Almighty Array Notation

Almighty Array Notation (AAN) is a notation made by googology wiki user HaydenTheGoogologist2009.Hayden's Big Numbers - Introduction to Almighty Array Notation. Retrieved 2022-07-21. It has the format of a*(#) where # is an array. It comes in multiple parts. An example of a valid array is a*(10, 4, 1, 3, 5). Unfortunately, the notation is ill-defined as we will explained in #Issues section.
Unfortunately, the creator abandoned this notation on October 23rd, 2023 with the statement "This notation is a chaos, and I don't think it's possible to fix the entire notation. I should probably take a break from making notations..."Hayden's Big Numbers - Introduction to Almighty Array Notation. Retrieved 2023-10-23.
== Parts ==
We explain parts of notations together with the creator's expectation of the limits. We note that since this notation is ill-defined as we clarified above, this expectation does not mathematically make sense.
* Linear Array Notation (LAN):LAN Retrieved 2022-07-22 2:00. takes linear arrays. It's equivalent to linear BEAF. The limit of this is approximated to \(\omega^{\omega}\) in the fast-growing hierarchy.
* Extended Array Notation (ExAN):ExAN Retrieved 2022-07-22 2:00. uses planar and dimensional arrays. The limit of this is approximated to \(\omega^{\omega^{\omega}}\) in the fast-growing hierarchy.
* Cascading Array Notation (CAN):CAN Retrieved 2022-07-22 2:00. uses hyperdimensional arrays. The limit of this is approximated to \(\omega^{\omega^{\omega^{\omega}}}\) in the fast-growing hierarchy.
* Nested Array Notation (NAN):NAN Retrieved 2022-07-22 2:00. uses nested arrays to complete the tetrational arrays. The limit of this is approximated to \(\varepsilon_0\) in the fast-growing hierarchy.
For those beyond NAN, those notations are still incomplete, partially because we currently are in not-so-defined BEAF territory, as pentational arrays are not so well-defined.Hayden's Big Numbers - Introduction to Almighty Array Notation. Retrieved 2022-07-23.
The original source only included the definitions up to NAN, i.e. did not include the definition of the latter parts except for the names and the statements on the strength.
* First-order Array Notation (FoAN):FoAN Retrieved 2022-07-22 2:00.FoAN Retrieved 2022-07-23 10:00. adds a separator called the first-order hyper-separators (! is the shorthand for <1> separator, a fixed-point of {X} = {1{X}2}, a tetrational space, compared to Bird's [1\2] and DeepLineMadom's {1/2} fixed-points). The limit of this is approximated to \(\zeta_0\) in the fast-growing hierarchy.
Later, the creator updated the source to include the definition of FoAN.
* Extended First-order Array Notation (ExFoAN):ExFoAN Retrieved 2022-07-22 2:00.Hayden's Big Numbers - Extended First-order Array Notation. Retrieved 2022-07-23 12:23. uses linear expanding arrays with the first-order hyper-separators. The limit of this is approximated to \(\varphi(\omega,0)\) in the fast-growing hierarchy.
Later, the creator updated the source to include the definition of ExFoAN.
* Hyper-Extended First-order Array Notation (HExFoAN):HExFoAN Retrieved 2022-07-22 2:00.Hayden's Big Numbers - Hyper-Extended First-order Array Notation. Retrieved 2022-09-17 13:19. uses more than one copies of the first-order hyper-separators. The limit of this is approximated to \(\Gamma_0\) in the fast-growing hierarchy.
Later, the creator updated the source to include the definition of HExFoAN.
* Mega-Extended First-order Array Notation (MExFoAN):MExFoAN Retrieved 2022-07-22 2:00.Hayden's Big Numbers - Mega-Extended First-order Array Notation. Retrieved 2023-06-24 11:55. adds inner hyper-separators to extend the first-order hyper-separators. The limit of this is approximated to \(\psi_0(\varepsilon_{\Omega+1}) = \psi_0(\Omega_2)\) with respect to the (extended) Buchholz's function in the fast-growing hierarchy.
