Omega

The \(\omega\) (pronounced "omega") is the first ordinal satisfying the equation \(1+\alpha = \alpha\). The growth rates of finite forms of that ordinal in different hierarchies are shown below:

\(f_\omega(n) \approx 2 \uparrow^{n-1} n\) (fast-growing hierarchy)

\(H_\omega(n) = 2n\) (Hardy hierarchy)

\(g_\omega(n) = n\) (slow-growing hierarchy)

In the fast-growing hierarchy it is on par with Ackermann and weak Goodstein functions, and therefore it is the first non-recursive function in this hierarchy.