A dependent theory is a (first order complete theory) T which does not have the independence property. A main result here is: if we expand a model of T by …
A dependent theory is a (first order complete theory) T which does not have the independence property. A main result here is: if we expand a model of T by the traces on it of sets definable in a bigger model then we preserve its being dependent. Another one justifies the cofinality restriction in the theorem (from a previous work) saying that pairwise perpendicular indiscernible sequences, can have arbitrary dual-cofinalities in some models containing them.
A dependent theory is a (first order complete theory) T which does not have the independence property. A main result here is: if we expand a model of T by …
A dependent theory is a (first order complete theory) T which does not have the independence property. A main result here is: if we expand a model of T by the traces on it of sets definable in a bigger model then we preserve its being dependent. Another one justifies the cofinality restriction in the theorem (from a previous work) saying that pairwise perpendicular indiscernible sequences, can have arbitrary dual-cofinalities in some models containing them.
A dependent theory is a (first order complete theory) which does not have the independence property. A major result here is: if we expand a model of by the traces …
A dependent theory is a (first order complete theory) which does not have the independence property. A major result here is: if we expand a model of by the traces on it of sets definable in a bigger model then we preserve its being dependent. Another one justifies the cofinality restriction in the theorem (from a previous work) saying that pairwise perpendicular indiscernible sequences, can have arbitrary dual-cofinalities in some models containing them. We introduce strongly dependent T and look at definable groups in such models; also look at forking and relatives.
We further investigate the class of models of a strongly dependent (first order complete) theory T, continuing math.LO/0406440. If |A|+|T| =beth_{|T|^+}(mu) then some J subseteq I of cardinality mu^+ is …
We further investigate the class of models of a strongly dependent (first order complete) theory T, continuing math.LO/0406440. If |A|+|T| =beth_{|T|^+}(mu) then some J subseteq I of cardinality mu^+ is an indiscernible sequence over A .
We further investigate the class of models of a strongly dependent (first order complete) theory T, continuing math.LO/0406440. If |A|+|T|<= mu, I subseteq C, |I| >=beth_{|T|^+}(mu) then some J subseteq …
We further investigate the class of models of a strongly dependent (first order complete) theory T, continuing math.LO/0406440. If |A|+|T|<= mu, I subseteq C, |I| >=beth_{|T|^+}(mu) then some J subseteq I of cardinality mu^+ is an indiscernible sequence over A .
The present status of Unified Theories is summarized with special emphasis on their possible experimental tests. Outline: i) Unification of couplings; ii) Where can a positive signal come from? iii) …
The present status of Unified Theories is summarized with special emphasis on their possible experimental tests. Outline: i) Unification of couplings; ii) Where can a positive signal come from? iii) HERA anomaly and Unification; iv) Recent progress in model building; v) Flavour and Unification.
This chapter starts the second part of the book, where neutral type time-delay systems are studied. Issues related to the existence, uniqueness, and continuation of the solutions of an initial …
This chapter starts the second part of the book, where neutral type time-delay systems are studied. Issues related to the existence, uniqueness, and continuation of the solutions of an initial value problem for such systems are discussed. In addition, stability concepts and basic stability results obtained with the use of the Lyapunov–Krasovskii approach, mainly in the form of necessary and sufficient conditions, are presented.
In this chapter we discuss the notion of an induced representation and the structure of the commutant of a representation, and we present a new approach to Clifford theory. We …
In this chapter we discuss the notion of an induced representation and the structure of the commutant of a representation, and we present a new approach to Clifford theory. We assume the reader to be familiar with the basic rudiments of the representation theory of finite groups. We refer to the monographs by Bump [7], Fulton and Harris [29], Isaacs [36], Serre [67], Simon [68] and Sternberg [73] as basic references; see also our monograph [15].
We study definably amenable NIP groups. We develop a theory of generics showing that various definitions considered previously coincide, and we study invariant measures. As applications, we characterize ergodic measures, …
We study definably amenable NIP groups. We develop a theory of generics showing that various definitions considered previously coincide, and we study invariant measures. As applications, we characterize ergodic measures, give a proof of the conjecture of Petrykowski connecting existence of bounded orbits with definable amenability in the NIP case, and prove the Ellis group conjecture of Newelski and Pillay connecting the model-theoretic connected component of an NIP group with the ideal subgroup of its Ellis enveloping semigroup.
We investigate distality and existence of distal expansions in valued fields and related structures. In particular, we characterize distality in a large class of ordered abelian groups, provide an Ax-Kochen-Eršov-style …
We investigate distality and existence of distal expansions in valued fields and related structures. In particular, we characterize distality in a large class of ordered abelian groups, provide an Ax-Kochen-Eršov-style characterization for henselian valued fields, and demonstrate that certain expansions of fields, e.g., the differential field of logarithmic-exponential transseries, are distal. As a new tool for analyzing valued fields we employ a relative quantifier elimination for pure short exact sequences of abelian groups.
