User blog comment:Rgetar/Simple way to create lists of ordinals/@comment-35392788-20181008074754/@comment-35470197-20181009134205

Since your system gives a way to list up all ordinals below \(\beta\), it gives (not unique, various) ways to enumerate them.

For example, every \(\alpha\) corresponds to the finite array \(n[1], \ldots, n[i]\) of natural numbers in the way above. The set of finite arrays of natural numbers can be enumerated by any way which you like. For example, send it to the natural nunmber \(L(\alpha) = p_1^{n[1]+1} \cdots p_i^{n[i]+1}\), where \(p_k\) denotes the \(k\)-th prime number.

Then \(L\) is an injective map from ordinals below (\beta\) to positive integers, which gives you an explicit way to enumerate such ordinals. In this sense, \(i\)-th level corresponds to natural numbers whose prime divisor belongs to \(\{p_1,\ldots,p_i\}\).

I am sorry that my enumeration is not what you intended. Your enumeration looks natural because it directly uses the finiteness of each expansion.