For any ordinal Λ, we can define a polymodal logic GLPΛ, with a modality [ξ] for each ξ < Λ. These represent provability predicates of increasing strength. Although GLPΛ has no Kripke models, Ignatiev showed that indeed one can construct a Kripke model of the variable-free fragment with natural number modalities, denoted . Later, Icard defined a topological model for which is very closely related to Ignatiev's.

In this paper we show how to extend these constructions for arbitrary Λ. More generally, for each Θ, Λ we build a Kripke model and a topological model , and show that is sound for both of these structures, as well as complete, provided Θ is large enough.

We introduce the notions of copyable and diagonalizable classes of structures. We then show how these notions are connected to two other notions that had already been studied for some particular classes of structures, namely the listability property and the low property.

The main result of this paper is the characterizations of the classes of structures with the low property, that is, the classes whose low members all have computable copies. We characterize these classes as the ones whose structural jumps are listable.

For integers l ≥ 1, d ≥ 0 we study (undirected) graphs with vertices 1, …, n such that the vertices can be partitioned into l parts such that every vertex has at most d neighbours in its own part. The set of all such graphs is denoted Pn (l, d). We prove a labelled first-order limit law, i.e., for every first-order sentence φ, the proportion of graphs in Pn (l, d) that satisfy φ converges as n → ∞. By combining this result with a result of Hundack, Prömel and Steger [12] we also prove that if 1 ≤ s1 ≤ … ≤ sl are integers, then Forb() has a labelled first-order limit law, where Forb() denotes the set of all graphs with vertices 1, …, n, for some n, in which there is no subgraph isomorphic to the complete (l + 1 )-partite graph with parts of sizes 1, s1, …, sl. In the course of doing this we also prove that there exists a first-order formula ξ, depending only on l and d, such that the proportion of ∈ Pn (l, d) with the following property approaches 1 as n → ∞: there is a unique partition of {1, …, n} into l parts such that every vertex has at most d neighbours in its own part, and this partition, viewed as an equivalence relation, is defined by ξ.

We introduce a natural generalization of Borel's Conjecture. For each infinite cardinal number κ, let BCκ denote this generalization. Then BCℕ0 is equivalent to the classical Borel conjecture. Assuming the classical Borel conjecture, ¬BCℕ1 is equivalent to the existence of a Kurepa tree of height ℕ1. Using the connection of BCκ with a generalization of Kurepa's Hypothesis, we obtain the following consistency results:

(1) If it is consistent that there is a 1-inaccessible cardinal then it is consistent that BCℕ1.

(2) If it is consistent that BCℕ1, then it is consistent that there is an inaccessible cardinal.

(3) If it is consistent that there is a 1-inaccessible cardinal with ω inaccessible cardinals above it, then ¬BCℕω + (∀n < ω)BCℕn is consistent.

(4) If it is consistent that there is a 2-huge cardinal, then it is consistent that BCℕω

(5) If it is consistent that there is a 3-huge cardinal, then it is consistent that BCκ for a proper class of cardinals κ of countable cofinality.

This paper shows that the interpolation theorem fails in the intuitionistic logic of constant domains. This result refutes two previously published claims that the interpolation property holds.

An infinite binary sequence A is absolutely undecidable if it is impossible to compute A on a set of positions of positive upper density. Absolute undecidability is a weakening of bi-immunity. Downey, Jockusch and Schupp [2] asked whether, unlike the case for bi-immunity, there is an absolutely undecidable set in every non-zero Turing degree. We provide a positive answer to this question by applying techniques from coding theory. We show how to use Walsh–Hadamard codes to build a truth-table functional which maps any sequence A to a sequence B, such that given any restriction of B to a set of positive upper density, one can recover A. This implies that if A is non-computable, then B is absolutely undecidable. Using a forcing construction, we show that this result cannot be strengthened in any significant fashion.

We introduce the notions of a mutually algebraic structures and theories and prove many equivalents. A theory T is mutually algebraic if and only if it is weakly minimal and trivial if and only if no model M of T has an expansion (M, A) by a unary predicate with the finite cover property. We show that every structure has a maximal mutually algebraic reduct. and give a strong structure theorem for the class of elementary extensions of a fixed mutually algebraic structure.

We give a sharper version of a theorem of Rosický, Trnková and Adámek [13], and a new proof of a theorem of Rosický [12], both about colimits in categories of structures. Unlike the original proofs, which use category-theoretic methods, we use set-theoretic arguments involving elementary embeddings given by large cardinals such as α-strongly compact and C(n)-extendible cardinals.