We recast the problem of calculating Vapnik-Chervonenkis (VC) density into one of counting types, and thereby calculate bounds (often optimal) on the VC density for some weakly o-minimal, weakly quasi-o-minimal, …
We recast the problem of calculating Vapnik-Chervonenkis (VC) density into one of counting types, and thereby calculate bounds (often optimal) on the VC density for some weakly o-minimal, weakly quasi-o-minimal, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding="application/x-tex">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-minimal theories.
Abstract We show that a complete first-order theory T is distal provided it has a model M such that the theory of the Shelah expansion of M is distal.
Abstract We show that a complete first-order theory T is distal provided it has a model M such that the theory of the Shelah expansion of M is distal.
Our “long term and large scale” aim is to characterize the first order theories $T$ (at least the countable ones) such that for every ordinal $\alpha$ there are $\lambda$, $M_1$, …
Our “long term and large scale” aim is to characterize the first order theories $T$ (at least the countable ones) such that for every ordinal $\alpha$ there are $\lambda$, $M_1$, $M_2$ such that $M_1$ and $M_2$ are non-isomorphic models of $T$ of cardinality $\lambda$ which are EF$^+_{\alpha,\lambda}$-equivalent. We expect that as in the main gap [11, XII], we get a strong dichotomy, i.e., on the non-structure side we have stronger, better examples, and on the structure side we have an analogue of [11, XIII]. We presently prove the consistency of the non-structure side for $T$ which is $\aleph_0$-independent (= not strongly dependent), even for PC$(T_1,T)$.
We present an updated exposition of the classical theory of complete first order theories without the independence property (also called NIP theories or dependent theories).
We present an updated exposition of the classical theory of complete first order theories without the independence property (also called NIP theories or dependent theories).
The following conjecture is due to Shelah-Hasson: Any infinite strongly NIP field is either real closed, algebraically closed, or admits a non-trivial definable henselian valuation, in the language of rings. …
The following conjecture is due to Shelah-Hasson: Any infinite strongly NIP field is either real closed, algebraically closed, or admits a non-trivial definable henselian valuation, in the language of rings. We specialise this conjecture to ordered fields in the language of ordered rings, which leads towards a systematic study of the class of strongly NIP almost real closed fields. As a result, we obtain a complete characterisation of this class.
We study compressible types in the context of (local and global) NIP. By extending a result in machine learning theory (the existence of a bound on the recursive teaching dimension), …
We study compressible types in the context of (local and global) NIP. By extending a result in machine learning theory (the existence of a bound on the recursive teaching dimension), we prove density of compressible types. Using this, we obtain explicit uniform honest definitions for NIP formulas (answering a question of Eshel and the second author), and build compressible models in countable NIP theories.
The main result of this article is sub-additivity of the dp-rank. We also show that the study of theories of finite dp-rank cannot be reduced to the study of its …
The main result of this article is sub-additivity of the dp-rank. We also show that the study of theories of finite dp-rank cannot be reduced to the study of its dp-minimal types, and we discuss the possible relations between dp-rank and VC-density.
Abstract We provide a general theorem implying that for a (strongly) dependent theory T the theory of sufficiently well-behaved pairs of models of T is again (strongly) dependent. We apply …
Abstract We provide a general theorem implying that for a (strongly) dependent theory T the theory of sufficiently well-behaved pairs of models of T is again (strongly) dependent. We apply the theorem to the case of lovely pairs of thorn-rank one theories as well as to a setting of dense pairs of first-order topological theories.
(Bull. London Math. Soc. 42 (2010) 64–74) There is a serious mistake in the proof of Theorem 1 in the above mentioned paper. Consequently, we must withdraw the claim of …
(Bull. London Math. Soc. 42 (2010) 64–74) There is a serious mistake in the proof of Theorem 1 in the above mentioned paper. Consequently, we must withdraw the claim of having proved that theorem.
Let $K$ be an elementary extension of $\mathbb{Q}_p$, $V$ be the set of finite $a \in K$, $\mathrm{st}$ be the standard part map $K^m \to \mathbb{Q}^m_p$, and $X \subseteq K^m$ …
Let $K$ be an elementary extension of $\mathbb{Q}_p$, $V$ be the set of finite $a \in K$, $\mathrm{st}$ be the standard part map $K^m \to \mathbb{Q}^m_p$, and $X \subseteq K^m$ be $K$-definable. Delon has shown that $\mathbb{Q}^m_p \cap X$ is $\mathbb{Q}_p$-definable. Yao has shown that $\dim \mathbb{Q}^m_p \cap X \leq \dim X$ and $\dim \mathrm{st}(V^n \cap X) \leq \dim X$. We give new $\mathrm{NIP}$-theoretic proofs of these results and show that both inequalities hold in much more general settings. We also prove the analogous results for the expansion $\mathbb{Q}^{\mathrm{an}}_p$ of $\mathbb{Q}_p$ by all analytic functions $\mathbb{Z}^n_p \to \mathbb{Q}_p$. As an application we show that if $(X_k)_{k \in \mathbb{N}}$ is a sequence of elements of an $\mathbb{Q}^{\mathrm{an}}_p$-definable family of subsets of $\mathbb{Q}^m_p$ which converges in the Hausdroff topology to $X \subseteq \mathbb{Q}^m_p$ then $X$ is $\mathbb{Q}^{\mathrm{an}}_p$-definable and $\dim X \leq \limsup_{k \to \infty} \dim X_k$.
