  In each of the above cases, the second statement is more evident than the first and there is general agreement on this fact. For example, the Hilbert Basis Theorem is far from immediate. It is the sort of thing that we set out to prove from things that are more evident. In contrast, the statement that ωω is well-founded is something that becomes clear by reflecting on the concepts involved. It is not the sort of thing that we set out to prove from something more evident. This is what lies behind the fact that the Hilbert Basis Theorem is called a theorem and not an axiom. As one climbs up through the other examples, the asymmetry becomes even more dramatic. For example, the Hydra Theorem and Kruskal's Theorem are actually quite surprising and counter-intuitive. These are definitely the sort of statements that require proof on the basis of statements that are more evident. 