Hyper-Extended First-order Array Notation (HExFoAN) is the seventh part of almighty array notation. A valid expression in HExFoAN 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.
The Hyper-Extended First-order Array Notation has the following 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.
The separator can come in many forms, such as {3}, {1, 4, 5}, {1, 1, 3 {2} 2}.
The first entry of the array is the base, the number after it is the iterator.
The {1} separator stands for comma.
In this extension, the separators are the comma, the exclamation mark (!) [not to be confused with the factorial operation], and {x A n} where x is an expression, A is a separator, and n is an integer > 1.
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 1: a*(# 1 X #) = a*(#) where # indicates the rest of array (X can be any separators)
Tailing rule 2: a*(# X 1) = a*(#)
Tailing rule 3: a*(# {% 1} #) = a*(# {%} #) where % indicates the rest of array in the separator
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 - 7 do not apply: a*(a, b, c #) = a*(a, a*(a, b - 1, c #), c - 1 #)
Comma rule: a*(a, b {1} c #) = a*(a, b, c #)
If more than one rule(s) apply to an array, start from step 1, and find the nearest rule that applies to that array.
So a*(2, 1, 1, 1, 1, 1, 2, 2) = a*(2) = 2 (by rule 6) instead of a*(2, 2, 2, 2, ..., 2, a*(2, 0, 1, 1, ..., 1, 1, 2, 2), 1, 2) (by rule 7), which is ill-formed due to the occurrence of 0.
Followed by some previous rules by following:
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.
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. Nested arrays reuse the rules 2 through 5.
If n < m, a*(# {n} 1 {m} #) = a*(# {m} #).
At this part, we define the new rules by following:
Reuse the previous rules above and ignore the exclamation mark separators.
Define the simplest expression for that part as a*(a, b {1 ! 2} 2) = a*(a, a {1 { ... 1 {1, 2} 2 ... } 2} 2)
If the first entry before an exclamation mark is just 1, change the {1 ! n + 1 #} to Sb (b is the iterator), where S1 is {1 ! n #}, Sn + 1 = {1 {Sn} 2 ! n #}, then change the iterator to the base.
Now we need to introduce more high-ranking separators. After !, it comes to !! = <2>, !!! = <3>, !!!! = <4>, <1, 2>, <1 {2} 2>, <1 {1 ! 2} 2>, etc. using all arrays up to this point. The previous process rules still apply, so that <2> or !! decomposes to just !, !!! decomposes to !!, !!!! decomposes to !!!, and so on.
So, a*(3, 4 {1 <2> 2} 2) = a*(3, 4 {1 !! 2} 2) = a*(3, 3 {1 ! 1 ! 1 ! 1 ! 2} 2).
a*(a, b {1 !! 2} 2) = a*(a, a {1 ! 1 ! ... ! 1 ! 2} 2) with b string of 1's separated by !. FGH level φ(ω, 0).
a*(a, b {1 ! 2 !! 2} 2) ~ FGH level ε(φ(ω, 0) + 1).
a*(a, b {1 ! 1 ! 2 !! 2} 2) ~ FGH level ζ(φ(ω, 0) + 1).
a*(a, b {1 !! 3} 2) = a*(a, a {1 ! 1 ! ... ! 1 !! 2} 2) with b string of 1's separated by !. FGH level φ(ω, 1).
a*(a, b {1 !! 4} 2) = a*(a, a {1 ! 1 ! ... ! 1 !! 3} 2) with b string of 1's separated by !. FGH level φ(ω, 2).
a*(a, b {1 !! 1, 2} 2) = a*(a, a {1 !! b} 2). FGH level φ(ω, ω).
a*(a, b {1 !! 1 {1 ! 2} 2} 2) = a*(a, a {1 !! b} 2). FGH level φ(ω, ε0).
a*(a, b {1 !! 1 {1 !! 2} 2} 2) = a*(a, a {1 !! b} 2). FGH level φ(ω, φ(ω, 0)).
