1=n!11n!12=n!n!111Here, a is recursive notation to help us track the number of iteration (w) and write the full expression, since it will be stacked-up creating a tower that would be hard to write if we didn't invent some kind of notation helper.Example for n=3
3!12
=3!a1
with a
=3!11=3!=6
3!12
=3!61
=3↑6(2↑6 1)
=3↑6 2
=3↑5 3
>f𝜔6
<f𝜔7
Example for n=4
4!12
=4!a1
with a
=4!11=4!=24
4!12
=4!241
=4↑24(3↑24(2↑24 1))
=4↑24(3↑24 2)
=4↑24(3↑23 3)
2.3. Second Iteration of Ultra2-FactorialDefinition and notation
2.4. wth Iteration of Ultra2-FactorialDefinition and notation
n!w2
=n!aw1
Where aw=n!w-123. Ultra3-Factorial3.1. First Ultra3-FactorialDefinition and notation
n!13
=n!a12
a1
=n!m-1m-1
=n!22
Notes that m here is the level of ultrafactorial. Since we deal with Ultra3-Factorial, m=3.Example for n=4
413
=4!a12
a1
=4!22
4!22
=4!4!4!1111
Thus,
413
=4!24!4!4!1111
From definition of n!w2 in section 2.4,
4!24!4!4!1111
=4!1a4!4!4!1111
a4!4!4!1111-1
=4!1a4!4!4!1111-1
Ugh, my brain hurt. It's very beeeg. Let just settle here. We can proceed to write in "arrow notation", but it is very complicated. From (5) we know that4!22=4↑(4↑24(3↑23 3))(3↑(4↑24(3↑23 3)) 2↑(4↑24(3↑23 3)) 1) Therefore,4!13=4!4!222=4!(4↑(4↑24(3↑23 3))(3↑(4↑24(3↑23 3)) 2↑(4↑24(3↑23 3)) 1))2From these example, you can see that every time we move one level, it's gonna exploded! 4!13 is already humongous in 4!2 definition and we still don't know how monstrous it is if we proceed to write in 4!1 definition.3.2. Second Ultra3-FactorialDefinition and notation
n!23
=n!a22
a2
=n!a12
a1
=n!22
Comment:If the first iteration of ultra3-factorial is already humongous, how about the second one? Ah, I can't imagine it. I can't even imagine to unpack it. Let's just settle it here. 3.3. wth Ultra3-FactorialDefinition and notation
n!w3
=n!aw2
aw
=n!aw-12
⋮
⋮
a2
=n!a12
a1
=n!22
4. Generalized Ultram-FactorialDefinition and notation
n!wm
=n!awm-1
aw
=n!aw-1m-1
⋮
⋮
a2
=n!a1m-1
a1
=n!m-1m-1
Example for m=2 and w=2 (second iteration of level 2)
n!22
=n!a22-1=n!a21
a2
=n!a1m-1=n!a11
a1
=n!m-1m-1
=n!11
Thus,
n!22
=n!n!n!1111
As you can see, it's the same value with equation (4).5. Beyond Ultram-Factorial: Ultra-Ara-Factorial (UAF)5.1. IntroductionWe can write m as ultram-factorial, such as:n!wn!wmor n!wn!wn!wn!wmBut it is ugly, right? We need another system of notation that clear and neat. However, we have used up both of superscript (the iteration) and subscript (the level of ultra-factorial). We don't have place to write another symbol ... except if we use another symbol. I propose "{" as our new symbol. It's good and aesthetically pleasing.5.2. Ultra-Ara-Factorial
