It is proved that Vaught's conjecture is true for modules over an arbitrary countable hereditary noetherian prime ring.

We characterise strongly minimal groups interpretable in elementary extensions of compact complex analytic spaces.

We establish the following results:

1. In ZF (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC), for every set I and for every ordinal number α ≥ ω, the following statements are equivalent:

(a) The Tychonoff product of ∣α∣ many non-empty finite discrete subsets of I is compact.

(b) The union of ∣α∣ many non-empty finite subsets of I is well orderable.

2. The statement: For every infinite set I, every closed subset of the Tychonoff product [0, 1]Iwhich consists offunctions with finite support is compact, is not provable in ZF set theory.

3. The statement: For every set I, the principle of dependent choices relativised to I implies the Tychonoff product of countably many non-empty finite discrete subsets of I is compact, is not provable in ZF0 (i.e., ZF minus the Axiom of Regularity).

4. The statement: For every set I, every ℵ0-sized family of non-empty finite subsets of I has a choice function implies the Tychonoff product of ℵ0many non-empty finite discrete subsets of I is compact, is not provable in ZF0.

If Σ is the class of all fields and Σ* is the class of all algebraically closed fields, then it is well known that Σ* is characterized by the following properties:

(i) Σ* is the class of models of some first order theory K*.

(ii) If m1m2 are in Σ* and m1 ⊆ m2 then m1 ≺ m2 (m1 is an elementary substructure of m2, i.e. any first order sentence true in m1 is true in m2).

(iii) If m1 is in Σ then there is a structure m2 in Σ* such that m1 ⊆ m2.

If Σ is some other class of models of a first order theory K and a subclass Σ* of Σ exists satisfying (i)–(iii) then Σ* is uniquely determined and K* (which is unique up to logical equivalence) is called the model-companion of K. This notion is a generalization of the fundamental notion of model-completion introduced and extensively studied by A. Robinson [6], When the model-companion exists it provides the basis for a satisfactory treatment of the notion of an algebraically closed model of K.

Recently A. Robinson has developed a more general formulation of the notion of “algebraically closed” structures in Σ, which is applicable to any inductive elementary class Σ of structures (by elementary we always mean ECΔ). Condition (i) must be weakened to

(i′) Σ* is closed under elementary substructure (i.e. if m1 is in Σ* and m2 ≺ m1 then m2 is in Σ*).

I analyze the hierarchies of the bounded and the weak bounded forcing axioms, with a focus on their versions for the class of subcomplete forcings, in terms of implications and consistency strengths. For the weak hierarchy, I provide level-by-level equiconsistencies with an appropriate hierarchy of partially remarkable cardinals. I also show that the subcomplete forcing axiom implies Larson’s ordinal reflection principle at ω2, and that its effect on the failure of weak squares is very similar to that of Martin’s Maximum.

One concern of descriptive set theory is the classification of definable sets of reals. Taken loosely ‘definable’ can mean anything from projective to formally describable in the language of Zermelo-Fraenkel set theory (ZF). Recursiveness, in the case of Kleene recursion, can be a particularly informative notion of definability. Sets of integers, for example, are categorized by their positions in the upper semilattice of Turing degrees, and the algorithms for computing their characteristic functions may be taken as their defining presentations. In turn it is interesting to position the common fauna of descriptive set theory in the upper semilattice of Kleene degrees. In so doing not only do we gain a perspective on the complexity of those sets common to the study of descriptive set theory but also a refinement of the theory of analytic sets of reals. The primary concern of this paper is to calculate the relative complexity of several notable coanalytic sets of reals and display (under suitable set theoretic hypothesis) several natural solutions to Post's problem for Kleene recursion.

For sets of reals A and B one says A is Kleene recursive in B (written A ≤KB) iff there is a real y so that the characteristic function XA of A is recursive (in the sense of Kleene [1959]) in y, XB and the existential integer quantifier ∃; i.e. there is an integer e so that XA(x) ≃ {e}(x, y, XB, ≃). Intuitively, membership of a real x ϵ ωω in A can be decided from an oracle for x, y and B using a computing machine with a countably infinite memory and an ability to search and manipulate that memory in finite time. A set is Kleene semirecursive in B if it is the domain of an integer valued partial function recursive in y, XB and ≃ for some real y.

In ZFC, the class Ord of ordinals is easily seen to satisfy the definable version of strong inaccessibility. Here we explore deeper ZFC-verifiable combinatorial properties of Ord, as indicated in Theorems A & B below. Note that Theorem A shows the unexpected result that Ord is never definably weakly compact in any model of ZFC.

Theorem A. Let ${\cal M}$ be any model of ZFC.

1. (1) The definable tree property fails in ${\cal M}$: There is an ${\cal M}$-definable Ord-tree with no ${\cal M}$-definable cofinal branch.

2. (2) The definable partition property fails in ${\cal M}$: There is an ${\cal M}$-definable 2-coloring $f:{[X]^2} \to 2$ for some ${\cal M}$-definable proper class X such that no ${\cal M}$-definable proper classs is monochromatic for f.

