1. The problem statement, all variables and given/known data How many different ways can one well-order the natural numbers? Different orders are those which are NOT order isomorphic. 2. Relevant equations 3. The attempt at a solution My approach thus far has been to examine a well-ordering on N. Clearly, any well-ordering is one that satisfies that our well ordering P(a,b), returns an element for the pair (a,b) for a,b in N where the value returned is the least element of the pair. So it must be the case that any well-ordering is at the very least a function from NxN into N. We can take it as known (by a previous problem I solved) that the cardinality of the set of functions from NxN into N [ denoted as N^(NxN) ] is the same as the cardinality of (N^N)^N or the cardinality of R. My problem here is that the set of orderings I have are all linear orders. I seem to think that the set of P that I have define well-orderings as well but I'm not sure that I'm even right or if I am correct, how to show that rigorously. Does anyone have any thoughts on this approach I'm taking to the problem?