Let A be an abelian group and B a pure injective pure extension of A. Then there is a homomorphic image C of B over A which is a pure injective hull of A; C can be constructed by using Zorn's lemma to find a suitable congruence on B. In a paper [4] which greatly generalises this and related facts about pure injectives, Walter Taylor asks (Problem 1.5) whether one can find a “construction” of C which is more concrete than the one mentioned above; he asks also whether the points of C can be explicitly described. In this note I return the answer No.

Let T be a second-order arithmetical theory, Λ a well-order, λ < Λ and X ⊆ ℕ. We use $[\lambda |X]_T^{\rm{\Lambda }}\varphi$ as a formalization of “φ is provable from T and an oracle for the set X, using ω-rules of nesting depth at most λ”.

For a set of formulas Γ, define predicative oracle reflection for T over Γ (Pred–O–RFNΓ(T)) to be the schema that asserts that, if X ⊆ ℕ, Λ is a well-order and φ ∈ Γ, then

$$\forall \,\lambda < {\rm{\Lambda }}\,([\lambda |X]_T^{\rm{\Lambda }}\varphi \to \varphi ).$$

In particular, define predicative oracle consistency (Pred–O–Cons(T)) as Pred–O–RFN{0=1}(T).

Our main result is as follows. Let ATR0 be the second-order theory of Arithmetical Transfinite Recursion, ${\rm{RCA}}_0^{\rm{*}}$ be Weakened Recursive Comprehension and ACA be Arithmetical Comprehension with Full Induction. Then,

$${\rm{ATR}}_0 \equiv {\rm{RCA}}_0^{\rm{*}} + {\rm{Pred - O - Cons\ }}\left( {{\rm{RCA}}_0^{\rm{*}} } \right) \equiv {\rm{RCA}}_0^{\rm{*}} + \,{\rm{Pred - O - Cons\ }}\left( {{\rm{RCA}}_0^{\rm{*}} } \right) \equiv {\rm{RCA}}_0^{\rm{*}} + \,{\rm{Pred - O - RFN}}\,_{{\bf{\Pi }}_2^1 } \left( {{\rm{ACA}}} \right).$$

We may even replace ${\rm{RCA}}_0^{\rm{*}}$ by the weaker ECA0, the second-order analogue of Elementary Arithmetic.

Thus we characterize ATR0, a theory often considered to embody Predicative Reductionism, in terms of strong reflection and consistency principles.

In [1], examples of types of U-rank 1 (i.e., minimal types) in the theories of separably closed fields were constructed, en route to displaying certain dimension phenomena. We construct here additional examples with U-rank 1 and of various transcendence degrees over arbitrary separably closed fields. Our examples include ones which are minimal but of infinite transcendence degree, i.e., not thin. Our interest in building new examples was piqued after seeing the role played by minimal types over separably closed fields in Hrushovski's analysis of abelian varieties. This article is the result of several working sessions between the authors at Wesleyan University and Paris 7, and was completed during the Model Theory of Fields program at MSRI in 1998. We are grateful for the hospitality and support of all three institutions. We thank Elisabeth Bouscaren and Françoise Delon for reading an earlier version of this paper, providing useful suggestions and corrections.

If G is a group with a supersimple theory having a finite SU-rank, then the subgroup of G generated by all of its normal nilpotent subgroups is definable and nilpotent. This answers a question asked by Elwes, Jaligot, Macpherson and Ryten. If H is any group with a supersimple theory, then the subgroup of H generated by all of its normal soluble subgroups is definable and soluble.

Using side-by-side Sacks forcing, it is shown that it is consistent that 2ω be large and that there be many types of ultrafilters of character ω1.

Cherlin introduced the concept of bad groups (of finite Morley rank) in [Ch1]. The existence of such groups is an open question. If they exist, they will contradict the Cherlin-Zil'ber conjecture that states that an infinite simple group of finite Morley rank is a Chevalley group over an algebraically closed field. In this paper, we prove that bad groups of finite Morley rank 3 act on a natural geometry Γ (namely on a special pseudoplane; see Corollary 20) sharply flag-transitively.We show that Γ is not very far from being a projective plane and when it is so rk(Γ) = 2 and Γ is not Desarguesian (Theorem 2). Baldwin [Ba] recently discovered non-Desarguesian projective planes of Morley rank 2. This discovery, together with this paper, makes the existence of bad groups (also of bad fields) more plausible. A bad field is a pair (K, A) of finite Morley rank, where K is an algebraically closed field, A <≠K* and A is infinite. There existence is also unknown.

