User blog comment:Luckyluxiuz/Question/@comment-37212455-20191219030832/@comment-37212455-20191219050307

Not quite. Remember, each time the index, n, of an FGH expression increases by one, that is equivalent to iteration of the last FGH expression, n-1. So if \(S(n)\) (which we know to be comparable to \(f_{\omega+1}(n)\) was iterated twice, it would just be a function similar in growth to \(f_{\omega+1}(f_{\omega+1}(n))\).

\(S_2(n)\) however is similar to \(f_{\omega+2}(n)\) because it is a function which iterates \(S(n)\) (or \(S_1(n)\)), and since a function dependent on the iteration of \(f_{a}(n)\) must have strength at least \(f_{a+1}(n)\) it means \(S_2(n)\) is comparable to \(f_{\omega+2}(n)\).

If I understand \(S_{n+1}(m)\) as being defined as \(S_n^m(m)\), then \(S_{n}(m)\) would in general be a similar strength to \(f_{\omega+n}(m)\).