User blog comment:LittlePeng9/ITTM galore/@comment-24509095-20140326102311/@comment-5529393-20140326121738

For example, to implement w+1, we could use the order 1,2,3,..., 0. So to determine the  bit (call it R()), we can use the following algorithm:

If a = 0, R() = 0 Else if b = 0, R() = 1 Else if a < b, R() = 1 Else R() = 0

For w*2, we could use 0,2,4,..., 1,3,5,..., using

If a is even and b is odd, R() = 1 Else if a is odd and b is even, R() = 0 Else if a < b, R() = 1 Else R() = 0

Challenge: Find nice orderings of N with order types w^2, w^3, w^w, w^w^w, epsilon_0.