We introduce and discuss a concept of approximation of a topological algebraic system A by finite algebraic systems from a given class 𝔎. If A is discrete, this concept agrees with the familiar notion of a local embedding of A in a class 𝔎 of algebraic systems. One characterization of this concept states that A is locally embedded in 𝔎 iff it is a subsystem of an ultraproduct of systems from 𝔎. In this paper we obtain a similar characterization of approximability of a locally compact system A by systems from 𝔎 using the language of nonstandard analysis.

In the signature of A we introduce positive bounded formulas and their approximations; these are similar to those introduced by Henson [14] for Banach space structures (see also [15,16]). We prove that a positive bounded formula φ holds in A if and only if all precise enough approximations of φ hold in all precise enough approximations of A.

We also prove that a locally compact field cannot be approximated arbitrarily closely by finite (associative) rings (even if the rings are allowed to be non-commutative). Finite approximations of the field ℝ can be considered as possible computer systems for real arithmetic. Thus, our results show that there do not exist arbitrarily accurate computer arithmetics for the reals that are associative rings.

We investigate the relation between intermediate predicate logics based on countable linear Kripke frames with constant domains and Gödel logics. We show that for any such Kripke frame there is a Gödel logic which coincides with the logic defined by this Kripke frame on constant domains and vice versa. This allows us to transfer several recent results on Gödel logics to logics based on countable linear Kripke frames with constant domains: We obtain a complete characterisation of axiomatisability of logics based on countable linear Kripke frames with constant domains. Furthermore, we obtain that the total number of logics defined by countable linear Kripke frames on constant domains is countable.

The pointwise ergodic theorem is nonconstructive. In this paper, we examine origins of this non-constructivity, and determine the logical strength of the theorem and of the auxiliary statements used to prove it. We discuss properties of integrable functions and of measure preserving transformations and give three proofs of the theorem, though mostly focusing on the one derived from the mean ergodic theorem. All the proofs can be carried out in ACA₀; moreover, the pointwise ergodic theorem is equivalent to (ACA) over the base theory RCA₀.

Let $M$ be a big o-minimal structure and $G$ a type-definable group in $M^n$. We show that $G$ is a type-definable subset of a definable manifold in $M^n$ that induces on $G$ a group topology. If $M$ is an o-minimal expansion of a real closed field, then $G$ with this group topology is even definably isomorphic to a type-definable group in some $M^k$ with the topology induced by $M^k$. Part of this result holds for the wider class of so-called invariant groups: each invariant group $G$ in $M^n$ has a unique topology making it a topological group and inducing the same topology on a large invariant subset of the group as $M^n$.

A $\Sigma^0_1$ class is the set of paths through a computable tree. Given classes $P$ and $Q$, $P$ is Medvedev reducible to $Q, P \leq_{M} Q$, if there is a computably continuous functional mapping $Q$ into $P$. We look at the lattice formed by $\Sigma^0_1$ subclasses of $2^\omega$ under this reduction. It is known that the degree of a splitting class of c.e. sets is non-branching. We further characterize non-branching degrees, providing two additional properties which guarantee non-branching: inseparable and hyperinseparable. Our main result is to show that non-branching iff inseparable if hyperinseparable if homogeneous and that all unstated implications do not hold. We also show that inseparable and not hyperinseparable degrees are downward dense.

In this paper we define intensional models for the classical theory of types, thus arriving at an intensional type logic ITL. Intensional models generalize Henkin's general models and have a natural definition. As a class they do not validate the axiom of Extensionality. We give a cut-free sequent calculus for type theory and show completeness of this calculus with respect to the class of intensional models via a model existence theorem. After this we turn our attention to applications. Firstly, it is argued that, since ITL is truly intensional, it can be used to model ascriptions of propositional attitude without predicting logical omniscience. In order to illustrate this a small fragment of English is defined and provided with an ITL semantics. Secondly, it is shown that ITL models contain certain objects that can be identified with possible worlds. Essential elements of modal logic become available within classical type theory once the axiom of Extensionality is given up.

