Omega

\(\omega\) (pronounced "omega") is the first limit ordinal, or informally "the smallest number 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)
 * \(g_\omega(n) = n\) (slow-growing hierarchy), the first function in SGH which is not constant.

Arithmetic
this section may need cleaning up

Arithmetic with ordinals is not like arithmetic with real numbers. The main difference is that addition and multiplication are not commutative.

While $$\omega+1>\omega$$, $$1+\omega=\omega$$.

Likewise, $$\omega*2>\omega$$, $$2*\omega=\omega$$.

We can have $$\omega*3$$, $$\omega*4$$, $$\omega*5$$, and even $$\omega*\omega=\omega^2$$.

Then we could have $$\omega*\omega*\omega=\omega^3$$, $$\omega^4$$, and even $$\omega^\omega$$.