3. (3) The definable compactness property for ${{\cal L}_{\infty ,\omega }}$ fails in ${\cal M}$: There is a definable theory ${\rm{\Gamma }}$ in the logic ${{\cal L}_{\infty ,\omega }}$ (in the sense of ${\cal M}$) of size Ord such that every set-sized subtheory of ${\rm{\Gamma }}$ is satisfiable in ${\cal M}$, but there is no ${\cal M}$-definable model of ${\rm{\Gamma }}$.

Theorem B. The definable ⋄Ord principle holds in a model ${\cal M}$ of ZFC iff ${\cal M}$ carries an ${\cal M}$-definable global well-ordering.

Theorems A and B above can be recast as theorem schemes in ZFC, or as asserting that a single statement in the language of class theory holds in all ‘spartan’ models of GB (Gödel-Bernays class theory); where a spartan model of GB is any structure of the form $\left( {{\cal M},{D_{\cal M}}} \right)$, where ${\cal M} \models {\rm{ZF}}$ and ${D_{\cal M}}$ is the family of${\cal M}$-definable classes. Theorem C gauges the complexity of the collection GBspa of (Gödel-numbers of) sentences that hold in all spartan models of GB.

Theorem C. GBspa is ${\rm{\Pi }}_1^1$-complete.

Let (M, ≤,…) denote a Boolean ordered o-minimal structure. We prove that a Boolean subalgebra of M determined by an algebraically closed subset contains no dense atoms. We show that Boolean algebras with finitely many atoms do not admit proper expansions with o-minimal theory. The proof involves decomposition of any definable set into finitely many pairwise disjoint cells, i.e., definable sets of an especially simple nature. This leads to the conclusion that Boolean ordered structures with o-minimal theories are essentially bidefinable with Boolean algebras with finitely many atoms, expanded by naming constants. We also discuss the problem of existence of proper o-minimal expansions of Boolean algebras.

We answer a question of Garcia-Ferreira and Hrušák by consistently constructing a MAD family maximal in the Katětov order. We also answer several questions of Garcia-Ferreira.

For every partial combinatory algebra (pca), we define a hierarchy of extensionality relations using ordinals. We investigate the closure ordinals of pca’s, i.e., the smallest ordinals where these relations become equal. We show that the closure ordinal of Kleene’s first model is ${\omega _1^{\textit {CK}}}$ and that the closure ordinal of Kleene’s second model is $\omega _1$ . We calculate the exact complexities of the extensionality relations in Kleene’s first model, showing that they exhaust the hyperarithmetical hierarchy. We also discuss embeddings of pca’s.

The theory of objects recursive in functions of higher types was developed originally by Kleene [Kl 1,2]. The sets of objects of finite types are defined by Kleene as follows:

T0 = ω = the set of nonnegative integers,

and, for all i ∈ ω,

Ti+1 = Tiω = the set of total functions from Ti to ω.

In order to study and classify the functions recursive in Kolmogorov's R-operator, it is necessary to discuss recursion in partial functions of higher types. Accordingly, we define the partial objects of higher finite types by

and, for all i ∈ ω,

= the set of partial functions from Ti to ω.

In addition to considering the partial functions from Ti to ω, the “hereditary partial” functions also have to be considered. For example, Hinman has studied objects of type 2 which act on partial functions from ω to ω. In case of dealing with a function F which has partial functions in its domain, we usually require that F be consistent, i.e. if F(f) is defined and g is an object of the same type as f, defined at least at all points where f is defined (in short, g extends f—notation: g ⊇ f), then F(g) is defined and is equal to F(f). Now we can define the higher type partial objects which are consistent by

and, for all i ∈ ω,

= the set of consistent partial functions from to ω.

We prove that every Turing degree a bounding some non-GL2 degree is recursively enumerable in and above (r.e.a.) some 1-generic degree.

This paper is based on the notions originally described by Dekker [2], [3], and the reader is referred to these for explanation of notation etc. Briefly, we are concerned with a countably infinite dimensional countable vector space Ū with recursive operations, regarded as being coded as a set of natural numbers. Necessarily, then, Ū must be a vector space over a field which itself is in some sense recursively enumerable and has recursive operations.

Our method is based on the same principles which we have assumed in a paper about the foundations of metamathematics which we published in collaboration with J. Herzberg some years ago. Nevertheless we have introduced some important modifications which enable us to use a very restricted preliminary language and to avoid any artificial construction.

The problem of a restricted language in mathematics appears to be a very important one, as our ordinary language is really ambiguous and obscure. On the other hand there is the problem of restricting the intuitive logic to be used in our construction. We shall see that for this purpose we need only substitutions and modus ponens.

If we avoided any abbreviation, we could reduce our preliminary language to propositions of the following forms:

X is a theorem.

If P, then Q.

But we have to do with very long symbolic expressions and as there is not yet any machine to build up such expressions we must use abbreviations. For this purpose we shall use propositions of the following forms in the preliminary language:

X is an expression.