We use a new construction of an o-minimal structure, due to Lipshitz and Robinson, to answer a question of van den Dries regarding the relationship between arbitrary o-minimal expansions of real closed fields and structures over the real numbers. We write a first order sentence which is true in the Lipshitz-Robinson structure but fails in any possible interpretation over the field of real numbers.

We give four examples of theories in which Kreisel's Conjecture is false: (1) the theory PA(-) obtained by adding a function symbol minus, ‘-’, to the language of PA, and the axiom ∀ x∀ y ∀ z (x-y=z) ≡ (x=y+z ∨ (x < y ∧ z=0)); (2) the theory 𝒵 of integers; (3) the theory PA(q) obtained by adding a function symbol q (of arity ≥ 1) to PA, assuming nothing about q; (4) the theory PA(N) containing a unary predicate N(x) meaning ‘x is a natural number’. In Section 6 we suggest a counterexample to the so called Sharpened Kreisel's Conjecture.

The forcing construction ℛ_max, invented by W. Hugh Woodin, produces a model whose collection of subsets of ω₁ is in some sense maximal. In this paper we study the Boolean algebra induced by the nonstationary ideal on ω₁ in this model. Among other things we show that the induced quotient does not have a simply definable form. We also prove several results about saturation properties of the ideal in this extension.

We examine ‘weak-distributivity’ as a rewriting rule $\underset{\leadsto{}}{WD}$ defined on multiplicative proof-structures (so, in particular, on multiplicative proof-nets: MLL). This rewriting does not preserve the type of proof-nets, but does nevertheless preserve their correctness. The specific contribution of this paper, is to give a direct proof of completeness for $\underset{\leadsto{}}{WD}$: starting from a set of simple generators (proof-nets which are a $n$-ary $\otimes$ of $\parr$-ized axioms), any mono-conclusion MLL proof-net can be reached by $\underset{\leadsto{}}{WD}$ rewriting (up to $\otimes$ and $\parr$ associativity and commutativity).

We investigate the complexity of various combinatorial theorems about linear and partial orders, from the points of view of computability theory and reverse mathematics. We focus in particular on the principles ADS (Ascending or Descending Sequence), which states that every infinite linear order has either an infinite descending sequence or an infinite ascending sequence, and CAC (Chain-AntiChain), which states that every infinite partial order has either an infinite chain or an infinite antichain. It is well-known that Ramsey's Theorem for pairs (RT₂²) splits into a stable version (SRT₂²) and a cohesive principle (COH). We show that the same is true of ADS and CAC, and that in their cases the stable versions are strictly weaker than the full ones (which is not known to be the case for RT₂² and SRT₂²). We also analyze the relationships between these principles and other systems and principles previously studied by reverse mathematics, such as WKL₀, DNR, and BΣ₂. We show, for instance, that WKL₀ is incomparable with all of the systems we study. We also prove computability-theoretic and conservation results for them. Among these results are a strengthening of the fact, proved by Cholak, Jockusch, and Slaman, that COH is Π₁¹-conservative over the base system RCA₀. We also prove that CAC does not imply DNR which, combined with a recent result of Hirschfeldt, Jockusch, Kjos-Hanssen, Lempp, and Slaman, shows that CAC does not imply SRT₂² (and so does not imply RT₂²). This answers a question of Cholak, Jockusch, and Slaman. Our proofs suggest that the essential distinction between ADS and CAC on the one hand and RT₂² on the other is that the colorings needed for our analysis are in some way transitive. We formalize this intuition as the notions of transitive and semitransitive colorings and show that the existence of homogeneous sets for such colorings is equivalent to ADS and CAC, respectively. We finish with several open questions.

