Omega

\(\omega\) (pronounced "omega") is the first transfinite ordinal, the smallest ordinal greater than all the positive integers. In the von Neumann definition of ordinals, it is equal to the set of nonnegative integers \(\mathbb{N}\).

In the Wainer hierarchy, the fundamental sequence of \(\omega\) is \(0,1,2,\ldots\). Using this hierarchy, the following holds:


 * \(f_\omega(n) \approx 2 \uparrow^{n-1} n\) (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), the first function in Hardy hierarchy which is not a.
 * \(g_\omega(n) = n\) (slow-growing hierarchy), the first function in SGH which is not constant.

However, if we are not working in Wainer hierarchy, \(f_\omega(n)\) may be arbitrarily fast-growing. For example, letting \(\omega[n] = \Sigma(n)\) (where \(\Sigma(n)\) is a busy beaver function) makes \(f_\omega(n)\) uncomputable.