a*(a, b {1 !! 1 ! 2} 2) = a*(a, a {1 !! 1 {1 !! 1 ... {1 !! 2} ... 2} 2} 2) with b 1{1's inside the outermost {}. FGH level φ(ω + 1, 0).
a*(a, b {1 !! 1 ! 1 ! 2} 2) ~ FGH level φ(ω + 2, 0).
a*(a, b {1 !! 1 !! 2} 2) = a*(a, a {1 !! 1 ! 1 ! ... ! 1 ! 2} 2) with b string of 1's separated by !. FGH level φ(ω2, 0).
a*(a, b {1 !! 1 !! 1 !! 2} 2) ~ FGH level φ(ω3, 0).
a*(a, b {1 !! 1 !! 1 !! 1 !! 2} 2) ~ FGH level φ(ω4, 0).
a*(a, b {1 !!! 2} 2) = a*(a, a {1 !! 1 !! ... !! 1 !! 2} 2) with b string of 1's separated by !!. FGH level φ(ω^2, 0).
a*(a, b {1 !!!! 2} 2) = a*(a, a {1 !!! 1 !!! ... !!! 1 !!! 2} 2) with b string of 1's separated by !!!. FGH level φ(ω^3, 0).
a*(a, b {1 <1, 2> 2} 2) = a*(a, a {1 !!!! ... !!!! 2} 2) with b !'s. FGH level φ(ω^ω, 0).
a*(a, b {1 <1, 1, 2> 2} 2) ~ FGH level φ(ω^ω^2, 0).
a*(a, b {1 <1 {2} 2> 2} 2) ~ FGH level φ(ω^ω^ω, 0).
a*(a, b {1 <1 {1, 2} 2> 2} 2) ~ FGH level φ(ω^ω^ω^ω, 0).
a*(a, b {1 <1 {1 ! 2} 2> 2} 2) ~ FGH level φ(ε0, 0).
a*(a, b {1 <1 {1 ! 3} 2> 2} 2) ~ FGH level φ(ε1, 0).
a*(a, b {1 <1 {1 ! 1, 2} 2> 2} 2) ~ FGH level φ(εω, 0).
a*(a, b {1 <1 {1 ! 1 {1 ! 2} 2} 2> 2} 2) ~ FGH level φ(ε(ε0), 0).
a*(a, b {1 <1 {1 ! 1 ! 2} 2} 2> 2} 2) ~ FGH level φ(ζ0, 0).
a*(a, b {1 <1 {1 ! 1 ! 1 ! 2} 2} 2> 2} 2) ~ FGH level φ(η0, 0).
a*(a, b {1 <1 {1 !! 2} 2} 2> 2} 2) ~ FGH level φ(φ(ω, 0), 0).
a*(a, b {1 <1 {1 !! 1 !! 2} 2} 2> 2} 2) ~ FGH level φ(φ(ω2, 0), 0).
a*(a, b {1 <1 {1 !!! 2} 2} 2> 2} 2) ~ FGH level φ(φ(ω^2, 0), 0).
a*(a, b {1 <1 {1 <1, 2> 2} 2} 2> 2} 2) ~ FGH level φ(φ(ω^ω, 0), 0).
a*(a, b {1 <1 {1 <1 {1 ! 2} 2> 2} 2} 2> 2} 2) ~ FGH level φ(φ(ε0, 0), 0).
a*(a, b {1 <1 {1 <1 {1 !! 2} 2> 2} 2} 2> 2} 2) ~ FGH level φ(φ(φ(ω, 0), 0), 0).
a*(a, b {1 <1 {1 <1 {1 <1 {1 ! 2} 2> 2} 2> 2} 2} 2> 2} 2) ~ FGH level φ(φ(φ(ε0, 0), 0), 0).
And finally, the limit is now Γ0 (Feferman–Schütte ordinal) = φ(1, 0, 0) = ψ(Ω^Ω) = θ(Ω). The Γ0-function is defined at a*(a, b {1 <1 ! 2> 2} 2) = a*(a, a {1 <1 {1 <1 ... {1 ! 2} 2> 2} 2> 2} ... 2> 2} 2) with b - 1 1{1's.