ThaAwesome's Array Notation

ThaAwesome's Array NotationThaAwesome's Array Notation retrieved (UTC) 2022-05-01 16:00 is an array notation created by Googology wiki user ThaAwesome that is inspired by BEAF.
== Issues ==
The notation is ill-defined due to many errors such as an incomplete case classification, the lack of the precise quantification of variables, the lack of the precise definition of the domain, and infinite loops. For example, since there is no precise quantification of variables declaring the precise range of the valriables, there are three or more natural choices of the range, all of which cause distinct serious errors:
# If all variables are intended to be quantified by "for all integers", then the rule "a[b,n,y,d]c = a[b,n,x]c, where x is a[b,n,y,d - 1]c" causes an infinite loop 1[2,3,4,1]5 = 1[2,3,1[2,3,4,0]5]5 = 1[2,3,1[2,3,1[2,3,4,-1]5]5]5 = ….
# If all variables are intended to be quantified by "for all non-negative integers", then the same rule ensures 1[2,3,4,0]5 = 1[2,3,1[2,3,4,-1]5]5 but there is no rule applicable to 1[2,3,4,-1]5.
# If all variables are intended to be quantified by "for all positive integers", then the same rule ensures 1[2,3,4,1]5 = 1[2,3,1[2,3,4,0]5]5 but there is no rule applicable to 1[2,3,4,0]5.
As a conclusion, the notation will be still ill-defined due to an infinite loop or an incomplete case classification, if the creator simply add natural quantification. Note that the creator removed the description of the ill-definedness of 3[3,3,3,3]3 and insisted its well-definedness,A difference page of the article of ThaAwesome's Hexatri the same problem appears also for this expression.
The notation and was said to "go all the way up to Mahlo" in an old revision of this articleAn old version of this article. However, this approximation does not make sense, since the notation is ill-defined and they do not fix a specific OCF or system of fundamental sequences. The creator later insisted that the OCF was "Buchholz's extended psi function".A difference page of this article If it means Extended Buchholz's function, then the description still has issues:
# The expression \(\psi(M)\) is ill-formed in Extended Buchholz's function, because it requires two variables. (See also Buchholz's function#Common misconceptions for an issue on abbreviation of the first argument.)
# Extended Buchholz's function is eventually constant above the least omega fixed point, and hence the claim "go all the way up to Mahlo" which sounds like that it non-constantly grows up to \(M\) is inappropriate.
# The comparison of a well-defined OCF and an ill-defined notation does not make sense.
Therefore in this sense, the claim that it is of Mahlo level is inappropriate.
== Intended values ==
Although the notation itself does not have a precise definition, below are some intended values listed by the author:
a[b]c = a ↑ᵇ c
a[b,1]c = a followed by a[b]c up arrows to c
a[b,2]c  = a followed by a[b,1]c up arrows to c
a[b,3]c  = a followed by a[b,2]c up arrows to c
a[b,n]c  = a followed by a[b,n - 1]c up arrows to c
a[b,n,1]c = a[b,x]c, where x is a[b,n]c
a[b,n,2]c = a[b,x]c, where x is a[b,n,1]c
a[b,n,y]c = a[b,x]c, where x is a[b,n,y - 1]c
a[b,n,y,d]c = a[b,n,x]c, where x is a[b,n,y,d - 1]c
a[b,n,y,d,e] = a[b,n,d,x]c, where x is a[b,n,y,d,e - 1]c
a[[b]]c = a[b,b,b,b,b,...b,b]c, with b numbers of b's
a[[b,1]]c = a[[x]]c where x is a[[b]]c
a[[b,n]]c = a[[x,n - 1]]c where x is a[[b,n - 1]]c
a[[[b]]]c = a[[b,b,b,b,b,..b,b]]c with b numbers of b's
a[b/0]c = a[[[[...[[b]]]...]]c with b numbers of brackets
a[b/1]c = a[[[...[b,b,b,b,b,...b]]...]]c with b number of brackets & b's
a[b/2]c = a[[[...[[b,b,b,b,b,...b]]]...]]c with a[b/1]c number of brackets & b's
a[b/n]c = a[[[...[[b,b,b,b,b,...b]]]...]]c with a[b/n - 1]c number of brackets & b's
a[b,1/0]c = a[[[...[[b/b]]]...]]c with b number of brackets & b's
a[b,1/1]c = a[[[...[[b/b]]]...]]c with a[b,1/0]c number of brackets & b's
a[b,2/0]c = a[x/x]c where x is a[b,1/0]c
a[b,n/0]c = a[x/x]c where x is a[b,n - 1/0]
a[b,n,1/0]c = a[x,x - 1/x] where x is a[b,n/y]c
a[b\0]c = a[b,b,b,b,...b/b]c with b number of b's
a[b\1]c = a[b,b,b,b,...b\0]c with b number of b's
a[b\n]c = a[b,b,b,b,...b\n - 1]c with b number of b's
a[b,1\n]c = a[x\n] where x is a[b\n]c
a[b,2\n]c = a[x\n] where x is a[b,1\n]c
a[b,y\n]c = a[x\n] where x is a[b,y - 1\n]c
a[b,y,1\n]c = a[x,x\n] where x is a[b,y - 1\n]c
a[b,y,d\n]c = a[x,x\n] where x is a[b,y,d - 1\n]c
a[b$0]c = a[b,b,b,b,...b\b]c with b numbers of b's
a[b$1]c = a[x$0]c where x is a[b$0]c
a[b$n]c = a[x$n - 1]c where x is a[b$n - 1]c
a[b,1$0]c = a[b$x]c where x is a[b$b]c
a[b,n$0]c = a[b$x]c where x is a[b,n - 1$b]c
a[b,n,1$0] = a[b,x$0]c where x is a[b,n$0]c
a[b@0]c = a[b,b,b,b,...b$0] with b number of b's
Now, let's define another function.
