Using the description of the Ziegler spectrum we characterise modules with various stability-theoretic properties (ω-stability, superstability, categoricity) over certain classes of finite-dimensional algebras. We also show that, for modules over the algebras we consider, having few types is equivalent to being ω-stable.

In §3 we construct a universal, ℵ0-categorical recursively presented partial order with greatest lower bound operator. This gives us the unique structure which embeds every countable lower semilattice. In §§5 and 6 we investigate the recursive and recursively enumerable substructures of this structure, in particular finding a suitable definition for the simple-maximal hierarchy and giving an example of an infinite recursively enumerable substructure which does not contain any infinite recursive substructure.

The idea of looking at the lattice of recursively enumerable substructures of some recursive algebraic structure was introduced by Metakides and Nerode in [5], and since then many different kinds of algebraic structures have been studied in this way, including vector spaces, Boolean algebras, groups, algebraically closed fields, and equivalence relations. Since different algebraic structures have different recursion theoretic properties, one natural question is whether an algebraic structure with relatively little structure (such as a partial order or an equivalence relation) exhibits behavior more like classical recursion theory than one with more structure (such as vector spaces or algebraically closed fields).

In [6] and [7], Metakides and Remmel studied recursion theory on orderings, and, as they point out in [6], orderings differ from most other algebraic structures in that the algebraic closure operation on orderings is trivial; but this does not present a problem for them, given the questions they explore. Moreover, they take an approach of proving general theorems which can then be applied to specific orderings. Our tack is different, although also well-established (see, for example, [3]), in which a “largest” structure is defined (in §3) which corresponds to the natural numbers in classical recursion theory. In order to distinguish substructures from subsets, a function symbol is added, namely greatest lower bound. The greatest lower bound function is fundamental to the study of orderings and occurs naturally in many of them, and thus is an appropriate addition to the theory of orderings. In §4 we redefine the concepts of simple and maximal in a manner appropriate to this structure, and prove several existence theorems.

We let R be an o-minimal expansion of a field, V a convex subring, and (R0,V0) an elementary substructure of (R,V). Our main result is that (R,V) considered as a structure in a language containing constants for all elements of R0 is model complete relative to quantifier elimination in R, provided that kR (the residue field with structure induced from R) is o-minimal. Along the way we show that o-minimality of kR implies that the sets definable in kR are the same as the sets definable in k with structure induced from (R,V). We also give a criterion for a superstructure of (R,V) being an elementary extension of (R,V).

We classify, in a nontrivial amenable collection of functors, all 2-chains up to the relation of having the same 1-shell boundary. In particular, we prove that in a rosy theory, every 1-shell of a Lascar strong type is the boundary of some 2-chain, hence making the 1st homology group trivial. We also show that, unlike in simple theories, in rosy theories there is no upper bound on the minimal lengths of 2-chains whose boundary is a 1-shell.

The idea of this paper is to approach linear orderings as generalized ordinals and to study how they are made from their initial segments. First we look at how the equality of two linear orderings can be expressed in terms of equality of their initial segments. Then we shall use similar methods to define functions by recursion with respect to the initial segment relation. Our method is based on the use of a game where smaller and smaller initial segments of linear orderings are considered. The length of the game is assumed to exceed that of the descending sequences of elements of the linear orderings considered. By use of such game-theoretical methods we can for example extend the recursive definitions of the operations of sum, product and exponentiation of ordinals in a unique and natural way for arbitrary linear orderings. Extensions coming from direct limits do not satisfy our game-theoretic requirements in general. We also show how our recursive definitions allow very simple constructions for fixed points of functions, giving rise to certain interesting linear orderings.

The usual semantics for the modal systems T, S4, and S5 assumes that the set of possible worlds contains at least one member. Recently versions of these modal systems have been developed in which this assumption is dropped. The systems discussed here are obtained by slightly weakening the liberated versions of T and S4. The semantics does not assume the existence of possible worlds, and the accessibility relation between worlds is only required to be quasi-reflexive instead of reflexive. Completeness and independence results are established.

Let be a complete Boolean algebra and G a finite simple group in the Scott-Solovay -valued model V () of set theory. If we observe G outside V (), then we get a new group which is denoted by Ĝ. In general, Ĝ is not finite nor simple. Nevertheless Ĝ satisfies every property satisfied by a finite simple group with some translation. In this way, we can get a class of groups for which we can use a well-developed theory of the finite simple groups. We call Ĝ Boolean simple if G is simple in some V (). In the same way we define Boolean simple rings. The main purpose of this paper is a study of structures of Boolean simple groups and Boolean simple rings. As for Boolean simple rings, K. Eda previously constructed Boolean completion of rings with a certain condition. His construction is useful for our purpose.