We compare the strength of polychromatic and monochromatic Ramsey theory in several set-theoretic domains. We show that the rainbow Ramsey theorem does not follow from ZF, nor does the rainbow Ramsey theorem imply Ramsey's theorem over ZF. Extending the classical result of Erdős and Rado we show that the axiom of choice precludes the natural infinite exponent partition relations for polychromatic Ramsey theory. We introduce rainbow Ramsey ultrafilters, a polychromatic analogue of the usual Ramsey ultrafilters. We investigate the relationship of rainbow Ramsey ultrafilters with various special classes of ultrafilters, showing for example that every rainbow Ramsey ultrafilter is nowhere dense but rainbow Ramsey ultrafilters need not be rapid. This entails comparison of the polychromatic and monochromatic Ramsey theorems as combinatorial principles on ω.

We investigate the question “To what extent can random reals be used as a tool to establish number theoretic facts?” Let 2-RAN be the principle that for every real X there is a real R which is 2-random relative to X. In Section 2, we observe that the arguments of Csima and Mileti [3] can be implemented in the base theory RCA0 and so RCA0 + 2-RAN implies the Rainbow Ramsey Theorem. In Section 3, we show that the Rainbow Ramsey Theorem is not conservative over RCA0 for arithmetic sentences. Thus, from the Csima–Mileti fact that the existence of random reals has infinitary-combinatorial consequences we can conclude that 2-RAN has non-trivial arithmetic consequences. In Section 4, we show that 2-RAN is conservative over RCA0 + BΣ2 for -sentences. Thus, the set of first-order consequences of 2-RAN is strictly stronger than P− + IΣ1 and no stronger than P− + BΣ2.

We introduce martingales defined by probabilistic strategies, in which randomness is used to decide whether to bet. We show that different criteria for the success of computable probabilistic strategies can be used to characterize ML-randomness, computable randomness, and partial computable randomness. Our characterization of ML-randomness partially addresses a critique of Schnorr by formulating ML randomness in terms of a computable process rather than a computably enumerable function.

We examine a definition of the mutual information of two reals proposed by Levin in [5]. The mutual information is

where K(·) is the prefix-free Kolmogorov complexity. A real A is said to have finite self-information if I (A : A) is finite. We give a construction for a perfect Π10 class of reals with this property, which settles some open questions posed by Hirschfeldt and Weber. The construction produces a perfect set of reals with K(σ) ≤+KA (σ) + f (σ) for any given Δ20f with a particularly nice approximation and for a specific choice of f it can also be used to produce a perfect Π10 set of reals that are low for effective Hausdorff dimension and effective packing dimension. The construction can be further adapted to produce a single perfect set of reals that satisfy K(σ) ≤+KA (σ) + f (σ) for all f in a ‘nice’ class of Δ20 functions which includes all Δ20 orders.

The notion J is independent in (M, M0, N) was established by Shelah, for an AEC (abstract elementary class) which is stable in some cardinal λ and has a non-forking relation, satisfying the good λ-frame axioms and some additional hypotheses. Shelah uses independence to define dimension.

Here, we show the connection between the continuity property and dimension: if a non-forking satisfies natural conditions and the continuity property, then the dimension is well-behaved.

As a corollary, we weaken the stability hypothesis and two additional hypotheses, that appear in Shelah's theorem.

Every definably complete expansion of an ordered field satisfies an analogue of the Baire Category Theorem.

We answer some questions from [4] by giving suitable examples of small Polish structures. First, we present a class of small Polish group structures without generic elements. Next, we construct a first example of a small non-zero-dimensional Polish G-group.

S-measures are Loeb measures restricted to the sigma algebra generated by standard sets. This paper gives new extensions of the S-measure machinery, with applications to standard measure theory.

We prove that every diagonally noncomputable function computes a set A which is bi-immune, meaning that neither A nor its complement has an infinite computably enumerable subset.

We define a quasi-order on Borel functions from a zero-dimensional Polish space into another that both refines the order induced by the Baire hierarchy of functions and generalises the embeddability order on Borel sets. We study the properties of this quasi-order on continuous functions, and we prove that the closed subsets of a zero-dimensional Polish space are well-quasi-ordered by bi-continuous embeddability.

The implicational fragment of the logic of relevant implication, R→ is known to be decidable. We show that the implicational fragment of the logic of ticket entailment, T→ is decidable. Our proof is based on the consecution calculus that we introduced specifically to solve this 50-year old open problem. We reduce the decidability problem of T→ to the decidability problem of R→. The decidability of T→ is equivalent to the decidability of the inhabitation problem of implicational types by combinators over the base {B, B′, I, W}.

We give a full description of the structure under inclusion of all finite level Borel classes of functions, and provide an elementary proof of the well-known fact that not every Borel function can be written as a countable union of Σα0-measurable functions (for every fixed 1 ≤ α < ω1). Moreover, we present some results concerning those Borel functions which are ω-decomposable into continuous functions (also called countably continuous functions in the literature): such results should be viewed as a contribution towards the goal of generalizing a remarkable theorem of Jayne and Rogers to all finite levels, and in fact they allow us to prove some restricted forms of such generalizations. We also analyze finite level Borel functions in terms of composition of simpler functions, and we finally present an application to Banach space theory.