The Journal of Symbolic Logic

Research Article

Nonexistence of universal orders in many cardinals

Menachem Kojmana1 and Saharon Shelaha2

a1 Department of Mathematics, The Hebrew University, Jerusalem, Israel, E-mail: kojman@math.huji.ac.il

a2 Department of Mathematics, The Hebrew University, Jerusalem, Israel, E-mail: shelah@math.huji.ac.il

Abstract

Our theme is that not every interesting question in set theory is independent of ZFC. We give an example of a first order theory T with countable D(T) which cannot have a universal model at ℵ1; without CH; we prove in ZFC a covering theorem from the hypothesis of the existence of a universal model for some theory; and we prove—again in ZFC—that for a large class of cardinals there is no universal linear order (e.g. in every regular ). In fact, what we show is that if there is a universal linear order at a regular λ and its existence is not a result of a trivial cardinal arithmetical reason, then λ “resembles” ℵ1—a cardinal for which the consistency of having a universal order is known. As for singular cardinals, we show that for many singular cardinals, if they are not strong limits then they have no universal linear order. As a result of the nonexistence of a universal linear order, we show the nonexistence of universal models for all theories possessing the strict order property (for example, ordered fields and groups, Boolean algebras, p-adic rings and fields, partial orders, models of PA and so on).

(Received April 24 1991)

(Revised July 12 1991)