D’Aquino, Knight, and Starchenko classified the countable real closed fields with integer parts that are nonstandard models of Peano Arithmetic. We rule out some possibilities for extending their results to the uncountable and study real closures of ɷ1-like models of PA.

We investigate the algebraic structure of the upper semi-lattice formed by the recursively enumerable Turing degrees. The following strong generalization of Lachlan's Nonsplitting Theorem is proved: Given n ≥ 1, there exists an r.e. degree d such that the interval [d, 0′] ⊂ R admits an embedding of the n-atom Boolean algebra preserving (least and) greatest element, but also such that there is no (n + 1 )-tuple of pairwise incomparable r.e. degrees above d which pairwise join to 0′ (and hence, the interval [d, 0′] ⊂ R does not admit a greatest-element-preserving embedding of any lattice which has n + 1 co-atoms, including ). This theorem is the dual of a theorem of Ambos-Spies and Soare, and yields an alternative proof of their result that the theory of R has infinitely many one-types.

Answering a problem of Fuhrken we prove that for every κ, 1 ≤ κ ≤ ℵ0, there is a (countable) complete theory T, with no prime model, and exactly κ minimal models (up to isomorphism).

An infinite set X is given. D. Gale, in correspondence with J. Mycielski, described the following game in which players one and two play an inning per positive integer: In the nth inning one chooses a finite subset Xn of X, and two chooses a point xn from (X1∪ … ∪Xn)\{x1,…,xn−1}. A play

is won by two if . Gale asked whether two could have a winning strategy which depends for each n on knowledge of only the contents of the set

In mathematical terms, is there a function F defined on the collection of finite subsets of X such that:

for every sequence X1, x1, …, Xn, xn,…. where each Xn is a finite subset

of X and for each n

we have

We shall call a strategy of this sort a remainder strategy for two. If there is some finite subset U of X such that F(U) ∉ U, then F cannot be a winning remainder strategy for two, because one can defeat F by choosing U each inning. So, when studying remainder strategies for two we may as well assume that for each finite set U ⊂ X, F(U) ∈ U.

Using projectivity groups, we classify some polygons with strongly minimal point rows and show in particular that no infinite quadrangle can have sharply 2-transitive projectivity groups in which the point stabilizers are abelian. In fact, we characterize the finite orthogonal quadrangles Q(4,2). Q− (5.2) and Q(4,3) by this property. Finally we show that the sets of points, lines and flags of any ℵ1-categorical polygon have Morley degree 1.

Using convex subrings of *$\mathbb{C}$, a nonstandard extension of $\mathbb{C}$, we define several kinds of complex bounded polynomials and we provide their associated analytic functions obtained by taking the quasistandard part.

In Chapter XIII of [4], Prior discusses a system QKt, designed to stand to the “minimal” tense logic Kt as the modal system Q of [3] stands to S5. In this paper I provide semantics for a similar system, slightly weaker than QKt: the weakening is due to the fact that Prior's axioms are slightly too strong for a “minimal” system. An extended post-Henkin style completeness proof for the axiomatization with respect to the semantics provided is then given: the underlying three-valued nature of the semantics requires a twist in the proof which is new to its author at least, and also results in some details being set out which could well be glossed over in the two-valued case.

There is a comeager set contained in the set of 1-generic reals and a first order structure such that for any real number X. there is an element of which is recursive in X if and only if there is a presentation of which is recursive in X.

Modules over a ring R, when tensored with an (R, S)-bimodule, are converted to S-modules. Here I consider, from the standpoint of the model theory of modules, the effect of this operation. The primary motivation arises from questions concerning representation type of algebras and interpretability of modules, where such tensor functors play a key role, but this paper is devoted to more general considerations. For instance, the elementary duality of [2] and [1] is generalised here. It is also shown that, although tensor product does not preserve elementary equivalence, one can define the tensor product of two complete theories of R-modules. The results in Section 1 grew out of a number of discussions with T. Kucera.

This is our generic situation. We have an (R, S)-bimodule B and we are considering the functor –⊗RBS from Mod – R (the category of right R-modules) to Mod – S which is given on objects by MR ↦ (M ⊗RB)s and has the obvious action on morphisms. There is a somewhat more general situation: namely we may consider the effect of tensoring (T, R)-bimodules over R with B to obtain (T, S)-bimodules. For the intended applications this case is not needed. Moreover, although some results extend to this more general case, there are some which definitely do not (see the example after 2.1). Therefore we confine ourselves to the case first described. We will also consider B itself as a variable and so ask “What is the effect of tensoring R-modules with (R, S)-bimodules?”.

