We study first-order logic with two variables FO² and establish a small substructure property. Similar to the small model property for FO² we obtain an exponential size bound on embedded substructures, relative to a fixed surrounding structure that may be infinite. We apply this technique to analyse the satisfiability problem for FO² under constraints that require several binary relations to be interpreted as equivalence relations. With a single equivalence relation, FO² has the finite model property and is complete for non-deterministic exponential time, just as for plain FO². With two equivalence relations, FO² does not have the finite model property, but is shown to be decidable via a construction of regular models that admit finite descriptions even though they may necessarily be infinite. For three or more equivalence relations, FO² is undecidable.

We prove that the strong polarized relation (μ⁺ μ) → (μ⁺ μ)^1,1₂ is consistent with ZFC, for a singular μ which is a limit of measurable cardinals.

This paper compares the roles classical and intuitionistic logic play in restricting the free use of truth principles in arithmetic. We consider fifteen of the most commonly used axiomatic principles of truth and classify every subset of them as either consistent or inconsistent over a weak purely intuitionistic theory of truth.

Gödel's second incompleteness theorem is proved for Herbrand consistency of some arithmetical theories with bounded induction, by using a technique of logarithmic shrinking the witnesses of bounded formulas, due to Z. Adamowicz [Herbrand consistency and bounded arithmetic, Fundamenta Mathematicae, vol. 171 (2002), pp. 279—292]. In that paper, it was shown that one cannot always shrink the witness of a bounded formula logarithmically, but in the presence of Herbrand consistency, for theories IΔ₀+Ω_m with m ≥ 2, any witness for any bounded formula can be shortened logarithmically. This immediately implies the unprovability of Herbrand consistency of a theory T⊇ IΔ₀+Ω₂ in T itself.

In this paper, the above results are generalized for IΔ₀+Ω₁. Also after tailoring the definition of Herbrand consistency for IΔ₀ we prove the corresponding theorems for IΔ₀. Thus the Herbrand version of Gödel's second incompleteness theorem follows for the theories IΔ₀+Ω₁ and IΔ₀.

By operations on models we show how to relate completeness with respect to permissive-nominal models to completeness with respect to nominal models with finite support. Models with finite support are a special case of permissive-nominal models, so the construction hinges on generating from an instance of the latter, some instance of the former in which sufficiently many inequalities are preserved between elements. We do this using an infinite generalisation of nominal atoms-abstraction.

The results are of interest in their own right, but also, we factor the mathematics so as to maximise the chances that it could be used off-the-shelf for other nominal reasoning systems too. Models with infinite support can be easier to work with, so it is useful to have a semi-automatic theorem to transfer results from classes of infinitely-supported nominal models to the more restricted class of models with finite support.

In conclusion, we consider different permissive-nominal syntaxes and nominal models and discuss how they relate to the results proved here.

In this paper we study with proof-theoretic methods the function(al)s provably recursive relative to Ramsey's theorem for pairs and the cohesive principle (COH).

Our main result on COH is that the type 2 functionals provably recursive from RCA₀ + COH + Π⁰₁-CP are primitive recursive. This also provides a uniform method to extract bounds from proofs that use these principles. As a consequence we obtain a new proof of the fact that WKL₀+Π⁰₁-CP+COH is Π⁰₂-conservative over PRA.

Recent work of the first author showed that Π⁰₁-CP + COH is equivalent to a weak variant of the Bolzano—Weierstraß principle. This makes it possible to use our results to analyze not only combinatorial but also analytical proofs.

For Ramsey's theorem for pairs and two colors (RT²₂) we obtain the upper bounded that the type 2 functionals provable recursive relative to RCA₀+Σ⁰₂-IA + RT²₂ are in T₁. This is the fragment of Gödel's system T containing only type 1 recursion—roughly speaking it consists of functions of Ackermann type. With this we also obtain a uniform method for the extraction of T₁-bounds from proofs that use RT²₂. Moreover, this yields a new proof of the fact that WKL₀+Σ⁰₂-IA + RT²₂ is Π⁰₃-conservative over RCA₀+Σ⁰₂-IA.

