Let I be an ideal of subsets of a Polish space X, containing all singletons and possessing a Borel basis. Assuming that I does not satisfy ccc, we consider the following conditions (B), (M) and (D). Condition (B) states that there is a disjoint family F ⊆ P(X) of size ϲ, consisting of Borel sets which are not in I. Condition (M) states that there is a Borel function f : X → X with f−1[{x}] ∉ I for each x ∈ X. Provided that X is a group and I is invariant, condition (D) states that there exist a Borel set B ∉ I and a perfect set P ⊆ X for which the family {B+x : x ∈ P} is disjoint. The aim of the paper is to study whether the reverse implications in the chain (D) ⇒ (M) ⇒ (B) ⇒ not-ccc can hold. We build a σ-ideal on the Cantor group witnessing (M) & ¬(D) (Section 2). A modified version of that σ-ideal contains the whole space (Section 3). Some consistency results on deriving (M) from (B) for “nicely” defined ideals are established (Sections 4 and 5). We show that both ccc and (M) can fail (Theorems 1.3 and 5.6). Finally, some sharp version's of (M) for invariant ideals on Polish groups are investigated (Section 6).

Let λ be a singular cardinal of uncountable cofinality ν. Under various assumptions about the sizes of covering families for cardinals below λ, we prove upper bounds for the covering number COV(λ. λ. ν+. 2). This covering number is closely related to the cofinality of the partial order ([λ]ν. ⊆).

Hindman's Theorem is a prototypical example of a combinatorial theorem with a proof that uses the topology of the ultrafilters. We show how the methods of this proof, including topological arguments about ultrafilters, can be translated into second order arithmetic.

The main subject of the article is the finite submodel property for ℵ0-categorical structures, in particular under the additional assumptions that the structure is simple, 1-based and has trivial dependence. Here, a structure has the finite submodel property if every sentence which is true in the structure is true in a finite substructure of it. It will be useful to consider a couple of other finiteness properties, related to the finite submodel property, which are variants of the usual concept of saturation.

For the rest of the introduction we will assume that M is an ℵ0-categorical (infinite) structure with a countable language. We also assume that there is an upper bound to the arity of the function symbols in M:s language and that, for every 0 < n < ℵ0 and R ⊆ Mn which is definable in M without parameters, there exists a relation symbol, in the language of M, which is interpreted as R; these assumptions are not necessary for most results to be presented, but it simplifies the statement of a result which I mention in this introduction.

First we will consider ‘canonically embedded’ substructures of Meq. Here, a structure N is canonically embedded in Meq if N's universe is a subset of Meq which is definable without parameters and, for every 0 < n < ℵ0 and R ⊆ Nn which is ε-definable in Meq there is a relation symbol in the language of N which is interpreted as R; we also assume that the language of N has no other relation (or function or constant) symbols.

1. We show that if p is a real type which is internal in a set Σ of partial types in a simple theory, then there is a type p′ interbounded with p, which is finitely generated over Σ, and possesses a fundamental system of solutions relative to Σ.

2. If p is a possibly hyperimaginary Lascar strong type, almost Σ-internal, but almost orthogonal to Σω, then there is a canonical non-trivial almost hyperdefinable polygroup which multi-acts on p while fixing Σ generically In case p is Σ-internal and T is stable, this is the binding group of p over Σ.

Suppose D ⊂ M is a strongly minimal set definable in M with parameters from C. We say D is locally modular if for all X, Y ⊂ D, with X = acl(X ∪ C)∩D, Y = acl(Y ∪ C) ∩ D and X ∩ Y ≠ ∅,

We prove the following theorems.

Theorem 1. Suppose M is stable and D ⊂ M is strongly minimal. If D is not locally modular then inMeqthere is a definable pseudoplane.

(For a discussion of Meq see [M, §A].) This is the main part of Theorem 1 of [Z2] and the trichotomy theorem of [Z3].

Theorem 2. Suppose M is stable and D, D′ ⊂ M are strongly minimal and nonorthogonal. Then D is locally modular if and only if D′ is locally modular.

One of the main theorems of the paper states the following. Let R-K-M be finite extensions of a rational one variable function field R over a finite field of constants. Let S be a finite set of valuations of K. Then the ring of elements of K having no poles outside S has a Diophantine definition over its integral closure in M.

