Normal functions and constructive ordinal notations

An r-normal function is a strictly increasing continuous function from r to r where r is a regular ordinal > ω (identify an ordinal with the set of smaller ordinals). Given an r-normal function f one can form a sequence {f(x, −)}x<r of r-normal functions—the Veblen hierarchy [33] on f—as follows: f(0, −) = f and, for x > 0, f(x, −) enumerates in order {z ∣ f(y, z) = z for all y < x}, the common fixed points of the f(y, −)'s for y < x. In this paper we give as readable an exposition as we can of Veblen hierarchies and of Bachmann's and Isles's techniques in [3] and [15] of using higher finite number classes for forming sequences {f(x, −)}x<y where y > r of r-normal functions which extend the Veblen hierarchy on f. We will show how these sequences—Bachmann hierarchies—yield extremely natural constructive notations for ordinals in various initial segments of the second number class. We will also consider various other techniques for obtaining constructive ordinal notations and relate them to the notations obtained by Bachmann's and Isles's techniques. In particular, we will use these notations to characterize as directly and as usefully as we can various of Takeuti's systems of constructive ordinal notations, which he calls ordinal diagrams ([31], [32]).