This article is dedicated to Professor Burton Dreben on his coming of age. I owe him particular thanks for his careful reading and numerous suggestions for improvement. My thanks go also to Jose Ruiz and the referee for their helpful comments. Parts of this account were given at the 1995 summer meeting of the Association for Symbolic Logic at Haifa, in the Massachusetts Institute of Technology logic seminar, and to the Paris Logic Group. The author would like to express his thanks to the various organizers, as well as his gratitude to the Hebrew University of Jerusalem for its hospitality during the preparation of this article in the autumn of 1995.

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 :

Abstract. The ω-asymmetry induced by transfinite partitions makes it impossible for Jordan curves to have an infinite length. 1. ω-asymmetry As we known from the XVIII century, ω-partitions (as we call them nowadays) of finite line segments are only possible if the successive adjacent parts of the ω-partition are of a decreasing length. This inevitable restriction induces a huge asymmetry in the very partition. In fact, whatever be the length of the ω-partitioned line segment and whatever be the ω-partition, all its parts, except a finite number of them, will necessarily lie within an arbitrarily small final segment. For the sake of illustration, consider an ω-partition of a 10 30 light years length segment-the assumed diameter of the universe. Whatever be the ω-partition of this enormous line segment all its infinitely many parts, except a finite number of them, will inevitably lie within a final segment inconceivable less than, for instance, Planck length ( ∼ 10 −33 cm). There is no way of performing a more equitable partition if the partition has to be ω-ordered. Thus, ω-partitions are ω-asymmetrical. For the same reason it is impossible to consider two proper points in the real line R separated by an infinite euclidean distance, in spite of the assumed infiniteness of the real line. The above simply unaesthetic consequences of ω-partitions become a little more controversial if the partitioned object is a closed line as a Jordan curve. The objective of the following short discussion is just to examine one of those consequences. Figure 1. A cosmic ω-asymmetry. The ω-order makes it impossible a more equitable distribution of the available space. 1 2 Transfinite partitions of Jordan Curves 2. Transfinite partitions of Jordan Curves Let f(x) be a real valued function whose graph is a Jordan Curve J in the euclidean plane R2. If a and b are any two J’s points, we will write L(a, b) to denote the length of the J’s arc ãb whose endpoints are a and b. That is to say: b √ L(a, b) = 1 + (f(x)′) 2dx (1)