Later, the creator updated the source to include the definition of MExFoAN.
For parts beyond MExFoAN, the creator clarifies that he or she intends to create at least eleven other parts in the site and later abandoned by the creator:
* Second-order Array Notation (SoAN)
* Higher-order Array Notation (HoAN)
* Beyond-order Array Notation (BoAN)
* Primitive Ranking Array Notation (PRAN)
* Recursive Ranking Array Notation (RRAN)
* Exploding Array Notation (ExpAN)
* Detonating Array Notation (DAN)
* Nuclear Array Notation (NuAN)
* Solar Array Notation (SAN)
* Milky Array Notation (MAN)
* Galactic Array Notation (GAN)
== Etymology ==
The 'a' before the array in a*(#) stands for the word 'almighty'.
== Definition ==
=== Original version ===
A valid expression in the linear array notation is of the form a*(#), where # is a non-empty sequence of positive integers separated by comma, and the linear array notation uses pretty much the same rules as BEAF.
# Base rule: a*(a) = a and a*(a, b) = a^b
# Tailing rule: a*(# 1) = a*(#) where # indicates the rest of array
# Prime rule: a*(a, 1 #) = a
# Recursion rule: a*(a, b, 1, 1, ..., 1, 1, c #) = a*(a, a, a, a, ..., a, a*(a, b - 1, 1, 1, ..., 1, 1, c #), c - 1 #)
# If rules 1 - 4 do not apply: a*(a, b, c #) = a*(a, a*(a, b - 1, c #), c - 1 #)
A valid expression in ExAN is of the form a*(a, b X c), where a, b, c ≧ 1, and X are the separators. Here, a separator means a formal string denoted by {d} for a number d ≧ 1, and commas indicate {1} separator.
* Reuse the rules for linear arrays to that row (after separators other than commas)
* a*(a, b {2} c) a*(a, a, 1, 1, 1, ..., 1, 1, 2 {2} c - 1) with b string of 1's using comma as a separator.
* a*(a, b {2} 1 {2} ... {2} 1 {2} c) = a*(a, a, {2} 1 {2} ... {2} 1, 1, 1, ..., 1, 1, 2 {2} c - 1) with b string of 1's using comma as a separator.
* a*(a, b {d} 1 {d} ... {d} 1 {d} c) = a*(a, a, {d - 1} 1 {d - 1} ... {d - 1} 1, 1, 1, ..., 1, 1, 2 {d - 1} c - 1) with b string of 1's using {d - 1} as a separator.
In hyper-dimensional and nested arrays, the rules are the same as each other, arrays with separators that merely contain commas as separators inside {} separator are called hyper-dimensional arrays.
* a*(a, b {1, 1, 1, ..., 1, 1, c, d%} 2) = a*(a, a {1, 1, 1, ..., 1, b, c - 1, d%} 2), which is different from a*(a, b, 1, 1, ..., 1, 1, c #) = a*(a, a, a, a, ..., a, a*(a, b - 1, 1, 1, ..., 1, 1, c #), c - 1 #).
* a*(a, b {1 {2} c} 2) = a*(a, a {1 {1, 1, 1, ..., 1, 1, 2 {2} c - 1} 2), and so on.
Although NAN has a definition, there is no definition of notations beyond NAN.
=== Second version ===
A valid expression in the linear array notation is of the form a*(#), where # is a non-empty sequence of positive integers separated by comma, and the linear array notation uses pretty much the same rules as BEAF.Hayden's Big Numbers - Linear Array Notation. Retrieved 2022-07-23.