In this paper, we define the concept of a sharp-field as a pair (K, A), where K is a field, A < K*and

1. K = A − A,

2. If a + b − 1 ∈ A, a ∈ A, b ∈ A, then either a = 1 or b = 1.

If K is finite this is equivalent to 1 and

2.′ ∣K∣ = ∣A∣2 ∣A∣ + 1.

Finite sharp-fields are special cases of difference sets [De]

The full-background-extender Kc -construction of [2] has the property that, if it does not break down and produces a final model , then

Ή is Woodin in V ⇒ Ή is Woodin in ,

for all Ή. It is natural to ask whether

κ is strong in V ⇒ κ is λ-strong in ,

for all κ, or even better,

κ is λ-strong in V ⇒ κ is λ-strong in .

As one might suspect, the more useful answer would be “yes”.

For the Kc-construction of [2], this question is open. The problem is that the construction of [2] is not local: because of the full-background-extender demand, it may produce mice projecting to ρ at stages much greater than ρ. Because of this, there is no reason to believe that if E is a λ-strong extender of V, then The natural proof only gives that if κ is Σ2-strong, then Σ, is strong in .

We do not know how to get started on this question, and suspect that in fact strong cardinals in V may fail to be strong in , if is the output of the construction of [2]. Therefore, we shall look for a modification of the construction of [2]. One might ask for a construction with output such that

(1) iteration trees on can be lifted to iteration trees on V,

(2) ∀δ(δ is Woodin ⇒ δ is Woodin in ), and

(3) (a) ↾κ(κ is a strong cardinal ⇒ κ is strong in ), and (b) ↾κ↾λ(Lim(λ) Λ κ is λ-strong ⇒ κ is λ-strong in ).

The number of homogeneous models has been studied in [1] and other papers. But the number of countable homogeneous models of a countable theory T is not determined when dropping the GCH. Morley in [2] proves that if a countable theory T has more than ℵ1 nonisomorphic countable models, then it has such models. He conjectures that if a countable theory T has more than ℵ0 nonisomorphic countable models, then it has such models. In this paper we show that if a countable theory T has more than ℵ0 nonisomorphic countable homogeneous models, then it has such models.

We adopt the conventions in [1]–[3]. Throughout the paper T is a theory and the language of T is denoted by L which is countable.

Lemma 1. If a theory T has more than ℵ0types, then T hasnonisomorphic countable homogeneous models.

Proof. Suppose that T has more than ℵ0 types. From [2, Corollary 2.4] T has types. Let σ be a Ttype with n variables, and T′ = T ⋃ {σ(c1, …, cn)}, where c1, …, cn are new constants. T′ is consistent and has a countable model (, a1, …, an). From [3, Theorem 3.2.8] the reduced model has a countable homogeneous elementary extension . σ is realized in . This shows that every type σ is realized in at least one countable homogeneous model of T. But each countable model can realize at most ℵ0 types. Hence T has at least countable homogeneous models. On the other hand, a countable theory can have at most nonisomorphic countable models. Hence the number of nonisomorphic countable homogeneous models of T is .

In the following, we shall use the languages Lα (α = 0, 1, 2) defined in [2]. We give a brief description of them. For a countable theory T, let K be the class of all models of T. L = L0 is countable.

In this paper we define by means of a partition property a decreasing sequence N = ‹Nα: α is an ordinal› of classes of ordinals. This property is a generalization of the nonexistence of special Aronszajn trees: the successor cardinal κ+ is in N0 iff there does not exist a special Aronszajn κ+-tree.

