Let ℒ be a first-order language of cardinality κ⁺⁺ with a distinguished unary predicate symbol U. In this paper we prove, working on L, the two cardinal transfer theorem (κ⁺,κ) ⇒ (κ⁺⁺,κ⁺) for this language. This problem was posed by Chang and Keisler more than twenty years ago.

We prove quantifier elimination for the field ℚ_p((t^ℚ)) (the completion of the field of Puiseux series over ℚ_p) in Macintyre's language together with symbols for functions in a class containing both t-adically and p-adically overconvergent functions. We also show that the theory of ℚ_p((t^ℚ)) is b-minimal in this language.

We prove that in a continuous ℵ₀-stable theory every type-definable group is definable. The two main ingredients in the proof are:

1. Results concerning Morley ranks (i.e., Cantor-Bendixson ranks) from [Ben08], allowing us to prove the theorem in case the metric is invariant under the group action; and

2. Results concerning the existence of translation-invariant definable metrics on type-definable groups and the extension of partial definable metrics to total ones.

The paper is aimed at studying the topological dimension for sets definable in weakly o-minimal structures in order to prepare background for further investigation of groups, group actions and fields definable in the weakly o-minimal context. We prove that the topological dimension of a set definable in a weakly o-minimal structure is invariant under definable injective maps, strengthening an analogous result from [2] for sets and functions definable in models of weakly o-minimal theories. We pay special attention to large subsets of Cartesian products of definable sets, showing that if X,Y and S are non-empty definable sets and S is a large subset of X× Y, then for a large set of tuples 〈\overline{a}₁,…,\overline{a}_2^k〉 ∈ X^{2k}, where k=dim(Y), the union of fibers S_{\overline{a}₁}∪…∪ S_{\overline{a}2^k} is large in Y. Finally, given a weakly o-minimal structure ℳ, we find various conditions equivalent to the fact that the topological dimension in ℳ enjoys the addition property.

A set A of vertices of a graph G is called d-scattered in G if no two d-neighborhoods of (distinct) vertices of A intersect. In other words, A is d-scattered if no two distinct vertices of A have distance at most 2d. This notion was isolated in the context of finite model theory by Ajtai and Gurevich and recently it played a prominent role in the study of homomorphism preservation theorems for special classes of structures (such as minor closed classes). This in turn led to the notions of wide, almost wide and quasi-wide classes of graphs. It has been proved previously that minor closed classes and classes of graphs with locally forbidden minors are examples of such classes and thus (relativized) homomorphism preservation theorem holds for them. In this paper we show that (more general) classes with bounded expansion and (newly defined) classes with bounded local expansion and even (very general) nowhere dense classes are quasi wide. This not only strictly generalizes the previous results but it also provides new proofs and algorithms for some of the old results. It appears that bounded expansion and nowhere dense classes are perhaps a proper setting for investigation of wide-type classes as in several instances we obtain a structural characterization. This also puts classes of bounded expansion in the new context. Our motivation stems from finite dualities. As a corollary we obtain that any homomorphism closed first order definable property restricted to a bounded expansion class is a restricted duality.

A meadow is a commutative ring with an inverse operator satisfying 0^-1=0. We determine the initial algebra of the meadows of characteristic 0 and prove a normal form theorem for it. As an immediate consequence we obtain the decidability of the closed term problem for meadows and the computability of their initial object.

We show that for an uncountable κ in a suitable Cohen real model for any family { A_ν} _{ν <kappa} of sets of reals there is a σ-homomorphism h from the σ-algebra generated by Borel sets and the sets A_ν into the algebra of Baire subsets of 2^κ modulo meager sets such that for all Borel B,

B is meager iff h(B)=0.

The proof is uniform, works also for random reals and the Lebesgue measure, and in this way generalizes previous results of Carlson and Solovay for the Lebesgue measure and of Kamburelis and Zakrzewski for the Baire property.

Building on Hrushovski's work in [5], we study definable groupoids in stable theories and their relationship with 3-uniqueness and finite internal covers. We introduce the notion of retractability of a definable groupoid (which is slightly stronger than Hrushovski's notion of eliminability), give some criteria for when groupoids are retractable, and show how retractability relates to both 3-uniqueness and the splitness of finite internal covers. One application we give is a new direct method of constructing non-eliminable groupoids from witnesses to the failure of 3-uniqueness. Another application is a proof that any finite internal cover of a stable theory with a centerless liaison groupoid is almost split.

This paper provides a Bishop-style constructive analysis of the contrapositive of the statement that a continuous homomorphism of R onto a compact abelian group is periodic. It is shown that, subject to a weak locatedness hypothesis, if G is a complete (metric) abelian group that is the range of a continuous isomorphism from R, then G is noncompact. A special case occurs when G satisfies a certain local path-connectedness condition at 0. A number of results about one-one and injective mappings are proved en route to the main theorem. A Brouwerian example shows that some of our results are the best possible in a constructive framework.