The results are obtained in two steps: in the first step a term including Skolem functions for the above principles is extracted from a given proof. This is done using Gödel's functional interpretation. After this the term is normalized, such that only specific instances of the Skolem functions are used. In the second step this term is interpreted using Π⁰₁-comprehension. The comprehension is then eliminated in favor of induction using either elimination of monotone Skolem functions (for COH) or Howard's ordinal analysis of bar recursion (for RT²₂).

As proved recently, the satisfaction problem for all prenex formulae in the set-theoretic Bernays—Shönfinkel—Ramsey class is semi-decidable over von Neumann's cumulative hierarchy. Here that semi-decidability result is strengthened into a decidability result for the same collection of formulae.

We study definable irreducible actions of SL₂(𝕂) on an abelian group of Morley rank ≤ 3rk(𝕂) and prove they are rational representations of the group.

We show that given ω many supercompact cardinals, there is a generic extension in which the tree property holds at ℵ_ω²+1 and the SCH fails at ℵ_ω².

Dynamic Topological Logic (𝒟𝒯ℒ) is a multimodal system for reasoning about dynamical systems. It is defined semantically and, as such, most of the work done in the field has been model-theoretic. In particular, the problem of finding a complete axiomatization for the full language of 𝒟𝒯ℒ over the class of all dynamical systems has proven to be quite elusive.

Here we propose to enrich the language to include a polyadic topological modality, originally introduced by Dawar and Otto in a different context. We then provide a sound axiomatization for 𝒟𝒯ℒ over this extended language, and prove that it is complete. The polyadic modality is used in an essential way in our proof.

We show the first examples of recursively enumerable (even decidable) two-dimensional products of finitely axiomatisable modal logics that are not finitely axiomatisable. In particular, we show that any axiomatisation of some bimodal logics that are determined by classes of product frames with linearly ordered first components must be infinite in two senses: It should contain infinitely many propositional variables, and formulas of arbitrarily large modal nesting-depth.

We show that every two-dimensional class of topological similarity, and hence every diagonal conjugacy class of pairs, is meager in the group of order preserving bijections of the rationals and in the group of automorphisms of the ordered rational Urysohn space.

Let κ be an infinite cardinal. A subset of (^κκ)ⁿ is a Σ¹₁-subset if it is the projection p[T] of all cofinal branches through a subtree T of ( ^{< κ}κ)ⁿ⁺¹ of height κ. We define Σ¹_k-, Π¹_k- and Δ¹_k-subsets of (^κκ)ⁿ as usual.

Given an uncountable regular cardinal κ with κ=κ^{< κ} and an arbitrary subset A of ^κκ, we show that there is a < κ-closed forcing ℛ that satisfies the κ⁺-chain condition and forces A to be a Δ¹₁-subset of ^κκ in every ℛ-generic extension of V. We give some applications of this result and the methods used in its proof.

i) Given any set x, we produce a partial order with the above properties that forces x to be an element of L(𝒫(κ)).

ii) We show that there is a partial order with the above properties forcing the existence of a well-ordering of ^κκ whose graph is a Δ¹₂-subset of ^κκ×^κκ.

iii) We provide a short proof of a result due to Mekler and Väänänen by using the above forcing to add a tree T of cardinality and height κ such that T has no cofinal branches and every tree from the ground model of cardinality and height κ without a cofinal branch quasi-order embeds into T.

iv) We will show that generic absoluteness for Σ¹₃(^κκ)-formulae (i.e., formulae with parameters which define Σ¹₃-subsets of ^κκ) under < κ-closed forcings that satisfy the κ⁺-chain condition is inconsistent.

In another direction, we use methods from the proofs of the above results to show that Σ¹₁- and Δ¹₁-subsets have some useful structural properties in certain ZFC-models.

We prove that each ω-categorical, generically stable group is solvable-by-finite.