User blog:B1mb0w/Nested f omega functions

Lets compare different combinations of Fast-growing hierarchy Functions using nested \(f_{\omega}\) functions

One Example

\(f_{\omega+3}^2(3) = f_{\omega}^{f_{\omega+3}(3)}(f_{\omega}^{f_{\omega+2}^2(3)}(f_{\omega}^{f_{\omega+2}(3)}(f_{\omega}^{f_{\omega+1}^2(3)}(f_{\omega}^{f_{\omega+1}(3)}(f_{\omega}^{f_{\omega}^2(3)}(f_{\omega}^{f_{\omega}(3)}(f_{\omega}(3))))))))\)

\(= f_{\omega}^{f_{\omega+3}(3)}(f_{\omega}^{f_{\omega+2}^2(3)}(f_{\omega}^{f_{\omega+2}(3)}(f_{\omega}^{f_{\omega+1}^2(3)}(f_{\omega}^{f_{\omega+1}(3)}(f_{\omega}^{f_{\omega}^2(3)}(f_{\omega}^2(3)))))))\)

\(= f_{\omega}^{f_{\omega+3}(3)}(f_{\omega}^{f_{\omega+2}^2(3)}(f_{\omega}^{f_{\omega+2}(3)}(f_{\omega}^{f_{\omega+1}^2(3)}(f_{\omega}^{f_{\omega+1}(3)}(f_{\omega}^3(3))))))\)

\(= f_{\omega}^{f_{\omega+3}(3)}(f_{\omega}^{f_{\omega+2}^2(3)}(f_{\omega}^{f_{\omega+2}(3)}(f_{\omega}^{f_{\omega+1}^2(3)}(f_{\omega}^{f_{\omega+1}(3)}(f_{\omega+1}(3))))))\)

Some Comparisons with \(\omega\).n

24. \(f_{\omega}^{f_{\omega.n}^{a-1}(a).2}(a) >> f_{\omega.n+1}(a)\)

because \(f_{\omega.n}^{a-1}(a).2 = f_{\omega.n}^{a-1}(a) + f_{\omega.n}^{a-1}(a) >> f_{\omega.n}^{a-1}(a) + f_{\omega.n}^{a-2}(a).2\)

and \(f_{\omega.n}^2(a) >> f_{\omega.n}(a).3 = f_{\omega.n}(a) + f_{\omega.n}(a).2 = f_{\omega.n}(a) + f_{\omega.(n-1)+a}(a).2 = f_{\omega.n}(a) + f_{\omega.(n-1)+a-1}^a(a).2\)

and \(f_{\omega.(n-1)+a-1}^2(a) >> f_{\omega.(n-1)+a-1}(a).3 = f_{\omega.(n-1)+a-1}(a) + f_{\omega.(n-1)+a-1}(a).2\)

\(= f_{\omega.(n-1)+a-1}(a) + f_{\omega.(n-1)+a-2}^a(a).2\)

and \(f_{\omega}^2(a) >> f_{\omega}(a).2 = f_{\omega}(a) + f_{\omega}(a) = f_{\omega}(a) + f_{a}(a) >> f_{\omega}(a) + a\)

then

\(f_{\omega}^{f_{\omega.n}^{a-1}(a).2}(a) >> f_{\omega}^{f_{\omega.n}^{a-1}(a)+f_{\omega.n}^{a-2}(a).2}(a) >> f_{\omega}^{f_{\omega.n}^{a-1}(a)}(f_{\omega.n}^{a-1}(a)) >> f_{\omega.n}^{a}(a) = f_{\omega.n+1}(a)\)

25. \(f_{\omega}^{f_{\omega.n}^{m}(a).2}(a) >> f_{\omega.n}^{m+1}(a)\)

General Comparison Rule

G1. \(f_{\omega}^{f_b^c(a).2}(a) >> f_b^{c+1}(a)\)