Abstract For any $n<\omega $ we construct an infinite $(n+1)$ -generated Heyting algebra whose n -generated subalgebras are of cardinality $\leq m_n$ for some positive integer $m_n$ . From this we conclude that for every $n<\omega $ there exists a variety of Heyting algebras which contains an infinite $(n+1)$ -generated algebra, but which contains only finite n -generated algebras. For the case $n=2$ this provides a negative answer to a question posed by G. Bezhanishvili and R. Grigolia in [4].

Trying to interpret B. Zilber's project on model theory of quantum mechanics we study a way of building limit models from finite-dimensional approximations. Our point of view is that of metric model theory, and we develop a method of taking ultraproducts of unbounded operators. We first calculate the Feynman propagator for the free particle as defined by physicists as an inner product 〈x0|Kt|x1〉 of the eigenvector |x0〉 of the position operator with eigenvalue x0 and Kt(|x1〉), where Kt is the time evolution operator. However, due to a discretising effect, the eigenvector method does not work as expected, and straightforward calculations give the wrong value. We look at this phenomenon, and then complement this by showing how to instead correctly calculate the kernel of the time evolution operator (for both the free particle and the harmonic oscillator) in the limit model. We believe that our method of calculating these is new.

Abstract We define and study a metric independence notion in a homogeneous metric abstract elementary class with perturbations that is dp-superstable (superstable wrt. the perturbation topology), weakly simple and has complete type spaces and we give a new example of such a class based on B. Zilber’s approximations of Weyl algebras. We introduce a way to measure the dependence of a tuple a from a set B over another set A. We prove basic properties of the notion, e.g., that a is independent of B over A in the usual sense of homogeneous model theory if and only if the measure of dependence is < ε for all ε > 0. In well behaved situations, the measure corresponds to the distance to a free extension. As an example of our measure of dependence we show a connection between the measure and entropy in models from quantum mechanics in which the spectrum of the observable is discrete. As an application, we show that weak simplicity implies a very strong form of simplicity and study the question of when the dependence inside a set of all realisations of some type can be seen to arise from a pregeometry in cases when the type is not regular. In the end of the paper, we demonstrate our notions and results in one more example: a class built from the p-adic integers.

Abstract 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 , of some power π. We assume that in Gödel's constructive universe , π is a regular cardinal at least the successor of the first cardinal in which , is stable. We show that the collection of pairs of submodels in as above which are potentially isomorphic with respect to certain cardinal-preserving extensions of is equiconstructible with 0#. As 0# is highly “transcendental” over , 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.

Abstract 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 weakκ-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 ( ) be a simple finitary AEC, weakly categorical in some uncountableκ. Then ( ) is weakly categorical in eachλ≥ min . If the class ( ) is also -tame, weakκ-categoricity is equivalent withκ-categoricity in the usual sense. We also discuss the relation between finitary AECs and some other non-elementary frameworks and give several examples.

Abstract We show that any (atomic) excellent class can be expanded with hyperimaginaries to form an (atomic) excellent class which has canonical bases. When is, in addition, of finite U-rank, then is also simple and has a full canonical bases theorem. This positive situation contrasts starkly with homogeneous model theory for example, where the eq-expansion may fail to be homogeneous. However, this paper shows that expanding an ω-stable, homogeneous class gives rise to an excellent class, which is simple if is of finite U-rank.

We introduce Lascar strong types in excellent classes and prove that they coincide with the orbits of the group generated by automorphisms fixing a model. We define a new independence relation using Lascar strong types and show that it is well-behaved over models, as well as over finite sets. We then develop simplicity (when this independence relation has local character) and show that, under simplicity, the independence relation satisfies all the properties of nonforking in a stable first order theory. Further, simplicity for an excellent class, as well as the independence relation itself, is uniquely determined. Finally, we show that an excellent class is simple if and only if it has extensible U-rank (excellence does not imply simplicity in general). We deduce that any excellent class of finite U-rank is simple, and that any uncountably categorical excellent class has an expansion with countably many constants which is simple.

Abstract We characterize the classifiability of a countable first-order theory T in terms of the solvability (in the sense of [2]) of the potential-isomorphism problem for models of T.

Abstract We study the class of elementary submodels of a large superstable homogeneous model. We introduce a rank which is bounded in the superstable case, and use it to define a dependence relation which shares many (but not all) of the properties of forking in the first order case. The main difference is that we do not have extension over all sets. We also present an example of Shelah showing that extension over all sets may not hold for any dependence relation for superstable homogeneous models.

We will answer questions due to Blass and Gurevich (2000) on definability of order in the first-order logic with Hilbert's epsilon operation. We show that a linear ordering is almost surely definable in models with random choice.

Abstract. We generalize the result of Mekler and Shelah [3] that the existence of a canary tree is independent of ZFC + GCH to uncountable regular cardinals. We also correct an error from the original proof.

Abstract We prove a main gap theorem for locally saturated submodels of a homogeneous structure. We also study the number of locally saturated models, which are not elementarily embeddable into each other.

Abstract In this paper we prove a strong nonstructure theorem for κ(T)-saturated models of a stable theory T with dop. This paper continues the work started in [1].

Abstract We study how equivalent nonisomorphic models of unsuperstable theories can be. We measure the equivalence by Ehrenfeucht-Fraïssé games. This paper continues [HS].

Abstract We study how equivalent nonisomorphic models an unsuperstable theory can have. We measure the equivalence by Ehrenfeucht-Fraisse games. This paper continues the work started in [HT].

Abstract Let and be two countable relational models of the same first order language. If the models are nonisomorphic, there is a unique countable ordinal α with the property that i.e. and are L∞ω-equivalent up to quantifier-rank α but not up to α + 1. In this paper we consider models and of cardinality ω1 and construct trees which have a similar relation to and as a above. For this purpose we introduce a new ordering T ≪ T′ of trees, which may have some independent interest of its own. It turns out that the above ordinal α has two qualities which coincide in countable models but will differ in uncountable models. Respectively, two kinds of trees emerge from α. We call them Scott trees and Karp trees, respectively. The definition and existence of these trees is based on an examination of the Ehrenfeucht game of length ω1 between and . We construct two models of power ω1 with mutually noncomparable Scott trees.