  In addition, one might raise skeptical worries about the evidentness order. For example, perhaps judgments of the form “X is prior to Y in the evidentness order” are really just reflections of our mathematical training. The best way to allay such concerns and test the robustness of the notion is to examine examples. Compare, for example, the statement of the Hydra Theorem and the statement that ϵ0 is well-ordered with respect to primitive recursive relations (a weak (semi-constructive) form of the statement that ϵ0 is well-ordered). If you describe the relevant notions to a mathematician or layperson who is not familiar with them then you will find the following: There will be no expectation that Hercules wins the Hydra. In fact, if you direct the person to one of the computer programs (available online) that simulates the Hydra you will find that they will be astounded at how fast the Hydra grows and if you assure them that if they kept on playing—for a period of more years than there are atoms in the physical universe (as currently estimated)—the tides will eventually turn, they will make gains and eventually cut the Hydra down to size, you will find that they will look at you in amazement. This is something that needs proof. Now if you describe to them the ordinals all the way up to ϵ0 and you describe the notion of a well-ordering and ask them whether ϵ0 is well-founded you will find that the answer rests solely on whether they have grasped the object in question. To come to the conclusion that ϵ0 is well-founded one does not search for a proof from some other statement which is more evident. Rather one works directly with the concepts involved. The statement, as it were, rests on the legitimacy of the underlying concepts and one comes to accept it by grasping those concepts. It is here that the buck stops in the process of seeking justification. And this is what we mean in saying that the statement “ϵ0 is well-ordered” is minimal in the evidentness order (of the degree of the Hydra Theorem). 