Where a is a Turing degree and ξ is an ordinal < (ℵ1)L1, the result of performing ξ jumps on a, a(ξ), is defined set-theoretically, using Jensen's fine-structure results. This operation appears to be the natural extension through (ℵ1)L1 of the ordinary jump operations. We describe this operation in more degree-theoretic terms, examine how much of it could be defined in degree-theoretic terms and compare it to the single jump operation.

We assume that the reader is familiar with the program of “reverse mathematics” and the development of countable algebra in subsystems of second order arithmetic. The subsystems we are using in this paper are RCA0, WKL0 and ACA0. (The reader who wants to learn about them should study [1].) In [1] it was shown that the statement “Every countable commutative ring has a prime ideal” is equivalent to Weak Konig's Lemma over RCA0, while the statement “Every countable commutative ring has a maximal ideal” is equivalent to Arithmetic Comprehension over RCA0. Our main result in this paper is that the statement “Every countable commutative ring has a minimal prime ideal” is equivalent to Arithmetic Comprehension over RCA0. Minimal prime ideals play an important role in the study of countable commutative rings; see [2, pp. 1–7].

There are two origins for first-order theories. One type of theory arises by generalizing the common features of a number of different structures, e.g. the theory of groups, and formulating a set of axioms to encode these common features. Here the set of axioms is well understood, frequently it is finite or at least recursive, but there usually is no clear understanding of all the logical consequences of these axioms. The second type of theory arises by considering the set, T = Th(A), of all sentences true in a fixed structure A,3 e.g. the theory of arithmetic (N, +, 0) or the theory of the field of complex numbers (alias: the theory of algebraically closed fields of characteristic zero). The second case gives little more insight as to the truth in A (i.e. membership in T) of a given sentence ∅. But it does guarantee that for a given sentence ∅, either ∅ or ¬∅ is in T, that is, that T is a complete theory. When does a theory T of the first type, i.e. with well-understood axioms, posses this completeness property? An obvious sufficient condition is that T be secretly of the second type, that it have only one model, or, in jargon, T is categorical. Unfortunately (or fortunately depending on your point of view) for any theory with an infinite model, the Löwenheim-Skolem theorem shows this to be impossible: The theory has a model in every infinite power. In the mid-50's Łoś and Vaught discovered that if a theory T with no finite models is categorical in some infinite power α (all models with cardinality α are isomorphic) then T is complete. We will be dealing below with countable complete theories and will assume, unless stated to the contrary, that each theory has no finite models.

It has been considered desirable by many set theorists to find maximality properties which state that the universe has in some sense “many sets”. The properties isolated thus far have tended to be consistent with each other (as far as we know). For example it is a widely held view that the existence of a supercompact cardinal is consistent with the axiom of determinacy holding in L(R). This consistency has been held to be evidence for the truth of these properties. It is with this in mind that the first author suggested the following:

Maximality Principle If P is a partial ordering and G ⊆ P is a V-generic ultrafilter then either

• a) there is a real number r ∈ V [G] with r ∉ V, or

• b) there is an ordinal α such that α is a cardinal in V but not in V[G].

This maximality principle applied to garden variety partial orderings has startling results for the structure of V.

For example, if for some , then P = 〈{p: p ⊆ κ, ∣p∣ < κ}, ⊆〉 neither adds a real nor collapses a cardinal. Thus from the maximality principle we can deduce that the G. C. H. fails everywhere and there are no inaccessible cardinals. (Hence this principle contradicts large cardinals.) Similarly one can show that there are no Suslin trees on any cardinal κ. These consequences help justify the title “maximality principle”.

Since the maximality principle implies that the G. C. H. fails at strong singular limit cardinals it has consistency strength at least that of “many large cardinals”. (See [M].) On the other hand it is not known to be consistent, relative to any assumptions.

In this paper we study the varieties of Stone algebras (S, ∧, ∨, *, 0, 1) and double Stone algebras (D, ∧, ∨, *, +, 0, 1). Our primary interest is to give a structural description of the algebraically and existentially closed members of both classes. Our technique is an application of the natural dualities of Davey [6] and Clark and Krauss [5]. This approach gives a description of the desired models as the algebras of all continuous structure-preserving maps from certain structured Boolean spaces into the generating algebra for the variety. In each case the resulting description can be converted in a natural way into a finite ∀∃-axiomatization for these models. For Stone algebras these axioms appeared earlier in Schmid [20], [21] and in Schmitt [22].

