We study dp-minimal and strongly dependent theories and investigate connections between these notions and weight.

We continue our study of finitary abstract elementary classes, defined in [7]. In this paper, we prove a categoricity transfer theorem for a case of simple finitary AECs. We introduce the concepts of weak κ-categoricity and f-primary models to the framework of ℵ₀-stable simple finitary AECs with the extension property, whereby we gain the following theorem: Let (𝕂,≼_𝕂) be a simple finitary AEC, weakly categorical in some uncountable κ. Then (𝕂,≼_𝕂) is weakly categorical in each λ≥min{κ,ℶ_(2^ℵ0)⁺}. If the class (𝕂,≼_𝕂) is also LS(𝕂)-tame, weak κ-categoricity is equivalent with κ-categoricity in the usual sense.

We also discuss the relation between finitary AECs and some other non-elementary frameworks and give several examples.

Traditional combinatory logic uses combinators S and K to represent all Turing-computable functions on natural numbers, but there are Turing-computable functions on the combinators themselves that cannot be so represented, because they access internal structure in ways that S and K cannot. Much of this expressive power is captured by adding a factorisation combinator F. The resulting SF-calculus is structure complete, in that it supports all pattern-matching functions whose patterns are in normal form, including a function that decides structural equality of arbitrary normal forms. A general characterisation of the structure complete, confluent combinatory calculi is given along with some examples. These are able to represent all their Turing-computable functions whose domain is limited to normal forms. The combinator F can be typed using an existential type to represent internal type information.

We obtain lower bounds for the consistency strength of fully nonregular ultrafilters on ω₂.

The first main result isolates some conditions which fail for the class of graphs and hold for the class of Abelian p-groups, the class of Abelian torsion groups, and the special class of “rank-homogeneous” trees. We consider these conditions as a possible definition of what it means for a class of structures to have “Ulm type”. The result says that there can be no Turing computable embedding of a class not of Ulm type into one of Ulm type. We apply this result to show that there is no Turing computable embedding of the class of graphs into the class of “rank-homogeneous” trees. The second main result says that there is a Turing computable embedding of the class of rank-homogeneous trees into the class of torsion-free Abelian groups. The third main result says that there is a “rank-preserving” Turing computable embedding of the class of rank-homogeneous trees into the class of Boolean algebras. Using this result, we show that there is a computable Boolean algebra of Scott rank ω₁^CK.

It is known that for all finite n ≥ 5, there are relation algebras with n-dimensional relational bases but no weak representations. We prove that conversely, there are finite weakly representable relation algebras with no n-dimensional relational bases. In symbols: neither of the classes RA_n and wRRA contains the other.

We employ an infinite-signature Hrushovski amalgamation construction to yield two results in Recursive Model Theory. The first result, that there exists a strongly minimal theory whose only recursively presentable models are the prime and saturated models, adds a new spectrum to the list of known possible spectra. The second result, that there exists a strongly minimal theory in a finite language whose only recursively presentable model is saturated, gives the second non-trivial example of a spectrum produced in a finite language.

A set of natural numbers B is computably enumerable in and strictly above (or c.e.a. for short) another set C if C <_T B and B is computably enumerable in C. A Turing degree b is c.e.a. c if b and c respectively contain B and C as above. In this paper, it is shown that if b is c.e.a. c then b is c.e.a. some 1-generic g.

This paper studies several systems of the transfinite iteration and autonomous progression of self-applicable truth and determines their proof-theoretic strength.

We introduce a natural strengthening of prompt simplicity which we call strong promptness, and study its relationship with existing lowness classes. This notion provides a ≤_wtt version of superlow cuppability. We show that every strongly prompt c.e. set is superlow cuppable. Unfortunately, strong promptness is not a Turing degree notion, and so cannot characterize the sets which are superlow cuppable. However, it is a wtt-degree notion, and we show that it characterizes the degrees which satisfy a wtt-degree notion very close to the definition of superlow cuppability.

Further, we study the strongly prompt c.e. sets in the context of other notions related promptness, superlowness, and cupping. In particular, we show that every benign cost function has a strongly prompt set which obeys it, providing an analogue to the known result that every cost function with the limit condition has a prompt set which obeys it. We also study the effect that lowness properties have on the behaviour of a set under the join operator. In particular we construct an array noncomputable c.e. set whose join with every low c.e. set is low.

An argument of A. Borel [Bor-61, Proposition 3.1] shows that every compact connected Lie group is homeomorphic to the Cartesian product of its derived subgroup and a torus. We prove a parallel result for definably compact definably connected groups definable in an o-minimal expansion of a real closed field. As opposed to the Lie case, however, we provide an example showing that the derived subgroup may not have a definable semidirect complement.

The results herein form part of a larger project to characterize the classification properties of the class of submodels of a homogeneous stable diagram in terms of the solvability (in the sense of [1]) of the potential isomorphism problem for this class of submodels.

We restrict ourselves to locally saturated submodels of the monster model $\mathfrak{M}$ of some power π. We assume that in Gödel's constructible universe 𝕃, π is a regular cardinal at least the successor of the first cardinal in which $\mathfrak{M}$ is stable.

We show that the collection of pairs of submodels in 𝕃 as above which are potentially isomorphic with respect to certain cardinal-preserving extensions of 𝕃 is equiconstructible with 0^#. As 0^# is highly “transcendental” over 𝕃, this provides a very strong statement to the effect that potential isomorphism for this class of models not only fails to be set-theoretically absolute, but is of high (indeed of the highest possible) complexity.

The proof uses a novel method that does away with the need for a linear order on the skeleton.

We consider actions of completely metrisable groups on simplicial trees in the context of the Bass—Serre theory. Our main result characterises continuity of the amplitude function corresponding to a given action. Under fairly mild conditions on a completely metrisable group G, namely, that the set of elements generating a non-discrete or finite subgroup is somewhere dense, we show that in any decomposition as a free product with amalgamation, G=A∗_CB, the amalgamated groups A, B and C are open in G.

We observe simple links between equivalence relations, groups, fields and groupoids (and between preorders, semi-groups, rings and categories), which are type-definable in an arbitrary structure, and apply these observations to the particular context of small and simple structures. Recall that a structure is small if it has countably many n-types with no parameters for each natural number n. We show that a ∅-type-definable group in a small structure is the conjunction of definable groups, and extend the result to semi-groups, fields, rings, categories, groupoids and preorders which are ∅-type-definable in a small structure. For an A-type-definable group G_A (where the set A may be infinite) in a small and simple structure, we deduce that

1. if G_A is included in some definable set X such that boundedly many translates of G_A cover X, then G_A is the conjunction of definable groups.

2. for any finite tuple \bar{g} in G_A, there is a definable group containing \bar{g}.

We present a characterization of weakly compact cardinals in terms of generalized stationarity. We apply this characterization to construct a model with no partial square sequences.

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 ℬ. For such subalgebras 𝒜, we show that it is possible for the spectrum of the unary relation 𝒜 on ℬ to contain a low₅ degree without containing 0.

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 P_I adds a Cohen real, or else I has the 1—1 or constant property.