Do any of you know of any authors of honors calculus texts who take as their completeness axiom the assertion that, if one partitions the reals into two non-empty sets A,B with a<b for all a in A and b in B, then there exists a cutpoint c such that either A=(-\infty,c] and B=(c,\infty) or A=(-\infty,c) and B=[c,\infty)? These partitions are close in spirit to Dedekind cuts, but here we are partitioning R, not Q, and we are axiomatizing the reals, not constructing them. Other popular completeness axioms for the reals are the least upper bound property and the bounded monotone sequence convergence property, but I think these are harder for undergraduates to grasp. It's fun to instigate a class discussion of "How can you partition the real line into two sets, one of which is completely to the left of the other?" at the start of the term; once the students have wrestled with it for a while, they're glad to accept the axiom. Then, a week or two later, when continuity is introduced, one can use the cut axiom to prove the Intermediate Value Theorem, and for a challenge problem on the homework one can ask the students to prove (with a few hints) that the IVT in turn implies the cut axiom. (If you could cut the reals into two non-empty sets A,B with A<B where A has no greatest element and B has no least element, then the function that's -1 on A and +1 on B is continuous.) Another approach some people take is to defer the proof of the IVT until the second semester, when sequences are traditionally introduced, so that the bounded monotone sequence convergence property won't seem so strange, and when the students have seen more proofs, but I like the idea of showing the students a non-trivial proof fairly early, and having them help turn an incomplete proof into a rigorous one. So, would one call this a small but genuine innovation in honors calculus pedagogy (perhaps worth publishing in a short note somewhere), or too minor a tweak to merit discussion, or a well-trodden pedagogical path that lots of teachers already know about? Jim