f(1) = a[b]c
f(2) = a[[b]]c
f(3) = a[b/1]c
f(4) = a[b\0]c
f(5) = a[b$0]c
f(6) = a[b@0]c
f(7) = a[b,b,b,b,...b@0]c with b number of b's
a[b`0]c = f(b)
a[b~0]c = f(x) where x is f(b)
a[b;0]c = f(x) where x is a[b~0]c
Now, let's define yet another function:
f(1) = a[b]c
f(2) = a[b`0]c
f(3) = a[b~0]c
f(4) = a[b;0]c
f(5) = f(x) where x is a[b;0]c
a[b:0]c = f(b)
a[b?0]c = f(x) where x is f(b)
Let's Define Another Function:
f(1) = a[b]c
f(2) = a[b`0]c
f(3) = a[b:0]c
a[b"0]c = f(b)
a[b|0]c = f(x) where x is f(b)
a[b(0)]c = f(x) where x is a[b|0]c
Let's Define Another Function:
f(1) = a[b]c
f(2) = a[b"0]c
f(3) = a[b|0]c
f(4) = a[b(0)]c
f(5) = f(x) in the previous function, where x is a[b(0)]c
a[b←0]c = f(b)
a[b%0]c = f(x) where x is a[b←0]c
If we keep defining functions like this, and then combine them all into one enormous function, we can create much more enormous numbers.
f(1) = a[b]c
f(2) = a[b`0]c
f(3) = a[b:0]c
f(4) = a[b"0]c
f(5) = a[b←0]c
a[b₁]c = f(b)
a[b₂]c = f(f(f(f(f(...f(f(b)))...)) with b number of f's
a[b₃]c = f(f(f(f(f(...f(f(b)))...)) with a[b₂]c number of f's
a[bₙ]c = f(f(f(f(f(...f(f(b)))...)) with a[bₙ ₋ ₁]c number of f's
a[b₁,₁]c = a[bₓ]c where x is a[bb]c
a[b₁,₂]c = a[bₓ]c where x is a[b₁,₁]c
a[b₁,ₙ]c = a[bₓ]c where x is a[b₁,ₙ ₋ ₁]c
a[b₂,₀]c = a[b₁,ₓ]c where x is a[b₁,b]c
a[bₙ,₀]c = a[bₙ ₋ ₁,ₓ]c where x is a[bₙ ₋ ₁,b]c
a[b₁,₁,₁]c = a[bₓ,ₓ]c where x is a[bb,b]c
a[b₁,₁,ₙ]c = a[b₁,ₓ]c where x is a[b₁,₁,ₙ ₋ ₁]c
a[b₁,ₙ,ₙ]c = a[bₓ]c where x is a[b₁,ₙ₋₁,ₙ₋₁]c
a[bₙ,₀,₀]c = a[bₙ ₋ ₁,ₓ,ₓ]c where x is a[bₙ ₋ ₁,b]c
a[b¹]c = a[bb,b,b,b,b,...b]c with b number of subscripted entries
a[b²]c = a[x¹]c where x is a[b¹]c
a[bⁿ]c = a[xⁿ⁻¹]c where x is a[bⁿ⁻¹]c
a[b¹,⁰]c = a[bˣ]c where x is a[bᵇ]c
We can even do a[b¹,⁰,⁰,⁰,⁰,⁰,⁰,⁰,⁰,⁰]c, and that is the limit of the notation.
== Sources ==
Category:Notations
Category:Incomplete
Category:Humor
Category:Array Notations