Introduction to the fast-growing hierarchy

The fast-growing hierarchy is a hierarchy of functions based on the ordinal numbers. It is particularly notable in googology not only in the growth rates of the numbers produced, but also its very simple formal definition.

Rule 1
The first rule is simple, \(f_0(n) = f(n) = n+1\).

For example: \(f_0(10^{100}) = f(10^{100}) = 10^{100}+1\).

Rule 2
For second rule, we need new function rule, \(f^n(m) = f(f(...(f(f(m))...))\) (n nested).

We need ordinal numbers.

Ordinal numbers
The German mathematician invented an extension to the nonnegative numbers, called the ordinal numbers. An ordinal number is defined as the set of all smaller ordinal numbers. Formally, an ordinal \(\alpha\) is \(\{\beta: \beta < \alpha\}\), and the successor to an ordinal is defined as \(\alpha + 1 = \alpha \cup \{\alpha\}\).

From 0 to \(\varepsilon_0\)
We start with \(0 := \{\}\) &mdash; zero is the empty set, since no ordinal number is smaller than zero. We continue with \(1 := \{0\} = \{\{\}\}\), \(2 := \{0, 1\} = \{\{\}, \{\{\}\}\}\), \(3 := \{0, 1, 2\} = \{\{\}, \{\{\}\}, \{\{\}, \{\{\}\}\}\), etc.

The next ordinal after all these is \(\{0, 1, 2, 3, \ldots\}\), the set of all nonnegative integers. We represent this with the Greek letter \(\omega\) (omega), the least transfinite ordinal. Next, we can define \(\omega + 1 = \{0, 1, 2, 3, \ldots, \omega\}\), \(\omega + 2 = \{0, 1, 2, 3, \ldots, \omega, \omega + 1\}\), etc.

The specifics of are not detailed here, but the general idea should be fairly clear from these examples. It is important to note that addition on the ordinals is often not commutative. \(\omega + 1 > \omega\), but \(1 + \omega = \omega\).

The supremum of all the \(\omega + n\) is \(\{0, 1, 2, \ldots, \omega, \omega + 1, \omega + 2, \ldots\} = \omega + \omega = \omega \times 2\). Next we have the series \(\omega + \omega + \omega = \omega \times 3, \omega \times 4, \omega \times 5, \omega \times 6, \ldots, \omega \times \omega = \omega^2, \omega^3, \omega^4, \ldots, \omega^\omega, \omega^{\omega^\omega} = {}^3\omega, \omega^{\omega^{\omega^\omega}} = {}^4\omega, \ldots\)

From \(\varepsilon_0\) to \(\Gamma_0\)
Eventually we reach \(^\omega\omega = \omega^{\omega^{\omega^{.^{.^.}}}}\), famously known as \(\varepsilon_0\) (epsilon-zero). (To googologists, the exponentiation seems to be a fairly arbitrary choice, but in fact it is not &mdash; it is important in induction proofs, for example.) \(\varepsilon_0\) is a fixed point such that \(\varepsilon_0 = \omega^{\varepsilon_0}\); \(\varepsilon_1\) can be defined as fixed point: \(\varepsilon_1 = \varepsilon_0^{\varepsilon_1}\). In general, \(\varepsilon_{\alpha+1}\) is a fixed point for \(\varepsilon_{\alpha+1} = \varepsilon_{\alpha}^{\varepsilon_{\alpha+1}}\). Examples of "epsilon numbers" are \(\varepsilon_0, \varepsilon_1, \varepsilon_2, \varepsilon_3, \varepsilon_\omega, \varepsilon_{\omega^\omega}, \varepsilon_{\varepsilon_0} \ldots\).

The limit of the hierarchy of \(\varepsilon_\alpha\) leads us to \(\varepsilon_{\varepsilon_{\varepsilon_{._{._.}}}}\), sometimes called \(\zeta_0\). The "zeta numbers" are the solutions to the equation \(\zeta = \varepsilon_\zeta\). We can also have \(\eta_0 = \zeta_{\zeta_{\zeta_{._{._.}}}}\) and the "eta numbers" \(\eta = \zeta_\eta\).

We only have finitely many Greek letters, so the next step is to generalize these. Define the Veblen hierarchy as \(\phi_0(\alpha) = \omega^\alpha\), and define \(\phi_{\beta + 1}(\alpha)\) as the \(\alpha\)th solution (starting from 0th) \(\gamma\) to the equation \(\phi_\beta(\gamma) = \gamma\). This gives us \(\phi_1(\alpha) = \varepsilon_\alpha, \phi_2(\alpha) = \zeta_\alpha, \phi_3(\alpha) = \eta_\alpha, \ldots\).

What would happen if the subscript is a limit ordinal? Obviously, we could define \(\phi_\omega(0)\) as the limit of \(\phi_1(0), \phi_2(0), \phi_3(0), \ldots\), but what is \(\phi_\omega(1)\)? We could define as a similar way as \(\phi_\omega(0)\), so \(\phi_\omega(1)\) is the limit of \(\phi_1(\phi_\omega(0)+1), \phi_2(\phi_\omega(0)+1), \phi_3(\phi_\omega(0)+1), \ldots\) (the +1 is to avoid fixed points).

Generally, when the subscript is a limit ordinal, we have \(\phi_\alpha(\beta)[n]\) = \(\phi_{\alpha[n]}(\phi_\alpha(\beta - 1) + 1)\).

The limit of the Veblen hierarchy is \(\phi_{\phi_{\phi_{._{._..}.}(0)}(0)}(0) = \Gamma_0\), the famous Feferman–Schütte ordinal. This is the solution to the equation \(\Gamma\) to \(\phi_\Gamma(0) = \Gamma\).