We show, relative to the base theory RCA₀: A nontrivial tree satisfies Ramsey's Theorem only if it is biembeddable with the complete binary tree. There is a class of partial orderings for which Ramsey's Theorem for pairs is equivalent to ACA₀. Ramsey's Theorem for singletons for the complete binary tree is stronger than BΣ⁰₂, hence stronger than Ramsey's Theorem for singletons for ω. These results lead to extensions of results, or answers to questions, of Chubb, Hirst, and McNicholl [3].

We provide a first order axiomatization of the expansion of the complex field by the exponential function restricted to the subring of integers modulo the first order theory of (Z, +, ·).

We say a countable model 𝒜 has a 0-basis if the types realized in 𝒜 are uniformly computable. We say 𝒜 has a (d-)decidable copy if there exists a model ℬ≅𝒜 such that the elementary diagram of ℬ is (d-)computable. Goncharov, Millar, and Peretyat'kin independently showed there exists a homogeneous model 𝒜 with a 0-basis but no decidable copy. We extend this result here. Let d≤0' be any low₂ degree. We show that there exists a homogeneous model 𝒜 with a 0-basis but no d-decidable copy. A degree d is 0-basis homogeneous bounding if any homogenous 𝒜 with a 0-basis has a d-decidable copy. In previous work, we showed that the nonlow₂ Δ₂⁰ degrees are 0-basis homogeneous bounding. The result of this paper shows that this is an exact characterization of the 0-basis homogeneous bounding Δ₂⁰ degrees.

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]ⁱ which consists of functions 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).

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 ℵ₀ many non-empty finite discrete subsets of I is compact, is not provable in ZF⁰.

We consider valued fields with a distinguished isometry or contractive derivation as valued modules over the Ore ring of difference operators. Under certain assumptions on the residue field, we prove quantifier elimination first in the pure module language, then in that language augmented with a chain of additive subgroups, and finally in a two-sorted language with a valuation map. We apply quantifier elimination to prove that these structures do not have the independence property.

Let λ denote a singular cardinal. Zeman, improving a previous result of Shelah, proved that □^*_λ together with 2^λ=λ⁺ implies ♢_S for every S⊆λ⁺ that reflects stationarily often. In this paper, for a set S⊆λ⁺, a normal subideal of the weak approachability ideal is introduced, and denoted by I[S;λ]. We say that the ideal is fat if it contains a stationary set. It is proved:

1. if I[S;λ] is fat, then NS_λ⁺↾ S is non-saturated;

2. if I[S;λ] is fat and 2^λ=λ⁺, then ♢_S holds;

3. □^*_λ implies that I[S;λ] is fat for every S⊆λ⁺ that reflects stationarily often;

4. it is relatively consistent with the existence of a supercompact cardinal that □^*_λ fails, while I[S;λ] is fat for every stationary S⊆λ⁺ that reflects stationarily often.

The stronger principle ♢^*_λ⁺ is studied as well.

We determine exact consistency strengths for various failures of the Singular Cardinals Hypothesis (SCH) in the setting of the Zermelo-Fraenkel axiom system ZF without the Axiom of Choice (AC). By the new notion of parallel Prikry forcing that we introduce, we obtain surjective failures of SCH using only one measurable cardinal, including a surjective failure of Shelah's pcf theorem about the size of the power set of ℵ_ω. Using symmetric collapses to ℵ_ω, ℵ_ω₁, or ℵ_ω₂, we show that injective failures at ℵ_ω, ℵ_ω₁, or ℵ_ω₂ can have relatively mild consistency strengths in terms of Mitchell orders of measurable cardinals. Injective failures of both the aforementioned theorem of Shelah and Silver's theorem that GCH cannot first fail at a singular strong limit cardinal of uncountable cofinality are also obtained. Lower bounds are shown by core model techniques and methods due to Gitik and Mitchell.

It is proved that any countable index, universally measurable subgroup of a Polish group is open. By consequence, any universally measurable homomorphism from a Polish group into the infinite symmetric group S_∞ is continuous. It is also shown that a universally measurable homomorphism from a Polish group into a second countable, locally compact group is necessarily continuous.

Vector spaces over fields are pseudofinite, and this remains true for vector spaces over division rings that are finite-dimensional over their center. We also construct a division ring such that the nontrivial vector spaces over it are not pseudofinite, using Richard Thompson's group F. The idea behind the construction comes from a first-order axiomatization of the class of division rings all whose nontrivial vector spaces are pseudofinite.

Let E be a coanalytic equivalence relation on a Polish space X and (A_n)_{n ∈ ω} a sequence of analytic subsets of X. We prove that if lim sup_{n ∈ K} A_n meets uncountably many E-equivalence classes for every K ∈ [ω]^ω, then there exists a K ∈ [ω]^ω such that ⋂_{n ∈ K} A_n contains a perfect set of pairwise E-inequivalent elements.

For a finite von Neumann algebra factor M, the projections form a modular ortholattice L(M). We show that the equational theory of L(M) coincides with that of some resp. all L(ℂ^{n × n}) and is decidable. In contrast, the uniform word problem for the variety generated by all L(ℂ^{n × n}) is shown to be undecidable.

We develop several aspects of local and global stability in continuous first order logic. In particular, we study type-definable groups and genericity.