We apply Hrushovski-Fraïssé's amalgamation procedure to obtain a theory of fields of prime characteristic of Morley rank 2 equipped with a definable additive subgroup of rank 1.

Let μ be singular of uncountable cofinality. If μ > 2^cf(μ), we prove that in ℛ = ([μ]^μ,⊇) as a forcing notion we have a natural complete embedding of Levy(ℵ₀,μ⁺) (so ℛ collapses μ⁺ to ℵ₀) and even Levy(ℵ₀,U_J^bdκ(μ)). The “natural” means that the forcing ({p ∈ [μ]^μ:p closed},⊇) is naturally embedded and is equivalent to the Levy algebra. Also if ℛ fails the χ-c.c. then it collapses χ to ℵ₀ (and the parallel results for the case μ > ℵ₀ is regular or of countable cofinality). Moreover we prove: for regular uncountable κ, there is a family P of 𝔟_κ partitions Ā=〈 A_α:α<κ〉 of κ such that for any A∈ [κ]^κ for some 〈 A_α:α<κ〉 ∈ P we have α<κ→ |A_α ∩ A|=κ.

The paper gives a soundness and completeness proof for the implicative fragment of intuitionistic calculus with respect to the semantics of computability logic, which understands intuitionistic implication as interactive algorithmic reduction. This concept — more precisely, the associated concept of reducibility — is a generalization of Turing reducibility from the traditional, input/output sorts of problems to computational tasks of arbitrary degrees of interactivity.

It is proved that the (2$p$)-c. e. e-degrees are not elementarily equivalent to the (2$p$+1)-c. e. e-degrees for each nonzero $p \in \omega$. It follows that $m$-c. e. e-degrees are not elementarily equivalent to the n-c. e. e-degrees if $1 < m < n$.

We give a necessary and sufficient condition in order that a type-shifting automorphism be constructed on a model of the Theory of Simple Types (TST) by forcing. Namely it is proved that, if for every $n \geq 1$ there is a model of TST in the ground model $M$ of ZFC that contains an $n$-extendible coherent pair, then there is a generic extension $M[G]$ of $M$ that contains a model of TST with a type-shifting automorphism, and hence $M[G]$ contains a model of NF. The converse holds trivially. It is also proved that there exist models of TST containing 1-extendible coherent pairs.

A Turing degree d is homogeneous bounding if every complete decidable (CD) theory has a d-decidable homogeneous model 𝒜, i.e., the elementary diagram D^e(𝒜) has degree d. It follows from results of Macintyre and Marker that every PA degree (i.e., every degree of a complete extension of Peano Arithmetic) is homogeneous bounding. We prove that in fact a degree is homogeneous bounding if and only if it is a PA degree. We do this by showing that there is a single CD theory T such that every homogeneous model of T has a PA degree.

We consider embeddings of structures which preserve spectra: if g:ℳ →𝒮 with 𝒮 computable, then ℳ should have the same Turing degree spectrum (as a structure) that g(ℳ) has (as a relation on 𝒮). We show that the computable dense linear order ℒ is universal for all countable linear orders under this notion of embedding, and we establish a similar result for the computable random graph 𝔖. Such structures are said to be spectrally universal. We use our results to answer a question of Goncharov, and also to characterize the possible spectra of structures as precisely the spectra of unary relations on 𝔖. Finally, we consider the extent to which all spectra of unary relations on the structure ℒ may be realized by such embeddings, offering partial results and building the first known example of a structure whose spectrum contains precisely those degrees c with c' ≥_T 0''.

Various theorems for the preservation of set-theoretic axioms under forcing are proved, regarding both forcing axioms and axioms true in the Lévy collapse. These show in particular that certain applications of forcing axioms require to add generic countable sequences high up in the set-theoretic hierarchy even before collapsing everything down to ℵ₁. Later we give applications, among them the consistency of MM with ℵ_ω not being Jónsson which answers a question raised in the set theory meeting at Oberwolfach in 2005.