Let κ be the cardinality of some saturated model of Peano Arithmetic. There is a set of ${2^{{\aleph _0}}}$ saturated models of PA, each having cardinality κ, such that whenever M and N are two distinct models from this set, then Aut(${\cal M}$) ≇ Aut ($${\cal N}$$).

Borovik proposed an axiomatic treatment of Morley rank in groups, later modified by Poizat, who showed that in the context of groups the resulting notion of rank provides a characterization of groups of finite Morley rank [2]. (This result makes use of ideas of Lascar, which it encapsulates in a neat way.) These axioms form the basis of the algebraic treatment of groups of finite Morley rank undertaken in [1].

There are, however, ranked structures, i.e., structures on which a Borovik-Poizat rank function is defined, which are not ℵ0-stable [1, p. 376]. In [2, p. 9] Poizat raised the issue of the relationship between this notion of rank and stability theory in the following terms: “… un groupe de Borovik est une structure stable, alors qu'un univers rangé n'a aucune raison de l'être …” (emphasis added). Nonetheless, we will prove the following:

Theorem 1.1. A ranked structure is superstable.

An example of a non-ℵ0-stable structure with Borovik-Poizat rank 2 is given in [1, p. 376]. Furthermore, it appears that this example can be modified in a straightforward way to give ℵ0-stable structures of Borovik-Poizat rank 2 in which the Morley rank is any countable ordinal (which would refute a claim of [1, p. 373, proof of C.4]). We have not checked the details. This does not leave much room for strenghthenings of our theorem. On the other hand, the proof of Theorem 1.1 does give a finite bound for the heights of certain trees of definable sets related to unsuperstability, as we will see in Section 5.

The study of countable theories categorical in some uncountable power was initiated by Łoś and Vaught and developed in two stages. First, Morley proved (1962) that a countable theory categorical in some uncountable power is categorical in every uncountable power, a conjecture of Łoś. Second, Baldwin and Lachlan confirmed (1969) Vaught's conjecture that a countable theory categorical in some uncountable power has either one or countably many isomorphism types of countable models. That result was obtained by pursuing a line of research developed by Marsh (1966). For certain well-behaved theories, which he called strongly minimal, Marsh's method yielded a simple proof of Łoś's conjecture and settled Vaught's conjecture.

In recent years efforts have been made to extend these results to uncountable theories. The generalized Łoś conjecture states that a theory T categorical in some power greater than ∣T∣ is categorical in every such power. It was settled by Shelah (1970). Shelah then raised the question of the models in power ∣T∣ = ℵα of a theory T categorical in ∣T∣+, conjecturing in [S3] that there are exactly ∣α∣ + ℵ0 such models, up to isomorphism. This conjecture provided the initial motivation for the present work. We define and study semi-minimal theories analogous in some ways to Marsh's strongly minimal (countable) theories. We describe the models of a semi-minimal theory T which contain an infinite indiscernible set. Besides throwing some light on Shelah's conjecture, our method gives simple proofs of the Łoś conjecture and of the Morley conjecture on categoricity in ∣T∣, in the case of a semi-minimal theory T. Other results as well as some examples are provided.

In this paper we deal with the model theory of differentially closed fields of characteristic zero with finitely many commuting derivations. First we observe that the only known lower bound for the Lascar rank of types in differentially closed fields, announced in a paper of McGrail, is false. This gives us a new class of regular types which are orthogonal to fields. Then we classify the subgroups of the additive group of Lascar rank omega with differential-type 1 which are nonorthogonal to fields. The last parts consist of an analysis of the quotients of the heat variety. We show that the generic type of such a quotient is locally modular. Finally, we answer a question of Phylliss Cassidy about the existence of certain Jordan-Hölder type series in the negative.

Suppose M is a countable standard transitive model of set theory. P. J. Cohen [2] showed that if κ is an infinite cardinal of M then there is a one-to-one function Fκ from κ into the set of real numbers such that M[Fκ] is a model of set theory with the same cardinals as M.

If Tκ is the range of Fκ then Cohen also showed [2] that M[Tκ] fails to satisfy the axiom of choice. We will give an easy proof of this fact.

If κ, λ are infinite we will also show that M[Tκ] is elementarily equivalent to M[Tλ] and that (] in M[Fλ]) is elementarily equivalent to (] in M[FK]).

Finally we show that there may be an N ∈ M[GK] which is a standard model of set theory (without the axiom of choice) and which has, from the viewpoint of M[GK], more real numbers than ordinals.

We write ZFC and ZF for Zermelo-Fraenkel set theory, respectively with and without the axiom of choice (AC). GBC is Gödel-Bernays' set theory with AC. DC and ACℵo are respectively the axioms of dependent choice and of countable choice defined in [6].

Lower case Greek characters (other than ω) are used as variables over ordinals. When α is an ordinal, R(α) is the set of all sets with rank less than α.