A field K is regularly closed if every absolutely irreducible affine variety defined over K has K-rational points. This notion was first isolated by Ax [A] in his work on the elementary theory of finite fields. Later Jarden [J2] and Jarden and Kiehne [JK] extended this in different directions. One of the primary results in this area is that the elementary properties of a regularly closed field K with a free Galois group (on either finitely or countably many generators) are determined by the set of integer polynomials in one indeterminate with a zero in K. The method of proof employed in [J1], [J2] and [JK] is unusual for algebra since it is a measure-theoretic argument. In this brief summary we have not made any attempt at completeness. We refer the reader to the recent paper of Cherlin, van den Dries, and Macintyre [CDM] and to the forthcoming book by Fried and Jarden [FJ] for a more thorough discussion of the latest results. We would like to thank Moshe Jarden, Angus Macintyre, and Zoe Chatzidakis for their comments on an earlier version of this paper.

A countable field K is ω-free if the absolute Galois group , where is the algebraic closure of K and is the free profinite group on ℵ0 generators.

The central notion of this paper is that of a κ-elementary end extension of a model of set theory. A model is said to be a κ-elementary end extension of a model of set theory if > and κ, which is a cardinal of , is end extended in the passage from to , i.e., enlarges κ without enlarging any of its members (see §0 for more detail). This notion was implicitly introduced by Scott in [Sco] and further studied by Keisler and Morley in [KM], Hutchinson in [H] and recently by the author in [E]. It is not hard to see that if has a κ-elementary end extension then κ must be regular in . Keisler and Morley [KM] noticed that this has a converse if is countable, i.e., if κ is a regular cardinal of a countable model then has a κ-elementary end extension. Later Hutchinson [H] refined this result by constructing κ-elementary end extensions 1 and 2 of an arbitrary countable model in which κ is a regular uncountable cardinal, such that 1 adds a least new element to κ while 2 adds no least new ordinal to κ. It is a folklore fact of model theory that the Keisler-Morley result gives soft and short proofs of countable compactness and abstract completeness (i.e. recursive enumera-bility of validities) of the logic L(Q), studied extensively in Keisler's [K2]; and Hutchinson's refinement does the same for stationary logic L(aa), studied by Barwise et al. in [BKM]. The proof of Keisler-Morley and that of Hutchinson make essential use of the countability of since they both rely on the Henkin-Orey omitting types theorem. As pointed out in [E, Theorem 2.12], one can prove these theorems using “generic” ultrapowers just utilizing the assumption of countability of the -power set of κ. The following result, appearing as Theorem 2.14 in [E], links the notion of κ-elementary end extension to that of measurability of κ. The proof using (b) is due to Matti Rubin.

It is shown that there are exactly six normal DeMorgan monoids generated by the idntity element alone. The free DeMorgan monoid with no generators but the identity is characterised and shown to have exactly three thousand and eighty-eight elements. This result solves the “Ackermann constant problem” of describing the structure of sentential constants in the logic R.

T. Jech [4] and M. Takahashi [7] proved that given any partial ordering R in a model of ZFC there is a symmetric submodel of a generic extension of where R is isomorphic to the injective ordering on a set of cardinals.

The authors raised the question whether the injective ordering of cardinals can be universal, i.e. whether the following axiom of “cardinal universality” is consistent:

CU. For any partially ordered set (X, ≼) there is a bijection f:X → Y such that

(i.e. x ≼ y iff ∃g: f(x) → f(y) injective). (See [1].)

The consistency of CU relative to ZF0 (Zermelo-Fraenkel set theory without foundation) is proved in [2], but the transfer method of Jech-Sochor-Pincus cannot be applied to obtain consistency with full ZF (including foundation), since CU apparently is not boundable.

In this paper the authors define a model of ZF + CU as a symmetric submodel of a generic extension obtained by forcing “à la Easton” with a class of conditions which add κ generic subsets to any regular cardinal κ of a ground model satisfying ZF + V = L.

This paper extends earlier work by its authors on formal aspects of the processes of contracting a theory to eliminate a proposition and revising a theory to introduce a proposition. In the course of the earlier work, Gärdenfors developed general postulates of a more or less equational nature for such processes, whilst Alchourrón and Makinson studied the particular case of contraction functions that are maximal, in the sense of yielding a maximal subset of the theory (or alternatively, of one of its axiomatic bases), that fails to imply the proposition being eliminated.

In the present paper, the authors study a broader class, including contraction functions that may be less than maximal. Specifically, they investigate “partial meet contraction functions”, which are defined to yield the intersection of some nonempty family of maximal subsets of the theory that fail to imply the proposition being eliminated. Basic properties of these functions are established: it is shown in particular that they satisfy the Gärdenfors postulates, and moreover that they are sufficiently general to provide a representation theorem for those postulates. Some special classes of partial meet contraction functions, notably those that are “relational” and “transitively relational”, are studied in detail, and their connections with certain “supplementary postulates” of Gàrdenfors investigated, with a further representation theorem established.

A very fruitful line of research in recent years has been the application of techniques in large cardinals and forcing to the production of models in which certain consequences of the axiom of determinateness (AD) are true or in which certain “AD-like” consequences are true. Numerous results have been published on this subject, among them the papers of Bull and Kleinberg [4], Bull [3], Woodin [15], Mitchell [11], and [1], [2].

Another such model will be constructed in this paper. Specifically, the following theorem is proven.

Theorem 1. Con(ZFC + There are cardinals κ < δ < λ so that κ is a supercompact limit of supercompact cardinals, λ is a measurable cardinal, and δ is λ supercompact) ⇒ Con(ZF + ℵ1 and ℵ2 are Ramsey cardinals + The ℵn for 3 ≤ n ≤ ω are singular cardinals of cofinality ω each of which carries a Rowbottom filter + ℵω + 1 is a Ramsey cardinal + ℵω + 2 is a measurable cardinal).

It is well known that under AD + DC, ℵ2 and ℵ2 are measurable cardinals, the ℵn for 3 ≤ n < ω are singular Jonsson cardinals of cofinality ℵ2, ℵω is a Rowbottom cardinal, and ℵω + 1 and ℵω + 2 are measurable cardinals.

The proof of the above theorem will use the existence of normal ultrafilters which satisfy a certain property (*) (to be defined later) and an automorphism argument which draws upon the techniques developed in [9], [2], and [4] but which shows in addition that certain supercompact Prikry partial orderings are in a strong sense “homogeneous”. Before beginning the proof of the theorem, however, we briefly mention some preliminaries.