Generalising Hrushovski’s fusion technique we construct the free fusion of two strongly minimal theories T₁,T₂ intersecting in a totally categorical sub-theory T₀. We show that if, e.g., T₀ is the theory of infinite vector spaces over a finite field then the fusion theory T_ω exists, is complete and ω-stable of rank ω. We give a detailed geometrical analysis of T_ω, proving that if both T₁, T₂ are 1-based then, T_ω can be collapsed into a strongly minimal theory, if some additional technical conditions hold—all trivially satisfied if T₀ is the theory of infinite vector spaces over a finite field 𝔽_q.

We give resource bounded versions of the Completeness Theorem for propositional and predicate logic. For example, it is well known that every computable consistent propositional theory has a computable complete consistent extension. We show that, when length is measured relative to the binary representation of natural numbers and formulas, every polynomial time decidable propositional theory has an exponential time (EXPTIME) complete consistent extension whereas there is a nondeterministic polynomial time (NP) decidable theory which has no polynomial time complete consistent extension when length is measured relative to the binary representation of natural numbers and formulas. It is well known that a propositional theory is axiomatizable (respectively decidable) if and only if it may be represented as the set of infinite paths through a computable tree (respectively a computable tree with no dead ends). We show that any polynomial time decidable theory may be represented as the set of paths through a polynomial time decidable tree. On the other hand, the statement that every polynomial time decidable tree represents the set of complete consistent extensions of some theory which is polynomial time decidable, relative to the tally representation of natural numbers and formulas, is equivalent to P=NP. For predicate logic, we develop a complexity theoretic version of the Henkin construction to prove a complexity theoretic version of the Completeness Theorem. Our results imply that that any polynomial space decidable theory Δ possesses a polynomial space computable model which is exponential space decidable and thus Δ has an exponential space complete consistent extension. Similar results are obtained for other notions of complexity.

Let o(κ) denote the Mitchell order of κ. We show how to reduce long games which run to the first ordinal admissible in the play, to iteration games on models with a cardinal κ so that (1) κ is a limit of Woodin cardinals; and (2) o(κ)=κ⁺⁺. We use the reduction to derive several optimal determinacy results on games which run to the first admissible in the play.

We introduce and study the notions of a PAC-substructure of a stable structure, and a bounded substructure of an arbitrary substructure, generalizing [10]. We give precise definitions and equivalences, saying what it means for properties such as PAC to be first order, study some examples (such as differentially closed fields) in detail, relate the material to generic automorphisms, and generalize a “descent theorem” for pseudo-algebraically closed fields to the stable context. We also point out that the elementary invariants of pseudo-algebraically closed fields from [6] are also valid for pseudo-differentially closed fields.

We show that the Proper Forcing Axiom implies the Singular Cardinal Hypothesis. The proof uses the reflection principle MRP introduced by Moore in [11].

Using the Friedman-Koepke Hyperfine Structure Theory of [2], we provide a short construction of a gap 1 morass in the constructible universe.

By using some classical reasoning we show that any countably presented formal topology, namely, a formal topology with a countable axiom set, is spatial.

A recursive enumerator for a function h is an algorithm f which enumerates for an input x finitely many elements including h(x). f is a k(n)-enumerator if for every input x of length n, h(x) is among the first k(n) elements enumerated by f. If there is a k(n)-enumerator for h then h is called k(n)-enumerable. We also consider enumerators which are only A-recursive for some oracle A.

We determine exactly how hard it is to enumerate the Kolmogorov function, which assigns to each string x its Kolmogorov complexity:

• For every underlying universal machine U, there is a constant a such that C is k(n)-enumerable only if k(n) ≥ n/a for almost all n.

• For any given constant k, the Kolmogorov function is k-enumerable relative to an oracle A if and only if A is at least as hard as the halting problem.

• There exists an r.e., Turing-incomplete set A such for every non-decreasing and unbounded recursive function k, the Kolmogorov function is k(n)-enumerable relative to A.

The last result is obtained by using a relativizable construction for a nonrecursive set A relative to which the prefix-free Kolmogorov complexity differs only by a constant from the unrelativized prefix-free Kolmogorov complexity.

Although every 2-enumerator for C is Turing hard for K, we show that reductions must depend on the specific choice of the 2-enumerator and there is no bound on the quantity of their queries. We show our negative results even for strong 2-enumerators as an oracle where the querying machine for any x gets directly an explicit list of all hypotheses of the enumerator for this input. The limitations are very general and we show them for any recursively bounded function g:

• For every Turing reduction M and every non-recursive set B, there is a strong 2-enumerator f for g such that M does not Turing reduce B to f.

• For every non-recursive set B, there is a strong 2-enumerator f for g such that B is not wtt-reducible to f.

Furthermore, we deal with the resource-bounded case and give characterizations for the class S₂^p introduced by Canetti and independently Russell and Sundaram and the classes PSPACE, EXP.

• S₂^p is the class of all sets A for which there is a polynomially bounded function g such that there is a polynomial time tt-reduction which reduces A to every strong 2-enumerator for g.

