Mathematics

Cohesive powers of computable structures are effective analogs of ultrapowers, where cohesive sets play the role of ultrafilters. Let ω , ζ , and η denote the respective order-types of the natural numbers, the integers, and the rationals when thought of as linear orders. We investigate the cohesive powers of computable linear orders, with special emphasis on computable copies of ω . If L is a computable copy of ω that is computably isomorphic to the standard presentation of ω , then every cohesive power of L has order-type ω+ζη . However, there are computable copies of ω , necessarily not computably isomorphic to the standard presentation, having cohesive powers not elementarily equivalent to ω+ζη . For example, we show that there is a computable copy of ω with a cohesive power of order-type ω+η . Our most general result is that if X⊆N∖{0} is either a Σ 2 set or a Π 2 set, thought of as a set of finite order-types, then there is a computable copy of ω with a cohesive power of order-type ω+σ(X∪{ω+ζη+ ω ∗ }) , where σ(X∪{ω+ζη+ ω ∗ }) denotes the shuffle of the order-types in X and the order-type ω+ζη+ ω ∗ . Furthermore, if X is finite and non-empty, then there is a computable copy of ω with a cohesive power of order-type ω+σ(X) .

We investigate the computability of algebraic closure and definable closure with respect to a collection of formulas. We show that for a computable collection of formulas of quantifier rank at most n , in any given computable structure, both algebraic and definable closure with respect to that collection are Σ 0 n+2 sets. We further show that these bounds are tight.

This paper engages the question "Does the consistency of a set of axioms entail the existence of a model in which they are satisfied?" within the frame of the Frege-Hilbert controversy. The question is related historically to the formulation, proof, and reception of Gödel's Completeness Theorem. Tools from mathematical logic are then used to argue that there are precise senses in which Frege was correct to maintain that demonstrating consistency is as difficult as it can be but also in which Hilbert was correct to maintain that demonstrating existence given consistency is as easy as it can be.

We prove that if a countable group is elementarily equivalent to a non-abelian free group and all of its abelian subgroups are cyclic, then the group is a union of a chain of regular NTQ groups (i.e., hyperbolic towers).

Abstract categorial grammars (ACG), as well as some other, closely related systems, are based on the ordinary, commutative implicational linear logic and linear λ -calculus in contrast to the better known "noncommutative" Lambek grammars and their variations. ACG seem attractive in many ways, not the least of which is the simplicity of the underlying logic. Yet it is known that ACG and their relatives behave poorly in modeling many natural language phenomena (such as, for example, coordination) compared to "noncommutative" formalisms. Therefore different solutions have been proposed in order to enrich ACG with noncommutative constructions. Tensor grammars of this work are another example of "commutative" grammars, based on the classical, rather than intuitionistic linear logic. They can be seen as a surface representation of ACG in the sense that derivations of ACG translate to derivations of tensor grammars and this translation is isomorphic on the level of string languages. An advantage of this representation, as it seems to us, is that the syntax becomes extremely simple and a direct geometric meaning is transparent. We address the problem of encoding noncommutative operations in our setting. This turns out possible after enriching the system with new unary operators. The resulting system allows representing both ACG and Lambek grammars as conservative fragments, while the formalism remains, as it seems to us, rather simple and intuitive.

We introduce the notion of first order amenability, as a property of a first order theory T : every complete type over ∅ , in possibly infinitely many variables, extends to an automorphism-invariant global Keisler measure in the same variables. Amenability of T follows from amenability of the (topological) group Aut(M) for all sufficiently large ℵ 0 -homogeneous countable models M of T (assuming T to be countable), but is radically less restrictive. First, we study basic properties of amenable theories, giving many equivalent conditions. Then, applying a version of the stabilizer theorem from [Amenability, connected components, and definable actions; E. Hrushovski, K. Krupiński, A. Pillay], we prove that if T is amenable, then T is G-compact, namely Lascar strong types and Kim-Pillay strong types over ∅ coincide. This extends and essentially generalizes a similar result proved via different methods for ω -categorical theories in [Amenability, definable groups, and automorphism groups; K. Krupiński, A. Pillay] . In the special case when amenability is witnessed by ∅ -definable global Keisler measures (which is for example the case for amenable ω -categorical theories), we also give a different proof, based on stability in continuous logic. Parallel (but easier) results hold for the notion of extreme amenability.

The notion of a categorical quotient can be generalized since its standard categorical concept does not recover the expected quotients in certain categories. We present a more general formulation in the form of F -quotients in a category C , which are relativized to a faithful functor F:C→D . The isomorphism theorems of universal algebras generalize to this setting, and we additionally find important links between F -quotients in the concrete category of first-order structures, and quotients defined for model-theoretic equivalence classes. By first working in this categorical setting, some quotient-related results for first-order structures can be naturally obtained. In particular, we are able to prove some isomorphism theorems in the context of model theory directly from their corresponding categorical isomorphism theorems.

We investigate several consequences of inclusion relations between quantified provability logics. Moreover, we give a necessary and sufficient condition for the inclusion relation between quantified provability logics with respect to Σ 1 arithmetical interpretations.

Let S be a set of states of a physical system. The probabilities p(s) of the occurrence of an event when the system is in different states s of S define a function from S to [0,1] called a numerical event or, more accurately, an S-probability. Sets of S-probabilities ordered by the partial order of functions give rise to so called algebras of S-probabilities, in particular to the ones that are lattice-ordered. Among these there are the sigma-algebras known from probability theory and the Hilbert-space logics which are important in quantum-mechanics. Any algebra of S-probabilities can serve as a quantum-logic, and it is of special interest when this logic turns out to be a Boolean algebra because then the observed physical system will be classical. Boolean algebras are in one-to-one correspondence to Boolean rings, and the question arises to find an analogue correspondence for lattice-ordered algebras of S-probabilities generalizing the correspondence between Boolean algebras and Boolean rings. We answer this question by defining ring-like structures of events (RLSEs). First, the structure of RLSEs is revealed and Boolean rings among RLSEs are characterized. Then we establish how RLSEs correspond to lattice-ordered algebras of numerical events. Further, functions for associating lattice-ordered algebras of S-probabilities to RLSEs are studied. It is shown that there are only two ways to assign lattice-ordered algebras of S-probabilities to RLSEs if one restricts the corresponding mappings to term functions over the underlying orthomodular lattice. These term functions are the very functions by which also Boolean algebras can be assigned to Boolean rings.

We explore "semibounded" expansions of arbitrary ordered groups; namely, expansions that do not define a field on the whole universe. We show that if R=⟨R,<,+,…⟩ is a semibounded o-minimal structure and P⊆R a set satisfying certain tameness conditions, then ⟨R,P⟩ remains semibounded. Examples include the cases when R=⟨R,<,+,(x↦λx ) λ∈R , ⋅ [0,1 ] 2 ⟩ , and P= 2 Z or P is an iteration sequence. As an application, we obtain that smooth functions definable in such ⟨R,P⟩ are definable in R .