The present work is a part of a series of systematic applications of Boolean valued method. The reader who is interested in this subject should consult with papers by Eda, Nishimura, Ozawa, and the author in the list of references.

In this essay we discuss Tarski's work on what he called the methodology of the deductive sciences, or more briefly, borrowing the terminology of Hilbert, metamathematics, The clearest statement of Tarski's views on this subject can be found in his textbook Introduction to logic [41m].1 Here he describes the tasks of metamathematics as “the detailed analysis and critical evaluation of the fundamental principles that are applied in the construction of logic and mathematics”. He goes on to describe what these fundamental principles are: All the expressions of the discipline under consideration must be defined in terms of a small group of primitive expressions that seem immediately understandable. Furthermore, only those statements of the discipline are accepted as valid that can be deduced by precisely defined and universally accepted means from a small set of axioms whose validity seems evident. The method of constructing a discipline in strict accordance with these principles is known as the deductive method, and the disciplines constructed in this manner are called deductive systems. Since contemporary mathematical logic is one of those disciplines that are subject to these principles, it itself is a deductive science. Tarski then goes on to say:

This identification of mathematics with the deductive sciences is in our view one of the distinctive aspects of Tarski's work. Another characteristic feature is his broad view of what constitutes the domain of metamathematical investigations. A clue to this aspect of his work can also be found in Chapter 6 of Introduction to logic . After a discussion of the notions of completeness and consistency, he remarks that the investigations concerning these topics were among the most important factors contributing to a considerable extension of the domain of methodological studies, and caused even a fundamental change in the whole character of the methodology of deductive sciences.

In this paper we present a negative solution of counting problems for some classes slightly different from bounded arithmetic (Δ 0sets). To get the results we study properties of chains of finite automata.

It is proved that all the isomorphisms in the cartesian category freely generated by a set of objects (i.e., a graph without arrows) can be written in terms of arrows from the symmetric monoidal category freely generated by the same set of objects. This proof yields an algorithm for deciding whether an arrow in this free cartesian category is an isomorphism.

We investigate in an algebraic setting the question of which logical languages can express the properties integral, permutational, and rigid for algebras of relations.

Via the formulas-as-types embedding certain extensions of Heyting Arithmetic can be represented in intuitionistic type theories. In this paper we discuss the embedding of ω-sorted Heyting Arithmetic HAω into a type theory WL, that can be described as Troelstra's system with so-called weak Σ-elimination rules. By syntactical means it is proved that a formula is derivable in HAω if and only if its corresponding type in WL is inhabited. Analogous results are proved for Diller's so-called restricted system and for a type theory based on predicate logic instead of arithmetic.

Fred Richman conjectured that the following principle is not constructive:

(*) If A is a decidable subset of the set N of natural numbers and if, for every decidable subset B of N, either A ⊆ B or A ⊆ N − B, then, for some n ∈ N, A ⊆ {n}.

Richman's conjecture was motivated by the problem of uniqueness of divisible hulls of abelian groups in constructive algebra. Richman showed that a countable discrete abelian p-group G has a unique (up to isomorphism over G) divisible hull if the subgroup pG is decidable. He also showed that the converse implies.

We confirm the nonconstructive nature of by showing (in §1) that it is not provable in intuitionistic set theory, IZF. Thus, in the models we construct, there are countable discrete abelian p-groups G whose divisible hulls are unique but whose subgroups pG are not decidable.

Our models do not satisfy further conditions imposed by Richman, namely Church's Thesis and Markov's Principle, so the full conjecture remains an open problem. We do, however, show (in §2) how to embellish our first model so that the fan theorem (i.e., compactness of 2N) fails. (Church's Thesis implies the stronger statement that the negation of the fan theorem holds.)

Our models will be constructed by the method of sheaf semantics [1], [3]. That is, we shall construct Grothendieck topoi in whose internal logic fails.

If L is a first order language and n is a natural number, then Ln is the set of formulas which only make use of the variables x 1,…,xn . While every finite structure is determined up to isomorphism by its theory in L, the same is no longer true in Ln . This simple observation is the source of a number of intriguing questions. For example, Poizat [2] has asked whether a complete theory in Ln which has at least two nonisomorphic finite models must necessarily also have an infinite one. The purpose of this paper is to present some counterexamples to this conjecture.