The interest in the classes Nα stems from their applicability in model theory, in particular to that aspect of model theory dealing with ordered and two-cardinal models. A model is κ-like iff < is a linear ordering of A of cardinality κ but such that every proper initial segment has cardinality < κ. is α-ordered iff ≼ is a reflexive, linear ordering of some subset of A with order type α. The sequence N can be characterized by a first-order sentence σ in the following manner: The sentence σ has a κ-like α-ordered model iff κ ∉ Nα. This characterization will allow us to translate various independence statements regarding the sequence N to statements about the independence of transfer properties. We say that the transfer property κ → λ holds iff every first-order sentence which has a κ-like model also has a λ-like model. κ ⇸ λ is the negation of κ → λ.

Let DO denote the principle: Every infinite set has a dense linear ordering. DO is compared to other ordering principles such as O, the Linear Ordering principle, KW, the Kinna-Wagner Principle, and PI, the Prime Ideal Theorem, in ZF, Zermelo-Fraenkel set theory without AC, the Axiom of Choice.

The main result is:

Theorem. AC ⇒ KW ⇒ DO ⇒ O, and none of the implications is reversible in ZF + PI.

The first and third implications and their irreversibilities were known. The middle one is new. Along the way other results of interest are established. O, while not quite implying DO, does imply that every set differs finitely from a densely ordered set. The independence result for ZF is reduced to one for Fraenkel-Mostowski models by showing that DO falls into two of the known classes of statements automatically transferable from Fraenkel-Mostowski to ZF models. Finally, the proof of PI in the Fraenkel-Mostowski model leads naturally to versions of the Ramsey and Ehrenfeucht-Mostowski theorems involving sets that are both ordered and colored.

We study the automorphism group of the algebraic closure of a substructure A of a pseudo-finite field F. We show that the behavior of this group, even when A is large, depends essentially on the roots of unity in F. For almost all completions of the theory of pseudofinite fields, we show that over A, algebraic closure agrees with definable closure, as soon as A contains the relative algebraic closure of the prime field.

We provide algebraic conditions ensuring the decidability of the theory of modules over effectively given Prüfer (in particular Bézout) domains with infinite residue fields in terms of a suitable generalization of the prime radical relation. For Bézout domains these conditions are also necessary.

In 1956 W. Ackermann proposed a new axiomatic set theory, that has received some attention in recent years. (See references [3], [4], [6], [7], and [9].) This theory distinguishes between sets and classes. In this paper we study mainly the natural models of this theory. We show, among other results, that the set-theoretical fragment of these models are also models of Zermelo-Fraenkel set theory. This result gives a partial answer to the question, raised by A. Levy, of the relative strength of Ackermann's set theory with respect of Zermelo-Fraenkel's.2

Our goal in this paper is to analyze the interpretation of arbitrary unsolvable λ-terms in a given model of λ-calculus. We focus on graph models and (a special type of) stable models. We introduce the syntactical notion of a decoration and the semantical notion of a critical sequence. We conjecture that any unsolvable term β-reduces to a term admitting a decoration. The main result of this paper concerns the interconnection between those two notions: given a graph model or stable model , we show that any unsolvable term admitting a decoration and having a non-empty interpretation in generates a critical sequence in the model.

In the last section, we examine three classical graph models, namely the model of Plotkin and Scott, Engeler's model and Park's model . We show that and do not contain critical sequences whereas does.

The Mycielski ideal is defined to consist of all sets A ⊆ ℕk such that {f ↾ X : f ∈ A} ≠ Xk for all . It will be shown that the covering numbers for these ideals are all equal. However, the covering numbers of the closely associated Rosłanowski ideals will be shown to be consistently different.

We show that for any uncountable cardinal λ, the category of sets of cardinality at least λ and monomorphisms between them cannot appear as the category of points of a topos, in particular is not the category of models of a ${L_{\infty ,\omega }}$-theory. More generally we show that for any regular cardinal $\kappa < \lambda$ it is neither the category of κ-points of a κ-topos, in particular, nor the category of models of a ${L_{\infty ,\kappa }}$-theory.

The proof relies on the construction of a categorified version of the Scott topology, which constitute a left adjoint to the functor sending any topos to its category of points and the computation of this left adjoint evaluated on the category of sets of cardinality at least λ and monomorphisms between them. The same techniques also apply to a few other categories. At least to the category of vector spaces of with bounded below dimension and the category of algebraic closed fields of fixed characteristic with bounded below transcendence degree.