According to [4, p. 1154], a complete L-theory T eliminates imaginaries just in case for every L-formula φ(x1,… , xm, y1, …, yn), every model M of T, and every ā Є Mn, there is a subset A of M's domain with the following property: if N ≽ M and f is an automorphism of N, then

if and only if

Among the several equivalent conditions discussed in [4, p. 1155], one may single out the following: if T is a complete theory in which two distinct objects are definable, T eliminates imaginaries just in case every T-definable n-ary equivalence relation may be defined by a formula

where g is a T-definable n-ary function taking k-tuples as values (for some natural number k).

Say that an L-structure M eliminates imaginaries just in case Th(M) does. If L is the language of rings with unit, [4, p. 1158] shows that any algebraically closed field eliminates imaginaries, and [2, p. 629] points out that any real-closed field eliminates imaginaries.

I solve here some problems left open in “T-convexity and Tame Extensions” [9]. Familiarity with [9] is assumed, and I will freely use its notations. In particular, T will denote a complete o-minimal theory extending RCF, the theory of real closed fields. Let (, V) ⊨ Tconvex, let = V/m(V) be the residue field, with residue class map x ↦ : V ↦ , and let υ: → Γ be the associated valuation. “Definable” will mean “definable with parameters”. The main goal of this article is to determine the structure induced by (, V) on its residue fieldand on its value group Γ. In [9] we expanded the ordered field to a model of T as follows. Take a tame elementary substructure ′ of such that R′ ⊆ V and R′ maps bijectively onto under the residue class map, and make this bijection into an isomorphism ′ ≌ of T-models. (We showed such ′ exists, and that this gives an expansion of to a T-model that is independent of the choice of ′.).

We study a generalization of the splitting number s to uncountable cardinals. We prove that 𝔰(κ) > κ+ for a regular uncountable cardinal κ implies the existence of inner models with measurables of high Mitchell order. We prove that the assumption 𝔰(ℵω) > ℵω+1 has a considerable large cardinal strength as well.

This article is concerned with functions k assigning a cardinal number to each infinite Boolean algebra (BA), and the behaviour of such functions under ultraproducts. For some common functions k we have

for others we have ≤ instead, under suitable assumptions. For the function π character we go into more detail. More specifically, ≥ holds when F is regular, for cellularity, length, irredundance, spread, and incomparability. ≤ holds for π. ≥ holds under GCH for F regular, for depth, π, πχ, χ, h-cof, tightness, hL, and hd. These results show that ≥ can consistently hold in ZFC since if V = L holds then all uniform ultrafilters are regular. For π-character we prove two more results: (1) If F is regular and ess , then

(2) It is relatively consistent to have , where A is the denumerable atomless BA.

A thorough analysis of what happens without the assumption that F is regular can be found in Rosłanowski, Shelah [8] and Magidor, Shelah [5]. Those papers also mention open problems concerning the above two possible inequalities.

This work is inspired by the correspondence of Malcev between rings and groups. Let A be a domain with unit, and S a multiplicative group of invertible elements. We define AS as the structure obtained from A by restraining the multiplication to A × S, and σ(AS) as the group of functions from A to A of the form x → xa + b, where (a, b) belongs to S × A. We show that AS and σ(As) are interpretable in each other, and then, that we can transfer some properties between classes (or theories) of “reduced” domains and corresponding groups, such as being elementary, axiomatisability (for classes), decidability, completeness, or, in some cases, existence of a model-completion (for theories).

We study the extensions of the additive group of A by the group S, acting by right multiplication, and show that sometimes σ(AS) is the unique extension of this type. We also give conditions allowing us to eliminate parameters appearing in interpretations.

We emphasize the case where the domain is a division ring K and S is its multiplicative group K×. Here, the interpretations can always be done without parameters. If the centre of K contains more than two elements, then σ(K) is the only extension of the additive group of K by its multiplicative group acting by right multiplication, and the class of all such σ(K)'s is elementary and finitely axiomatisable. We give, in particular, an axiomatisation for this class and for the class of σ(K)'s where K is an algebraically closed field of characteristic 0. From these results it follows that some classical model-companion results about theories of fields can be translated and restated as results about theories of solvable groups of class 2.

If a formal theory T is able to reason about its own syntax, then the diagonalizable algebra of T is defined as its Lindenbaum sentence algebra endowed with a unary operator □ which sends a sentence φ to the sentence □φ asserting the provability of φ in T. We prove that the first order theories of diagonalizable algebras of a wide class of theories are undecidable and establish some related results.

It is well known that if σ is a strongly compact cardinal and λ a regular cardinal ≥ σ, then for every stationary subset X of {α < λ: cof (α) = ω} there is some β < λ such that X ⋂ β is stationary in β. In fact the existence of a uniform, countably complete ultrafilter over λ is sufficient to prove the same conclusion about stationary subsets of {α < λ: cof (α) = ω}. See [13] or [10]. By analyzing the proof of this theorem as presented in [10], we realized the same conclusion will follow from the existence of a certain ideal, not necessarily prime, on . Throughout we will assume that σ is a regular uncountable cardinal and use the word “ideal” to mean fine ideal.

We show here that the first order theory of the positive integers equipped with multiplication remains decidable when one adds to the language the usual order restricted to the prime numbers. We see moreover that the complexity of the latter theory is a tower of exponentials, of height O(n).

Tense logics formulated in the bimodal propositional language are investigated with respect to Kripke-completeness (completeness) and decidability. It is proved that all minimal tense extensions of modal logics of finite width (in the sense of K. Kine) as well as all minimal tense extensions of cofinal subframe logics (in the sense of M. Zakharyaschev) are complete. The decidability of all finitely axiomatizable minimal tense extensions of cofinal subframe logics is shown. A number of variations and extensions of these results are also presented.

