User blog comment:Rgetar/Simple way to create lists of ordinals/@comment-35392788-20181008074754/@comment-32213734-20181009150555

So α uniquely corresponds to finite array, which uniquely corresponds to positive integer, which can be enumerated.

For example, α = β[3][1][2] corresponds to array

3, 1, 2

which corresponds to positive integer

2·2·2·2·3·3·5·5·5 = 18000

and such positive integers can be enumerated in increasing order.

Interestingly, we can easily convert 18000 back to β[3][1][2].