We generalize the model theory of small profinite structures developed by Newelski to the case of compact metric spaces considered together with compact groups of homeomorphisms and satisfying the existence of m-independent extensions (we call them compact e-structures). We analyze the relationships between smallness and different versions of the assumption of the existence of m-independent extensions and we obtain some topological consequences of these assumptions. Using them, we adopt Newelski's proofs of various results about small profinite structures to compact e-structures. In particular, we notice that a variant of the group configuration theorem holds in this context. A general construction of compact structures is described. Using it, a class of examples of compact e-structures which are not small is constructed. It is also noticed that in an m-stable compact e-structure every orbit is equidominant with a product of m-regular orbits. %which is new even for small profinite structures.

We show that the free pseudospace is a reduct of a 1-based theory. This answers a question of David M. Evans. The theory is superstable, but not ω-stable.

In this paper, we introduce the systems ns-ACA₀ and ns-WKL₀ of non-standard second-order arithmetic in which we can formalize non-standard arguments in ACA₀ and WKL₀, respectively. Then, we give direct transformations from non-standard proofs in ns-ACA₀ or ns-WKL₀ into proofs in ACA₀ or WKL₀.

We study the behaviour of stable types in rosy theories. The main technical result is that a non-þ-forking extension of an unstable type is unstable. We apply this to show that a rosy group with a þ-generic stable type is stable. In the context of super-rosy theories of finite rank we conclude that non-trivial stable types of U^þ-rank 1 must arise from definable stable sets.

We show that it is relatively consistent with ZFC to have PCF structures of height δ, for all ordinals δ < ω₃.

We initiate the study of reducts of relational structures up to primitive positive interdefinability: After providing the tools for such a study, we apply these tools in order to obtain a classification of the reducts of the logic of equality. It turns out that there exists a continuum of such reducts. Equivalently, expressed in the language of universal algebra, we classify those locally closed clones over a countable domain which contain all permutations of the domain.

We construct a model of ZFC satisfying the Dual Borel Conjecture in which there is a set of size ℵ₁ that does not have measure zero.

An open U⊆ ℝ is produced such that (ℝ,+,·,U) defines a Borel isomorph of (ℝ,+,·,ℕ) but does not define ℕ. It follows that (ℝ,+,·,U) defines sets in every level of the projective hierarchy but does not define all projective sets. This result is elaborated in various ways that involve geometric measure theory and working over o-minimal expansions of (ℝ,+,·). In particular, there is a Cantor set E⊆ ℝ such that (ℝ,+,·,E) defines a Borel isomorph of (ℝ,+,·,ℕ) and, for every exponentially bounded o-minimal expansion ℜ of (ℝ,+,·), every subset of ℝ definable in (ℜ,E) either has interior or is Hausdorff null.

An instance of Stratified Comprehension

∀ x₁…∀ x_n∃ y∀ x (x∈ y ↔ φ(x,x₁,…,x_n))

is called strictly impredicative iff, under minimal stratification, the type of x is 0. Using the technology of forcing, we prove that the fragment of NF based on strictly impredicative Stratified Comprehension is consistent. A crucial part in this proof, namely showing genericity of a certain symmetric filter, is due to Robert Solovay.

As a bonus, our interpretation also satisfies some instances of Stratified Comprehension which are not strictly impredicative. For example, it verifies existence of Frege natural numbers.

Apparently, this is a new subsystem of NF shown to be consistent. The consistency question for the whole theory NF remains open (since 1937).

We construct a high c.e. degree which is not the join of two minimal degrees and so refute Posner's conjecture that every high c.e. degree is the join of two minimal degrees. Additionally, the proof shows that there is a high c.e. degree a such that for any splitting of a into degrees b and c one of these degrees bounds a 1-generic degree.

We reformulate, in the context of continuous logic, an oscillation theorem proved by G. Hjorth and give a proof of the theorem in that setting which is similar to, but simpler than, Hjorth's original one. The point of view presented here clarifies the relation between Hjorth's theorem and first-order logic.

Bounded lattices (that is lattices that are both lower bounded and upper bounded) form a large class of lattices that include all distributive lattices, many nondistributive finite lattices such as the pentagon lattice N₅, and all lattices in any variety generated by a finite bounded lattice. Extending a theorem of Paris for distributive lattices, we prove that if L is an ℵ₀-algebraic bounded lattice, then every countable nonstandard model ℳ of Peano Arithmetic has a cofinal elementary extension 𝒩 such that the interstructure lattice Lt(𝒩/ℳ) is isomorphic to L.

We prove that if V=L then there is a Π¹₁ maximal orthogonal (i.e., mutually singular) set of measures on Cantor space. This provides a natural counterpoint to the well-known theorem of Preiss and Rataj [16] that no analytic set of measures can be maximal orthogonal.

For a first-order formula φ(x;y) we introduce and study the characteristic sequence 〈 P_n: n < ω 〉 of hypergraphs defined by P_n(y₁,…,y_n) := (∃ x) ⋁_{i ≤ n} φ(x;y_i). We show that combinatorial and classification theoretic properties of the characteristic sequence reflect classification theoretic properties of φ and vice versa. The main results are a characterization of NIP and of simplicity in terms of persistence of configurations in the characteristic sequence. Specifically, we show that some tree properties are detected by the presence of certain combinatorial configurations in the characteristic sequence while other properties such as instability and the independence property manifest themselves in the persistence of complicated configurations under localization.

We prove that given any first order formula φ in the language L'={+,·, <, (f_i)_{i ∈ I},(c_i)_{i ∈ I}}, where the f_i are unary function symbols and the c_i are constants, one can find an existential formula ψ such that φ and ψ are equivalent in any L'-structure 〈ℝ,+,·, <,(x^c_i)_{i ∈ I},(c_i)_{i ∈ I}〉.

This paper began as a generalization of a part of the author's PhD thesis about ACFA and ended up with a characterization of groups definable in T_A. The thesis concerns minimal formulae of the form x ∈ A ∧ σ(x) = f(x) for an algebraic curve A and a dominant rational function f: A → σ(A). These are shown to be uniform in the Zilber trichotomy, and the pairs (A,f) that fall into each of the three cases are characterized. These characterizations are definable in families. This paper covers approximately half of the thesis, namely those parts of it which can be made purely model-theoretic by moving from ACFA, the model companion of the class of algebraically closed fields with an endomorphism, to T_A, the model companion of the class of models of an arbitrary totally-transcendental theory T with an injective endomorphism, if this model-companion exists. A T_A analog of the characterization of groups definable in ACFA is obtained in the process. The full characterization of the cases of the Zilber trichotomy in the thesis is obtained from these intermediate results with heavy use of algebraic geometry.

We show that for no chain C there is a monadic-second order interpretation of the full binary tree in C.