this paper, the seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject. The motif is foreshadowed in Cantor's diagonal proof, and emerges in the interstices of the inclusion vs. membership distinction, a distinction only clarified at the turn of this century, remarkable though this may seem. Russell runs with this distinction, but is quickly caught on the horns of his well-known paradox, an early expression of our motif. The motif becomes fully manifest through the study of functions f :

hsimmons @ manchester.ac.uk I re-work Veblen’s ideas to show how hierarchies of normal functions can be generated merely by iterating certain higher order gadgets. As an illustration I show that an application of a Schütte bracket can be evaluated without the need of an intricate recursion.

Finding the phase transition for Friedman’s long finite sequences