User blog:89Pepsi Polka/the hyper-square hierarchy

Here is a notation that’s probably very weak, but exceeds \(f_{\varepsilon_0}(n)\), so that’s something. In the future i will post a full analysis and extension.

As a prelude to the actual notation, I’ll clear up some matters of labeling. Let C denote ‘’any’’ construction of squares, and \(\gamma_n(C)\) the concatenation of C n-times. In addition, the insertion of a construction C1 into another construction C2 n-levels deep is to be denoted \(C_1\in_n C_2\). \(C\equiv f(n)\) is the function to which \(C(n)\) is equivalent to.

Rule set
\(\square(n)\equiv n+1\)

\(\gamma_a(\square)(n)\equivn+a\)

\(\gamma_{\omega}(C)(n)=\square\in_1 C(n)\equiv min\;\;sup\{f(n):f(n)>\gamma_{m}(C),\; m\in\mathbb{N}\}\)

\(\square_{C\square}(n)=sup\left\{\square_C,\boxed{\square_C}_C,\boxed{\boxed{\square_C}_C}_C,\boxed{\boxed{\boxed{\square_C}_C}_C}_C,\ldots\right\}\)

Limit
The limit of this notation is most likely \(f_{\zeta_0}\) - this is only an educated guess and may be wrong.