X is an abbreviation of Y.

Any proposition which does not belong to one of our four forms will be excluded from our construction.

The existence of complete rigid Boolean algebras was first proved by McAloon [8] who also showed that every Boolean algebra can be completely embedded in a rigid complete Boolean algebra. McAloon was interested in consistency results on ordinal definable sets. His approach was based on forcing. Recently, Shelah [10] proved that for every uncountable cardinal κ there exists a Boolean algebra of power κ with rigid completion. Extending his method, we get the following theorems.

Theorem 1. Any Boolean algebra B can be completely embedded in a complete Boolean algebra C with no nontrivial σ-complete one-one endomor-phism. If B satisfies the κ-chain condition for an uncountable cardinal κ, the same holds true for C.

Since every automorphism is a complete endomorphism, it follows from Theorem 1 that C is rigid. The other extreme case of Boolean algebras are homogeneous algebras. It was proved by Kripke [7] that every Boolean algebra can be completely embedded in a homogeneous complete Boolean algebra. In his proof, the homogeneous algebra contains antichains of cardinality equal to the power of the embedded Boolean algebra. The following result shows that this is essential: the analogue of Theorem 1 is not provable in set theory even for Boolean algebras with a very weak homogeneity property. We use a Suslin tree with particular properties constructed by Jensen [6] in conjunction with a forcing argument.

BD-N is a weak principle of constructive analysis. Several interesting principles implied by BD-N have already been identified, namely the closure of the anti-Specker spaces under product, the Riemann Permutation Theorem, and the Cauchyness of all partially Cauchy sequences. Here these are shown to be strictly weaker than BD-N, yet not provable in set theory alone under constructive logic.

We examine the question of how many Boolean algebras, distinct up to isomorphism, that are quotients of the powerset of the naturals by Borel ideals, can be proved to exist in ZFC alone. The maximum possible value is easily seen to be the cardinality of the continuum ; earlier work by Ilijas Farah had shown that this was the value in models of Martin's Maximum or some similar forcing axiom, but it was open whether there could be fewer in models of the Continuum Hypothesis.

We develop and apply a new technique for constructing many ideals whose quotients must be nonisomorphic in any model of ZFC. The technique depends on isolating a kind of ideal, called shallow, that can be distinguished from the ideal of all finite sets even after any isomorphic embedding, and then piecing together various copies of the ideal of all finite sets using distinct shallow ideals. In this way we are able to demonstrate that there are continuum-many distinct quotients by Borel ideals, indeed by analytic P-ideals, and in fact that there is in an appropriate sense a Borel embedding of the Vitali equivalence relation into the equivalence relation of isomorphism of quotients by analytic P-ideals. We also show that there is an uncountable definable wellordered collection of Borel ideals with distinct quotients.

Throughout this paper κ denotes an infinite cardinal, S = Sym(κ) and G is a subgroup of S. We shall be seeking the subgroups G with [S: G] < 2κ. In [2], the following result was proved.

Theorem 1. If [S: G] ≤ κthen there exists a subset Δ of k such that ∣Δ∣ < k and S(Δ) ≤ G.

Here S(Δ) = Sym(K/⊿) is the pointwise stabilizer of Δ in S.

However, the converse of Theorem 1 is not true. For if cf(κ) ≤ ∣Δ∣ < κ, then [S: S(Δ)] ≥ κcf(κ) > κ. This suggests that a substantially sharpened version of Theorem 1 may be true.

Question 1 [2]. Is it provable in ZFC, or even in ZFC with GCH, that if [S: G] ≤ κ then there is a subset Δ of κ such that ∣Δ∣ < cf(κ) and S(Δ) ≤ G?

At least two of the authors of [2] made a serious attempt to answer the above question positively. In §3, we shall see that they were essentially trying to prove that measurable cardinals do not exist.

The following result, due independently to Semmes [5] and Neumann [2], suggests a second way in which Theorem 1 might be improved.

Theorem 2. If k = ℵ0andthen there is a finite subset Δ of k such thatS(Δ) ≤ G.

Question 2 [2]. Is it provable in ZFC that if [S: G] < 2κ then there is a subset Δ of κ such that ∣Δ∣ < κ and S(Δ) < G?

This question will be answered negatively in §4.

Monadic second order logic and linear temporal logic are two logical formalisms that can be used to describe classes of infinite words, i.e., first-order models based on the natural numbers with order, successor, and finitely many unary predicate symbols.

Monadic second order logic over infinite words (S1S) can alternatively be described as a first-order logic interpreted in ${\cal P}\left( \omega \right)$, the power set Boolean algebra of the natural numbers, equipped with modal operators for ‘initial’, ‘next’, and ‘future’ states. We prove that the first-order theory of this structure is the model companion of a class of algebras corresponding to a version of linear temporal logic (LTL) without until.

The proof makes crucial use of two classical, nontrivial results from the literature, namely the completeness of LTL with respect to the natural numbers, and the correspondence between S1S-formulas and Büchi automata.