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The Q-supersystem is a googological function by Wiki user Boboris02. It is a notation based on recursion and is very similar to many hierarchies.


Basics

The Q-supersystem is a system that follows a very simple fundamental rule and that is:

\(Q_n(a)\) is always at the next notation after \(Q_{n-1}(a)\).

It also always accounts that \(Q_{n,0\#}(a)=Q_{n-1,a\#}(a)\) no matter how much \(a\) is.

Whenever you have more than one zeros like in this example \(Q_{1,0,0,...,0}(n)\) then it can be simplified like so \(Q_{1,0,0,...,0}(n)=Q_{n,0,0,...,0}(n)\) where now the number of zeros is reduced by 1 and the first entry becomes the number,the operation is done on.

Also, the first number must always be reduced by one. \(Q_{n,0,0,...,0}(a)=Q_{n-1,a,0,0...0,0}(a)\)

Whenever things like \(Q_{n,0\#,b}(a)\) occur, it can be simplified through the following equation:

\(Q_{n,0\#,b}(a)=Q^{a}_{n,0\#,b-1}(a) \leadsto Q_{n,0\#,0}(c)=Q_{n-1,c,0(\#-1),0}(c)\).

When there is only one entry, \(Q_0(n)=n+1,Q_1(n)=n2,Q_2(n)=n^2\) and \(Q_{a+2}(n)=n\uparrow^a n\) for \(n\geq 1\).

The limit of the Q-supersystem (\(Q_{1,0,0,...,0,0}(n)\)) is approximately \(f_{\omega^{\omega}}(n)\) in the FGH.

Sources