Phenol notation

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Phenol notation
Growth rate	\(\approx f\)\(_{\psi_0(\varepsilon_{\Omega+1})}\)\((n)\) (the creator's expectation of the limit of this notation using Buchholz's function)
Notations	a[#]b
Author(s)	Googology wiki user DeepLineMadom
Year	2022

Phenol notation (also known as simply "phenol") is an array notation created by Wikia user DeepLineMadom.. It was originally started as an ill-definedness fixed version of Username5243's Array Notation, but later parts have diverged from UNAN and became semi-homogenous with DeepLineMadom's Array Notation.

Parts

These three extensions have been formalised:

• Extension 1 - Basic Array Notation (BAN) - Takes simple arrays with one entry a[c]b. FGH level \(\omega\).
• Extension 2 - Extended Array Notation (XAN) - Takes linear, dimensional, and hyperdimensional arrays. FGH level \(\varepsilon_0\).
• Extension 3 - First-order Array Notation (FoAN) - Adds the first-order separators. Expected FGH level \(\psi_0(\varepsilon_{\Omega+1})\).

These extensions have not been formalised, and are parts the creator has yet to work on; however, the creator has outlined the expected FGH limits for some of these, under the assumption that they have been formalised.

• Extension 4 - Multi-order Array Notation (MoAN) - Adds some greater nth-order separators. Expected limit \(\psi_0(\Omega_\omega)\).
• Extension 5 - Hyper-order Array Notation (HoAN) - Adds some arrays into separators' subscript called "hyper-nth-order separators". Expected limit \(\psi_0(\Lambda)\) (Countable limit of Extended Buchholz's function).
• Extension 6 - Expanding Array Notation (XpaAN) - Adds the super-order separators which ranks over the subscript fixed point.
• Extension 7 - Extended Expanding Array Notation (XXpaAN) - Adds separators with multiple characters and superscripts.

Extensions

Extension 1 - Basic Array Notation

Basic Definition

The basic array notation has the following form:

a[c]b

where all variables (a, b, c) are non-negative integers.

Rules

1. Base Rule: a[0]b = a×b
2. Prime Rule: a[c]1 = a, a[c]0 = 1 if c > 0, a[0]0 = 0
3. Recursion Rule: a[c]b = a[c-1](a[c](b-1)) if b > 1 and c > 0

If there are two or more distinct rules to apply to a single expression, the lowest-numbered rule which is applicable and whose result is a valid expression will be applied.

Explanation and Analysis

It's easy to see that the above three rules mirror the definition of up-arrow notation. So the limit of this notation is f_ω(n). A small example:

2[3]3
= 2[2]2[3]2
= 2[2]2[2]2[3]1
= 2[2]2[2]2
= 2[2]2[1]2[2]1
= 2[2]2[1]2
= 2[2]2[0]2[1]1
= 2[2]2[0]2
= 2[2]2*2
= 2[2]4
= 2[1]2[2]3
= 2[1]2[1]2[2]2
= 2[1]2[1]2[1]2[2]1
= 2[1]2[1]2[1]2
= 2[1]2[1]2[0]2[1]1
= 2[1]2[1]2[0]2
= 2[1]2[1]2*2
= 2[1]2[1]4
= 2[1]2[0]2[1]3
= 2[1]2[0]2[0]2[1]2
= 2[1]2[0]2[0]2[0]2[1]1
= 2[1]2[0]2[0]2[0]2
= 2[1]2[0]2[0]2*2
= 2[1]2[0]2[0]4
= 2[1]2[0]2*4
= 2[1]2[0]8
= 2[1]2*8
= 2[1]16
= 2[0]2[1]15
= 2[0]2[0]2[1]14
= ...
= 2[0]2[0]2[0]...[0]2[0]2[0]2 (with 16 2's)
= 2^16
= 65,536