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 the Wainer hierarchy, \(f_\omega\) may be arbitrarily fast-growing. For example, letting \(\omega[n] = \Sigma(n)\) (where \(\Sigma(n)\) is a busy beaver function) makes \(f_\omega\) uncomputable.

In fact, there are as many fundamental sequences for \(\omega\) (and for any other ordinal with \(\omega\)), as the number of monotonically increasing \(\mathbb{N} \to \mathbb{N}\) functions, i.e. uncountable continuum \(\mathfrak c\) of them.

\(\omega\) is the first admissible ordinal. It is also the first transfinite.