# Empty array rule: a*() = 1, since a*(1) = a*() using the tailing rule.
# Base rule: a*(a) = a and a*(a, b) = a^b
# Tailing rule: a*(# 1) = a*(#) where # indicates the rest of array
# Prime rule: a*(a, 1 #) = a
# Recursion rule: a*(a, b, 1, 1, ..., 1, 1, c #) = a*(a, a, a, a, ..., a, a*(a, b - 1, 1, 1, ..., 1, 1, c #), c - 1 #) (w/ b number of a's) for a, b, c > 1 and the length of the array ≥ 4
# If rules 1 - 5 do not apply: a*(a, b, c #) = a*(a, a*(a, b - 1, c #), c - 1 #)
A valid expression in ExAN is of the form a*(a, b X c X d X e X ... X n) where a, b, c, d, e, ..., n ≥ 1, and X are the separators.Hayden's Big Numbers - Extended Array Notation. Retrieved 2022-07-23. Here, a separator means a formal string denoted by {d} for a number d ≧ 1, and commas indicate {1} separator.
* Reuse the rules for linear arrays to that row (after separators other than commas)
* a*(a, b {2} c) a*(a, a, 1, 1, 1, ..., 1, 1, 2 {2} c - 1) with b string of 1's using comma as a separator.
* a*(a, b {2} 1 {2} ... {2} 1 {2} c) = a*(a, a, {2} 1 {2} ... {2} 1, 1, 1, ..., 1, 1, 2 {2} c - 1) with b string of 1's using comma as a separator.
* a*(a, b {d} 1 {d} ... {d} 1 {d} c) = a*(a, a, {d - 1} 1 {d - 1} ... {d - 1} 1, 1, 1, ..., 1, 1, 2 {d - 1} c - 1) with b string of 1's using {d - 1} as a separator.
In hyper-dimensional and nested arrays, the rules are the same as each other, arrays with separators that merely contain commas as separators inside {} separator are called hyper-dimensional arrays.
* a*(a, b {1, 1, 1, ..., 1, 1, c, d%} 2) = a*(a, a {1, 1, 1, ..., 1, b, c - 1, d%} 2), which is different from a*(a, b, 1, 1, ..., 1, 1, c #) = a*(a, a, a, a, ..., a, a*(a, b - 1, 1, 1, ..., 1, 1, c #), c - 1 #).
* a*(a, b {1 {2} c} 2) = a*(a, a {1, 1, 1, ..., 1, 1, 2 {2} c - 1} 2), and so on.
Although NAN and FoAN have definitions, there is no definition of notations beyond FoAN.
=== Third version ===
The third version adds an additional rule:
* If more than one rule(s) apply to an array, start from step 1, and find the nearest rule that applies to that array.Hayden's Big Numbers - Linear Array Notation. Retrieved 2022-11-12 13:27.Hayden's Big Numbers - Extended Array Notation. Retrieved 2022-11-12 13:27.Hayden's Big Numbers - Cascading Array Notation. Retrieved 2022-11-12 13:27.Hayden's Big Numbers - Nested Array Notation. Retrieved 2022-11-12 13:27.Hayden's Big Numbers - First-order Array Notation. Retrieved 2022-11-12 13:27.Hayden's Big Numbers - Extended First-order Array Notation. Retrieved 2022-11-12 13:27.Hayden's Big Numbers - Hyper Extended First-order Array Notation. Retrieved 2022-11-12 13:27.Hayden's Big Numbers - Mega Extended First-order Array Notation. Retrieved 2022-11-12 13:27.
Although NAN, FoAN, ExFoAN and HExFoAN have definitions, there is no definition of notations beyond HExFoAN.
== Issues ==
Since the symbol "a" is unquantified in the definition of a valid expression of LAN, the first symbol "a" of a valid expression "a*(#)" is perhaps different from other occurrence of the variable "a" in parentheses in the rules. This is informal abuse of a single symbol.
Even if we ignore this ambiguity, there are several issues:
# Both the base rule and tailing rule are applicable to the single valid expression a*(1):
## If we apply Base rule, then the resulting value is 1.
## If we apply Tailing rule, then the resulting value is a*(), which is ill-formed due to the emptiness of the sequence in the parenthesis. (We note that "#" in the definition of a valid expression is assumed to be non-empty, but "#" in Tailing rule is defined as the rest of the sequence in the parenthesis, which is empty when the array is of length \(1\). In order to avoid the abuse of "#" in two distinct meanings and conditions, one can use the case classification "If # is expressed as #' 1 for some non-empty sequence of positive integers by abbreviating the concatenation of sequences, then a(#) = a*(#')." instead of the partial specialization "a*(# 1) = a(#)".)