Since both cases we consider satisfy the amalgamation property, the existentially closed members form a model companion for the variety which is also its model completion. Moreover, it is also ℵ0 categorical and its countably infinite member is the unique countable homogeneous universal model for the variety. In the case of Stone algebras, explicit constructions for this model appear in Schmitt [22] and Weispfenning [23]. We give here an explicit construction for double Stone algebras of S. Hayes.

This work was motivated by a problem of Stanley Burris. In [4] Burris and Werner superseded many previous results by showing that for any finite algebra A, the universal Horn class ISP has a model companion. Weispfenning [24], [25] discovered that this model companion is always ℵ0 categorical and has a primitive recursive ∀∃-axiomatization. In spite of these very general theorems, there are few instances in which a structural description of the (any!) existentially closed members of ISP is available. Burris and Werner [4] solve this problem in a special setting.

A homomorphism from a completely metrizable topological group into a free product of groups whose image is not contained in a factor of the free product is shown to be continuous with respect to the discrete topology on the range. In particular, any completely metrizable group topology on a free product is discrete.

A calculus, or its formalization, is m-valued when its truth-functions are allowed to take truth-values ranging from 1 through m. It is customary to divide the m truth-values that are possible into those that are “designated” and those that are “undesignated.” Furthermore, it is usually desired of a formalization that provable formulas and only provable formulas take designated truth-values exclusively. For our present purposes, we shall suppose that the truth-values 1, …, 8 are designated and the truth-values 8+1, …, m are undesignated. When calculi differ in respect to the number of their designated truth-values, we shall consider them different calculi, even if they are otherwise similar.

We study the class of weakly locally modular geometric theories introduced in [4], a common generalization of the classes of linear SU-rank 1 and linear o-minimal theories. We find new conditions equivalent to weak local modularity: “weak one-basedness”, absence of type definable “almost quasidesigns”, and “generic linearity”. Among other things, we show that weak one-basedness is closed under reducts. We also show that the lovely pair expansion of a non-trivial weakly one-based ω-categorical geometric theory interprets an infinite vector space over a finite field.

Let CH be the class of compacta (i.e., compact Hausdorff spaces), with BS the subclass of Boolean spaces. For each ordinal α and pair (K, L) of subclasses of CH, we define Lev≥α(K, L), the class of maps of level at least α from spaces in K to spaces in L, in such a way that, for finite α, Lev≥α(BS, BS) consists of the Stone duals of Boolean lattice embeddings that preserve all prenex first-order formulas of quantifier rank α. Maps of level ≥ 0 are just the continuous surjections, and the maps of level ≥ 1 are the co-existential maps introduced in [8]. Co-elementary maps are of level ≥ω a for all ordinals α: of course in the Boolean context, the co-elementary maps coincide with the maps of level ≥ ω. The results of this paper include:

(i) every map of level ≥ ωis co-elementary;

(ii) the limit maps of an co-indexed inverse system of maps of level ≥ α are also of level ≥ α; and

(iii) if K is a co-elementary class, k > ω and Lev≥k(K,K) = Lev≥k+1(K,K), then Lev≥1(K,K) = Lev≥(K,K).

A space X ∈ K is co-existentially closed inK if Lev≥0(K, X) = Lev≥1(K,X). Adapting the technique of “adding roots,” by which one builds algebraically closed extensions of fields (and, more generally, existentially closed extensions of models of universal-existential theories), we showed in [8] that every infinite member of a co-inductive co-elementary class (such as CH itself, BS, or the class CON of continua) is a continuous image of a space of the same weight that is co-existentially closed in that class. We show here that every compactum that is co-existentially closed in CON (a co-existentially closed continuum) is both indecomposable and of covering dimension one.

A notion of interpretation between arbitrary logics is introduced, and the poset $\mathsf {Log}$ of all logics ordered under interpretability is studied. It is shown that in $\mathsf {Log}$ infima of arbitrarily large sets exist, but binary suprema in general do not. On the other hand, the existence of suprema of sets of equivalential logics is established. The relations between $\mathsf {Log}$ and the lattice of interpretability types of varieties are investigated.