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Here a example : \(f_{\omega+2}(6)\).

\(f_{\omega+2}(6)\) =

\(f_{\omega+1}^6(6)\) =

\(f_{\omega+1}(f_{\omega+1}(f_{\omega+1}(f_{\omega+1}(f_{\omega+1}(f_{\omega+1}(6))))))\) =

\(f_{\omega+1}(f_{\omega+1}(f_{\omega+1}(f_{\omega+1}(f_{\omega+1}(A)))))\), this function get too complex so here a A:

\(f_{\omega+1}(6)\) =

\(f_{\omega}^6(6)\) =

\(f_{\omega}(f_{\omega}(f_{\omega}(f_{\omega}(f_{\omega}(f_{\omega}(6))))))\) =

\(f_{\omega}(f_{\omega}(f_{\omega}(f_{\omega}(f_{\omega}(f_{6}(6))))))\) =

\(f_{\omega}(f_{\omega}(f_{\omega}(f_{\omega}(f_{\omega}(B)))))\), also A function get too complex so here a B:

\(f_{6}(6)\) =

\(f_{5}^6(6)\) =

\(f_{5}(f_{5}(f_{5}(f_{5}(f_{5}(f_{5}(6))))))\).

So \(f_{\omega+2}(6)\) > \(\{6,6,2,2\}\).

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