Introduction to the fast-growing hierarchy

The fast-growing hierarchy is a hierarchy of functions based on the. It is particularly notable in googology not only in the growth rates of the numbers produced, but also its very simple formal definition. BEAF, in contrast, notoriously lacks a clear definition for legion arrays and beyond.

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 \(\alpha + 1 = \alpha \union \{\alpha\}\).

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 \(\{0, 1, 2, 3, \ldots\}\). We represent this with the Greek letter \(\omega\) (omega), the least infinite ordinal.