T L  C S C W S?

Sets play a key role in foundations of mathematics. Why? To what extent is it an accident of history? Imagine that you have a chance to talk to mathematicians from a far away planet. Would their mathematics be setbased? What are the alternatives to the set-theoretic foundation of mathematics? Besides, set theory seems to play a significant role in computer science, in particular in database theory and formal methods. Is there a good justification for that? We discuss these and some related issues. 1 Sets in Computer Science Quisani: I wonder why sets play such a prominent role in foundations of mathematics. To what extent is it an accident of history? And I have questions about the role of sets in computer science. ∗Partially supported by NSF grant DMS–0070723 and by a grant from Microsoft Research. Address: Mathematics Department, University of Michigan, Ann Arbor, MI 48109–1109, U.S.A., [email protected]. This paper was written during a visit to Microsoft Research. Authors: Have you studied set theory? Q: Not really but I came across set theory when I studied discrete mathematics and logic, and I looked into Lévy’s book [38] years ago. I remember that ZFC, first-order Zermelo-Fraenkel set theory with the axiom of choice, became for all practical purposes the foundation of mathematics. I can probably reconstruct the ZFC axioms. A: Do you remember the intuitive model for ZFC. Q: Let me see. You consider the so-called cumulative hierarchy of sets. It is a transfinite hierarchy, so that you have levels 0, 1, . . . , ω, ω + 1, . . . On the level zero, you have the empty set and possibly some atoms. On any other level α you have the sets of objects that occur on levels < α. Intuitively the process never ends. To model ZFC, you just go far enough in this hierarchy so that all axioms are satisfied. Is that correct, more or less?