User blog comment:Mh314159/Alpha numbers (and beyond)/@comment-35470197-20191007022858/@comment-35470197-20191010011615

> Your quoted definitions don't seem to match the ones I used.

Sorry for the ambiguity. I did not mean that \(f\), \(g\), and \(h\) are specific functions in your definitions, but just stated a general fact which will be helpful to surpass the current growth rate.

The precise meaning is the following: For example, \(\alpha_b[x]\) is defined by \(\alpha_{b-1}^{\alpha_b[x-1]}[x]\), and hence corresponds to the first pattern. Therefore incrementing \(b\) corresponds to "\(+2\)" in terms of ordinals. On the other hand, \(\alpha[x]\) is defined by \(\alpha[x] = \alpha_{\alpha[x-1]}[x]\), and hence corresponds to the second pattern. Therefore its growth rate can be computed as the supremum of \(\alpha_m\)'s plus \(1\).
 * 1) If \(g(x)\) is an arbitrary funtion which is approximated to an ordinal \(\alpha\) with respect to FGH, then the new function \(f(x)\) defined by \(f(x) = g^{f(x-1)}(x)\) and a reasonable initial value \(f(0)\) is roughly bounded by \(\alpha+2\).
 * 2) If \((h_0(x), h_1(x), h_2(x), \ldots)\) is an arbitrary family of functions indexed by natural numbers such that which \(h_n(x)\) is approximated to an ordinal \(\alpha_n\) with respect to FGH for any natural number \(n\), then the new function \(g(x)\) defined by \(g(x) = h_{g(x-1)}(x)\) and a reasonable initial value \(g(0)\) is roughly bounded by \((\sup_{n \in \omega} \alpha_n) + 1\).

The growth rate of \(\alpha^x[x]\) corresponds to the successor ("+1") of the ordinal corresponding to the growth rate of \(\alpha[x]\). Even if \(\alpha^b[x]\) is much stronger than \(\alpha_b[x]\), its contribution to the ordinal is smaller than that of \(\alpha_b[x]\). It is partially because teh difference of \(\omega\) and \(\omega+1\) is much smaller than that of \(\omega+1\) and \(\omega+2\).

> If f(x) already has growth rate w+1, and if g(x) puts a functional power on f(x) that grows even faster than f(x), what is the growth rate?

It is bounded by w+3. Iterating power just contributes to the ordinal as "+2".

> And h does to g what g did to f.

In this case, the resulting growth rate is bounded by w+5.

> And alpha(n,x) extends the letters to numbers n that are themselves functions of letters.

Such a step is called "diagonalisation", which corresponds to the second pattern. It contributes to the ordinal as "sup + 1".

> I have a hard time imagining that with all the times the argument gets copied to the subscript that alpha2(x) doesn't grow much faster than only w^2.

The base strategies in your notation are the following: They are good. However, imagine that you diagonalise the first paterns. Say +2, +4, +6, and so on. The resulting contribution of the diagonalisation is +2w, which is +w as an ordinal. Even if you repeat this diagonalisation 100 times, it just contributes to the ordinal as +w×100.
 * 1) to define a recursive extension which contributes to the ordinal as "+2".
 * 2) to define a diagonalisation for them.

In order to go beyond w^2, you need to diagonalise diagonalisation steps. Say, diagonase steps corresponding to "+w", "+w×2", "+w×3", and so on. Then you will get "+w^2". Your definition includes only single diagonalisation of diagonalisation, and hence the resulting growth rate is bounded by w^2×2.

Of course, after understanding FGH and the diagonalisation, you will soon find a way to diagonalise steps corresponding to "+w^2"'s, which gives "+w^3". Then you will find a new wall, w^w. Then you need to exporer the realm of the diagonali sation of "+w", "+w^2", "+w^3", and so on.