Theorem. For each n ≤ 3 there are complete theories in L 2n−2 and L 2n−1 having exactly n + 1 models.

In our notation and definitions, we follow Poizat [2]. To test structures for elementary equivalence in Ln , we shall use the modified Ehrenfeucht-Fraïssé games of Immerman [1]. For convenience, we repeat his definition here.

Suppose that L is a purely relational language, each of the relations having arity at most n. Let and ℬ be two structures for L. Define the Ln game on and ℬ as follows. There are two players, I and II, and there are n pairs of counters a 1, b 1, …, an , bn . On each move, player I picks up any of the counters and places it on an element of the appropriate structure.

Let ω be the set of natural numbers, let be the lattice of recursively enumerable subsets of ω, and let A be the lattice of subsets of ω which are recursively enumerable in A. If U, V ⊆ ω, put U =* V if the symmetric difference of U and V is finite.

A natural and interesting question is then to discover what the relation is between the Turing degree of A and the isomorphism class of A. The first result of this form was by Lachlan, who proved [6] that there is a set A ⊆ ω such that A ≇ . He did this by finding a set A ⊆ ω and a set C ϵ A such that the structure ({W ϵ A∣W ⊇ C},∪,∩)/=* is a Boolean algebra and is not isomorphic to the structure ({W ϵ ∣W ⊇ D},∪,∩)/=* for any D ϵ . There is a nonrecursive ordinal which is recursive in the set A which he constructs, so his set A is not (see, for example, Shoenfield [11] for a definition of what it means for a set A ⊆ ω to be ). Feiner then improved this result substantially by proving [1] that for any B ⊆ ω, B′ ≇ B, where B′ is the Turing jump of B. To do this, he showed that for each X ⊆= ω there is a Boolean algebra which is but not and then applied a theorem of Lachlan [6] (definitions of and Boolean algebras will be given in §2). Feiner's result is of particular interest for the case B = ⊘, for it shows that the set A of Lachlan can actually be chosen to be arithmetical (in fact, ⊘′), answering a question that Lachlan posed in his paper. Little else has been known.

We show that there exist uncountably many (tall and nontall) pairwise nonisomorphic density-like ideals on $\omega $ which are not generalized density ideals. In addition, they are nonpathological. This answers a question posed by Borodulin-Nadzieja et al. in [this Journal, vol. 80 (2015), pp. 1268–1289]. Lastly, we provide sufficient conditions for a density-like ideal to be necessarily a generalized density ideal.

The smooth development of large parts of mathematics hinges on the idea that some sets are ‘small’ or ‘negligible’ and can therefore be ignored for a given purpose. The perhaps most famous smallness notion, namely ‘measure zero’, originated with Lebesgue, while a second smallness notion, namely ‘meagre’ or ‘first category’, originated with Baire around the same time. The associated Baire category theorem is a central result governing the properties of meagre (and related) sets, while the same holds for Tao’s pigeonhole principle for measure spaces and measure zero sets. In this paper, we study these theorems in Kohlenbach’s higher-order Reverse Mathematics, identifying a considerable number of equivalent and robust theorems. The latter involve most basic properties of semi-continuous and pointwise discontinuous functions, Blumberg’s theorem, Riemann integration, and Volterra’s early work circa 1881. All the aforementioned theorems fall (far) outside of the Big Five of Reverse Mathematics, and we investigate natural restrictions like Baire 1 and quasi-continuity that make these theorems provable again in the Big Five (or similar). Finally, despite the fundamental differences between measure and category, the proofs of our equivalences turn out to be similar.

After reviewing various natural bi-interpretations in urelement set theory, including second-order set theories with urelements, we explore the strength of second-order reflection in these contexts. Ultimately, we prove that second-order reflection with the abundant atom axiom is bi-interpretable and hence also equiconsistent with the existence of a supercompact cardinal. The proof relies on a reflection characterization of supercompactness, namely, a cardinal $\kappa $ is supercompact if and only if every $\Pi ^1_1$ sentence true in a structure M (of any size) containing $\kappa $ in a language of size less than $\kappa $ is also true in a substructure $m\prec M$ of size less than $\kappa $ with $m\cap \kappa \in \kappa $.