User:Vel!/Ordinal Hyper-E

I will define Hyper-E using strings now.

A hyper-E expression is a finite string of ordered pairs \((\alpha, n)\), where \(n\) is a positive integer and \(\alpha\) is an ordinal. In this page I will define a function \(S_b(A)\) that maps hyper-E expressions to positive integers. When I write \(S_b(((\alpha_1, n_1), (\alpha_2, n_2), (\alpha_3, n_3), \ldots))\) I will typically leave out the inner pair of parentheses: \(S_b((\alpha_1, n_1), (\alpha_2, n_2), (\alpha_3, n_3), \ldots)\)

Also, define \(||\) as concatenation. It's pretty obvious what it means in context.

Let \(A\) be such a string.


 * \(S_b((0, a)) = b^a\)
 * \(S_b(A || (\alpha, 1)) = S_b(A)\)