It is known that the spectrum of a Boolean algebra cannot contain a low4 degree unless it also contains the degree 0; it remains open whether the same holds for low5 degrees. We address the question differently, by considering Boolean subalgebras of the computable atomless Boolean algebra . For such subalgebras , we show that it is possible for the spectrum of the unary relation on to contain a low5 degree without containing 0.

We investigate the extension of monadic second-order logic of order with cardinality quantifiers “there exists uncountably many sets such that…” and “there exists continuum many sets such that … ”. We prove that over the class of countable linear orders the two quantifiers are equivalent and can be effectively and uniformly eliminated. Weaker or partial elimination results are obtained for certain wider classes of chains. In particular, we show that over the class of ordinals the uncountability quantifier can be effectively and uniformly eliminated. Our argument makes use of Shelah's composition method and Ramsey-like theorem for dense linear orders.

We study cut-elimination in first-order classical logic. We construct a sequence of polynomial-length proofs having a non-elementary number of different cut-free normal forms. These normal forms are different in a strong sense: they not only represent different Herbrand-disjunctions but also differ in their prepositional structure.

This result illustrates that the constructive content of a proof in classical logic is not uniquely determined but rather depends on the chosen method for extracting it.

We study relationships between certain algebraic properties of groups and rings definable in a first order structure or *-closed in a compact G-space. As a consequence, we obtain a few structural results about ω-categorical rings as well as about small, nm-stable compact G-rings, and we also obtain surprising relationships between some conjectures concerning small profinite groups.

The Infinite Time Turing Machine model [8] of Hamkins and Kidder is, in an essential sense, a “Σ2-machine” in that it uses a Σ2Liminf Rule to determine cell values at limit stages of time. We give a generalisation of these machines with an appropriate Σn rule. Such machines either halt or enter an infinite loop by stage , again generalising precisely the ITTM case.

The collection of such machines taken together computes precisely those reals of the least model of analysis.

We show that the theory of the real field with a generic real power function is decidable, relative to an oracle for the rational cut of the exponent of the power function. We also show the existence of generic computable real numbers, hence providing an example of a decidable o-minimal proper expansion of the real field by an analytic function.

Improving on a result of Arana, we construct an effective family (φr ∣ r ϵ ℚ ∩ [0,1]) of Σn-conservative Πn sentences, increasing in strength as r decreases, with the property that ¬φp is Πn-conservative over PA + φq whenever p < q. We also construct a family of Σn sentences with properties as above except that the roles of Σn and Πn are reversed. The latter result allows to re-obtain an unpublished result of Solovay, the presence of a subset order-isomorphic to the reals in every non-trivial end-segment of every branch of the E-tree, and to generalize it to analogues of the E-tree at higher levels of the arithmetical hierarchy.

With every σ-ideal I on a Polish space we associate the σ-ideal I* generated by the closed sets in I. We study the forcing notions of Borel sets modulo the respective σ -ideals I and I* and find connections between their forcing properties. To this end, we associate to a σ-ideal on a Polish space an ideal on a countable set and show how forcing properties of the forcing depend on combinatorial properties of the ideal.

We also study the 1–1 or constant property of σ-ideals, i.e., the property that every Borel function defined on a Borel positive set can be restricted to a positive Borel set on which it either 1–1 or constant. We prove the following dichotomy: if I is a σ-ideal generated by closed sets, then either the forcing P1 adds a Cohen real, or else I has the 1–1 or constant property.

The transitive closure of a binary relation R can be thought of as the best possible approximation of R “from above” by a transitive relation. We consider the question of approximating a relation from below by transitive relations. Our main result is that every thick relation (a relation whose complement contains no infinite chain) on a countable set has a transitive thick subrelation. This allows for a solution to a problem arising from previous work by the author and Alan Taylor. We also exhibit a thick relation on an uncountable set with no transitive thick subrelation.

Elementary Recursive Nonstandard Analysis, in short ERNA, is a constructive system of nonstandard analysis with a PRA consistency proof, proposed around 1995 by Patrick Suppes and Richard Sommer. Recently, the author showed the consistency of ERNA with several transfer principles and proved results of nonstandard analysis in the resulting theories (see [12] and [13]). Here, we show that Weak König's lemma (WKL) and many of its equivalent formulations over RCA0 from Reverse Mathematics (see [21] and [22]) can be ‘pushed down’ into the weak theory ERNA, while preserving the equivalences, but at the price of replacing equality with equality ‘up to infinitesimals’. It turns out that ERNA plays the role of RCA0 and that transfer for universal formulas corresponds to WKL.

It is shown that a stable division ring with positive characteristic has finite dimension over its centre. This is then extended to simple division rings.

We give some sufficient conditions for a predicate P in a complete theory T to be “stably embedded”. Let be P with its “induced ∅-definable structure”. The conditions are that (or rather its theory) is “rosy”. P has NIP in T and that P is stably 1-embedded in T. This generalizes a recent result of Hasson and Onshuus [6] which deals with the case where P is o-minimal in T. Our proofs make use of the theory of strict nonforking and weight in NIP theories ([3], [10]).

Hindman's Theorem is a prototypical example of a combinatorial theorem with a proof that uses the topology of the ultrafilters. We show how the methods of this proof, including topological arguments about ultrafilters, can be translated into second order arithmetic.

We show that MRP + MA implies that ITP(λ,ω2) holds for all cardinal λ ≥ ω2. This generalizes a result by Weiβ who showed that PFA implies that ITP(λ, ω2) holds for all cardinal λ ≥ ω2. Consequently any of the known methods to prove MRP + MA consistent relative to some large cardinal hypothesis requires the existence of a strongly compact cardinal. Moreover if one wants to force MRP + MA with a proper forcing, it requires at least a supercompact cardinal. We also study the relationship between MRP and some weak versions of square. We show that MRP implies the failure of □(λ, ω) for all λ ≥ ω2 and we give a direct proof that MRP + MA implies the failure of □(λ, ω1) for all λ ≥ ω2.

We build on an existing a term-sequent logic for the λ-calculus. We formulate a general sequent system that fully integrates αβη-reductions between untyped λ-terms into first order logic.

We prove a cut-elimination result and then offer an application of cut-elimination by giving a notion of uniform proof for λ-terms. We suggest how this allows us to view the calculus of untyped αβ-reductions as a logic programming language (as well as a functional programming language, as it is traditionally seen).

We clarify and correct some statements and results in the literature concerning unimodularity in the sense of Hrushovski [7], and measurability in the sense of Macpherson and Steinhorn [8], pointing out in particular that the two notions coincide for strongly minimal structures and that another property from [7] is strictly weaker, as well as “completing” Elwes' proof [5] that measurability implies 1-basedness for stable theories.