User blog comment:Ubersketch/What are some ways to map a well-order to the natural numbers without using a Godel coding/@comment-39541634-20190909210146/@comment-35470197-20190910144310

For example, you can map \(\omega^{\omega}\) into \(\mathbb{N}\) through the correspondence \begin{eqnarray*} 0 & \mapsto & f(0) := 0 \\ \omega^N + \sum_{n=0}^{N} \omega^n \times a_n & \mapsto & f(\omega^N + \sum_{n=0}^{N} \omega^n \times a_n) := 2^N(2f(\sum_{n=0}^{N} \omega^n \times a_n)+1), \end{eqnarray*} where \(N\) is a natural number, \((a_n)_{n=0}^{N}\) is a sequence of natural numbers, and \(\sum\) is the sum in the descending order. This is not derived from the methods using the well-ordering, but is derived from an obvious notation below \(\omega^{\omega}\). (You might regard this as a variant of Goedel numbering, though.)