In 1965, during the first All-Union Symposium on Group Theory, Kargapolov presented the following two problems: (a) describe the universal theory of free nilpotent groups of class m; (b) describe the universal theory of free groups (see [18, 1.28 and 1.27]). The first of these problems is still open and it is known [25] that a positive solution of this problem for an m ≤ 2 should imply the decidability of the universal theory of the field of the rationals (this last problem is equivalent to Hilbert's tenth problem for the field of the rationals which is a difficult open problem; see [17] and [20] for discussions on this problem). Regarding the second problem, Makanin proved in 1985 that a free group has a decidable universal theory (see [15] for stronger results), however, the problem of deriving an explicit description of the universal theory of free groups is open. To try to solve this problem Remeslennikov gave different characterization of finitely generated groups with the same universal theory as a noncyclic free group (see [21] and [22] and also [11]). Recently, the author proved in [8] that a free metabelian group has a decidable universal theory, but the proof of [8] does not give an explicit description of the universal theory of free metabelian groups.

In this paper we explicate a very weak version of the principle □ discovered by Jensen who proved it holds in the constructible universe L. This principle is strong enough to include many of the known applications of □, but weak enough that it is consistent with the existence of very large cardinals. In this section we show that this principle is equivalent to a common combinatorial device, which we call a Jensen matrix. In the second section we show that our principle is consistent with a supercompact cardinal. In the third section of this paper we show that this principle is exactly equivalent to the statement that every torsion free Abelian group has a filtration into σ-balanced subgroups. In the fourth section of this paper we show that this principle fails if you assume the Chang's Conjecture:

In the fifth section of the paper we review the proofs that the various weak squares we consider are strictly decreasing in strength. Section 6 was added in an ad hoc manner after the rest of the paper was written, because the subject matter of Theorem 6.1 fit well with the rest of the paper. It deals with a principle dubbed “Not So Very Weak Square”, which appears close to Very Weak Square but turns out not to be equivalent.

The complexity of priority proofs in recursion theory has been growing since the first priority proofs in [1] and [11]. Refined versions of classic priority proofs can be found in [18]. To this date, this part of recursion theory is at about the same stage of development as real analysis was in the early days, when the notions of topology, continuity, compactness, vector space, inner product space, etc., were not invented. There were no general theorems involving these concepts to prove results about the real numbers and the proofs were repetitive and lengthy.

The priority method contains an unprecedent wealth of combinatorics which is used to answer questions in recursion theory and is bound to have applications in many other fields as well. Unfortunately, very little progress has been made in finding theorems to formulate the combinatorial part of the priority method so as to answer questions without having to reprove the combinatorics in each case.

Lempp and Lerman in [10] provide an overview of the subject. The entire edifice of definitions and theorems which formulate the combinatorics of the priority method has acquired the name Priority Theory. From a different vein, Groszek and Slaman in [2] have initiated a program to classify priority constructions in terms of how much induction or collection is needed to carry them out. This program studies the complexity of priority proofs and can be called Complexity Theory of Priority Proofs or simply Complexity.

We consider the problem of finding and classifying representations in algebraic logic. This is approached by letting two players build a representation using a game. Homogeneous and universal representations are characterized according to the outcome of certain games. The Lyndon conditions defining representable relation algebras (for the finite case) and a similar schema for cylindric algebras are derived. Finte relation algebras with homogeneous representations are characterized by first order formulas. Equivalence games are defined, and are used to establish whether an algebra is ω-categorical. We have a simple proof that the perfect extension of a representable relation algebra is completely representable.

An important open problem from algebraic logic is addressed by devising another two-player game, and using it to derive equational axiomatisations for the classes of all representable relation algebras and representable cylindric algebras.

Other instances of this approach are looked at, and include the step by step method.

We show that all posets of uniform density ℵ1 may have to add a Cohen real and develop some forcing machinery for obtaining this sort of result.

The p-adic semianalytic sets are defined, locally, as boolean combinations of sets of the form over the p-adic fields ℚp, where f is an analytic function. A well-know example due to Osgood showed the projection of a semianalytic set need not be a semianalytic set. We call those sets that are, locally, the projections of p-adic semianalytic sets p-adic subanalytic sets. The theory of p-adic subanalytic sets was presented by Denef and Van den Dries in [5]. The basic tools are the quantifier elimination techniques together with the ultrametric Weierstrass Preparation Theorem. Simultaneously with their developments of the p-adic subanalytic sets, they established some basic properties of p-adic semianalytic sets.

In this paper, we prove that the closure of any p-adic semianalytic set is also a semianalytic set. The analogous property for real semianalytic sets was proved in [12] and that for rigid semianalytic sets, informed by the referee, has been proved recently by a quite different method in [14] (cf. [9]). The keys to the proof are a separation lemma (Lemma 2) and an analytic cell decomposition theorem (Theorem 2) which is an analytic version of Denef's cell decomposition theorem (see [3, 4]; A total different form of anayltic cells appeared in [13]). The analytic cell decomposition theorem allows us to partition certain kinds of basic subsets into analytic cells that possess the closure property (see §1 for the definition).

We show that the 0-1 law does not hold for the class (∀∃∀ without = ) by finding a sentence in this class which almost surely expresses parity. We also show that every recursive real in the unit interval is the asymptotic probability of a sentence in this class. This expands a result by Lidia Tendera, who in 1994 proved that every rational number in the unit interval is the asymptotic probability of a sentence in the class ∀∃∀ with equality.