##* Hayden later solved this problem, as one could see progressively in the second and third versions of the notation.
# The use of ellipses in Recursion rule is ambiguous, because it does not possess the precise condition on the length of the 1's. (Is \(0\) allowed? Or is it assumed to be greater than or equal to \(4\)?) Moreover, the first occurrence of the ellipsis in the right hand side is quite ambiguous, because there is no information about the length of a's. (Is it the same as 1 plus the length of 1's in the left hand side? Or is it the same as some other value such as a*(a, b - 1, 1, 1, ..., 1, 1, c #)?) Even if we ignore the ambiguity, there are multiple ways to apply Recursion rule to the single valid expression a*(2,2,1,1,1,1,1,2).
## If we apply Recursion rule for the case \(a = 2\), \(b = 2\), \(c = 1\), and \(\# = 2\), then the resulting expression is a*(2,2,2,2, ...,2, a*(2,1, 1, 1, ..., 1, 1,1,2),0,2), which is ill-formed due to the occurrence of \(0\).
## If we apply Recursion rule for the case \(a = 2\), \(b = 2\), \(c = 2\), and \(\#\) is empty, then the resulting expression is a*(2,2,2,2, ...,2, a*(2,1, 1, 1, ..., 1, 1,1,2),1).
##* Although quite a weird way to solve such a problem, Hayden half-solved this by saying "Recursion rule: a*(a, b, 1, 1, ..., 1, 1, c #) = a*(a, a, a, a, ..., a, a*(a, b - 1, 1, 1, ..., 1, 1, c #), c - 1 #) (w/ b number of a's) for a, b, c > 1 and the length of the array ≥ 4." (Why b number of a's?) However, the length of the string of ones is still not told (yes, it can be one or more entries, but that's basically saying a*(a,b,1...1,c) is always equal to a*(a...a,c-1) with b a's, which would be nonsensical).
# Both of Prime rule and Recursion rule are applicable to the single valid expression a*(2,1,1,1,1,1,2,2):
## If we apply Prime rule, then the resulting expression is a*(2).
## If we apply Recursion rule for the case \(a = 2\), \(b = 1\), \(c = 2\), and \(\# = 2\), then the resulting expression is a*(2,2,2,2, ...,2, a*(2,0, 1, 1, ..., 1, 1,2,2),1,2), which is ill-formed due to the occurrence of \(0\).
##* Hayden later solved this problem, as one could see in the new rule made in the third version.
Similarly, other parts of the notation includes many issues. For example, ExAN directly uses LAN, which is ill-defined, and the rules refer to ill-formed expressions a*(a, b {2} 1 {2} ... {2} 1 {2} c), as a valid expression is assumed to be of the form a*(a,b X c), which should include precisely two separators.
* Hayden later solved this problem by saying "A valid expression in ExAN is of the form a*(a, b X c X d X e X ... X n) where a, b, c, d, e, ..., n ≥ 1, and X are the separators." in the second version.
Moreover, since the sources for the parts past NAN do not include any information other than the names of the parts, people except for the creator cannot estimate the definitions. Although the creator insists that the notation is expected to reaches ordinal collapsing functions with weakly higher Mahlo cardinals and a weakly compact cardinal, there is no justification of the expectation because of the lack of the definitions.
Therefore, the first few parts are incomplete, and the latter parts are unformalized. Namely, the whole part of the notation is ill-defined. We note that personal works from a source which is not peer-reviewed frequently have such issues. In particular, even if the creator corrected all the issues which have been pointed out, there might be still other errors. Therefore, the reader should be careful that the future revision does not imply the well-definedness of the notation and the correctness of the analysis.
== Sources ==
Category:Notations
Category:Functions
Category:Array Notations
Category:Incomplete
Category:Unformalised