Omega

The \(\omega\) (pronounced "omega") is the first infinite ordinal, and is equal to \(\{0, 1, 2, \ldots \}\). It can be viewed equivalently as the set of nonnegative integers, or the first fixed point of \(\alpha \mapsto 1 + \alpha\), or informally "the smallest number greater than all the positive integers." 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) is on par with Ackermann function and weak Goodstein function. It is the first function in FGH that is not primitive-recursive.
 * \(H_\omega(n) = 2n\) (Hardy hierarchy)
 * \(g_\omega(n) = n\) (slow-growing hierarchy), the first function in SGH which is not constant.