• PSPACE is the class of all sets A for which there is a polynomially bounded function g such that there is a polynomial time Turing reduction which reduces A to every strong 2-enumerator for g. Interestingly, g can be taken to be the Kolmogorov function for the conditional space bounded Kolmogorov complexity.

• EXP is the class of all sets A for which there is a polynomially bounded function g and a machine M which witnesses A ∈ PSPACE^f for all strong 2-enumerators f for g.

Finally, we show that any strong O(log n)-enumerator for the conditional space bounded Kolmogorov function must be PSPACE-hard if P=NP.

We characterise strongly minimal groups interpretable in elementary extensions of compact complex analytic spaces.

We prove a categoricity transfer theorem for tame abstract elementary classes.

Theorem. Suppose that 𝔎 is a χ-tame abstract elementary class and satisfies the amalgamation and joint embedding properties and has arbitrarily large models. Let λ≥Max{χ,LS(𝔎)⁺}. If 𝔎 is categorical in λ and λ⁺, then 𝔎 is categorical in λ⁺⁺.

Combining this theorem with some results from [37], we derive a form of Shelah’s Categoricity Conjecture for tame abstract elementary classes:

Corollary. Suppose 𝔎 is a χ-tame abstract elementary class satisfying the amalgamation and joint embedding properties. Let μ₀:= Hanf(𝔎). If χ≤ℶ_(2^μ₀)⁺ and 𝔎 is categorical in some λ⁺>ℶ_(2^μ₀)⁺, then 𝔎 is categorical in μ for all μ>ℶ_(2^μ₀)⁺.

Let 𝒜 be the category of all reduced compact complex spaces, viewed as a multi-sorted first order structure, in the standard way. Let 𝒰 be a sub-category of 𝒜, which is closed under the taking of products and analytic subsets, and whose morphisms include the projections. Under the assumption that Th(𝒰) is unidimensional, we show that Morley rank is equal to Noetherian dimension, in any elementary extension of 𝒰. As a result, we are able to show that Morley degree is definable in Th(𝒰), when Th(𝒰) is unidimensional.

We prove several trichotomy results for ideals of compact sets. Typically, we show that a “sufficiently rich” universally Baire ideal is either Π⁰₃-hard, or Σ⁰₃-hard, or else a σ-ideal.

We prove conjugacy and generic disjointness of generous Carter subgroups in groups of finite Morley rank. We elaborate on groups with a generous Carter subgroup and on a minimal counterexample to the Genericity Conjecture.

Let T be a positive L₁-L_∞ contraction. We prove that the following statements are equivalent in constructive mathematics.

• 1. The projection in L₂ on the space of invariant functions exists;

• 2. The sequence (Tⁿ)_{n ∈ N} Cesáro-converges in the L₂ norm;

• 3. The sequence (Tⁿ)_{n ∈ N} Cesáro-converges almost everywhere.

Thus, we find necessary and sufficient conditions for the Mean Ergodic Theorem and the Dunford-Schwartz Pointwise Ergodic Theorem.

As a corollary we obtain a constructive ergodic theorem for ergodic measure-preserving transformations. This answers a question posed by Bishop.

Modifying the methods of Z. Adamowicz’s paper Herbrand consistency and bounded arithmetic [3] we show that there exists a number n such that ⋃_m S_m (the union of the bounded arithmetic theories S_m) does not prove the Herbrand consistency of the finitely axiomatizable theory Sⁿ₃.

In [12], T. Scanlon proved a quantifier elimination result for valued D-fields in a three-sorted language by using angular component functions. Here we prove an analogous theorem in a different language ℒ₂ which was introduced by F. Delon in her thesis. This language allows us to lift the quantifier elimination result to a one-sorted language by a process described in the Appendix. As a byproduct, we state and prove a “positivstellensatz” theorem for the differential analogue of the theory of real-series closed fields in the valued D-field setting.

A general method is provided whereby bizarre revisions of consistent theories with respect to contingent sentences that they refute can be delivered by revision-functions satisfying both the basic and the supplementary postulates of the AGM-theory of theory-revision.

We investigate axiomatizations of Kripke’s theory of truth based on the Strong Kleene evaluation scheme for treating sentences lacking a truth value. Feferman’s axiomatization KF formulated in classical logic is an indirect approach, because it is not sound with respect to Kripke’s semantics in the straightforward sense; only the sentences that can be proved to be true in KF are valid in Kripke’s partial models. Reinhardt proposed to focus just on the sentences that can be proved to be true in KF and conjectured that the detour through classical logic in KF is dispensable. We refute Reinhardt’s Conjecture, and provide a direct axiomatization PKF of Kripke’s theory in partial logic. We argue that any natural axiomatization of Kripke’s theory in Strong Kleene logic has the same proof-theoretic strength as PKF, namely the strength of the system RA_<ω^ω ramified analysis or a system of Tarskian ramified truth up to ω^ω. Thus any such axiomatization is much weaker than Feferman’s axiomatization KF in classical logic, which is equivalent to the system RA_<ε₀ of ramified analysis up to ε₀.

The existence and uniqueness of a maximum point for a continuous real—valued function on a metric space are investigated constructively. In particular, it is shown, in the spirit of reverse mathematics, that a natural unique existence theorem is equivalent to the fan theorem.