Abstract We prove that in theories without the tree property of the second kind (which include dependent and simple theories) forking and dividing over models are the same, and in …
Abstract We prove that in theories without the tree property of the second kind (which include dependent and simple theories) forking and dividing over models are the same, and in fact over any extension base. As an application we show that dependence is equivalent to bounded non-forking assuming NTP 2 .
We give a short proof of the Marker–Steinhorn theorem for o-minimal expansions of ordered groups. The key tool is Ramakrishnan’s classification of definable linear orders in such structures.
We give a short proof of the Marker–Steinhorn theorem for o-minimal expansions of ordered groups. The key tool is Ramakrishnan’s classification of definable linear orders in such structures.
Abstract We develop the theory of generically stable types, independence relation based on nonforking and stable weight in the context of dependent theories.
Abstract We develop the theory of generically stable types, independence relation based on nonforking and stable weight in the context of dependent theories.
Title of dissertation: ON DEFINABILITY OF TYPES IN DEPENDENT THEORIES Vincent Guingona, Doctor of Philosophy, 2011 Dissertation directed by: Professor Michael Chris Laskowski Department of Mathematics Using definability of types …
Title of dissertation: ON DEFINABILITY OF TYPES IN DEPENDENT THEORIES Vincent Guingona, Doctor of Philosophy, 2011 Dissertation directed by: Professor Michael Chris Laskowski Department of Mathematics Using definability of types for stable formulas, one develops the powerful tools of stability theory, such as canonical bases, a nice forking calculus, and stable embeddability. When one passes to the class of dependent formulas, this notion of definability of types is lost. However, as this dissertation shows, we can recover suitable alternatives to definability of types for some dependent theories. Using these alternatives, we can recover some of the power of stability theory. One alternative is uniform definability of types over finite sets (UDTFS). We show that all formulas in dp-minimal theories have UDTFS, as well as formulas with VC-density < 2. We also show that certain Henselian valued fields have UDTFS. Another alternative is isolated extensions. We show that dependent formulas are characterized by the existence of isolated extensions, and show how this gives a weak stable embeddability result. We also explore the idea of UDTFS rank and show how it relates to VC-density. Finally, we use the machinery developed in this dissertation to show that VCminimal theories satisfy the Kueker Conjecture. ON DEFINABILITY OF TYPES IN DEPENDENT THEORIES
We study combinatorial properties of convex sets over arbitrary valued fields.We demonstrate analogs of some classical results for convex sets over the reals (for example, the fractional Helly theorem and …
We study combinatorial properties of convex sets over arbitrary valued fields.We demonstrate analogs of some classical results for convex sets over the reals (for example, the fractional Helly theorem and Bárány's theorem on points in many simplices), along with some additional properties not satisfied by convex sets over the reals, including finite breadth and VC dimension.These results are deduced from a simple combinatorial description of modules over the valuation ring in a spherically complete valued field.
We prove that if ${\cal M}=(M,\leq,+,\ldots)$ is a weakly o-minimal non-valuational structure expanding an ordered group $(M,\leq,+)$, then its expansion by a family of “non-valuational” unary predicates remains non-valuational. The …
We prove that if ${\cal M}=(M,\leq,+,\ldots)$ is a weakly o-minimal non-valuational structure expanding an ordered group $(M,\leq,+)$, then its expansion by a family of “non-valuational” unary predicates remains non-valuational. The paper is b
We investigate definable topological dynamics of groups definable in an o‐minimal expansion of the field of reals. Assuming that a definable group G admits a model‐theoretic analogue of Iwasawa decomposition, …
We investigate definable topological dynamics of groups definable in an o‐minimal expansion of the field of reals. Assuming that a definable group G admits a model‐theoretic analogue of Iwasawa decomposition, namely the compact‐torsion‐free decomposition , we give a description of minimal subflows and the Ellis group of its universal definable flow in terms of this decomposition. In particular, the Ellis group of this flow is isomorphic to . This provides a range of counterexamples to a question by Newelski whether the Ellis group is isomorphic to . We further extend the results to universal topological covers of definable groups, interpreted in a two‐sorted structure containing the o‐minimal sort and a sort for an abelian group.
We investigate the notions of strict independence and strict non-forking, and establish basic properties and connections between the two. In particular, it follows from our investigation that in resilient theories …
We investigate the notions of strict independence and strict non-forking, and establish basic properties and connections between the two. In particular, it follows from our investigation that in resilient theories strict non-forking is symmetric. Based on this study, we develop notions of weight which characterize NTP 2 , dependence and strong dependence. Many of our proofs rely on careful analysis of sequences that witness dividing. We prove simple characterizations of such sequences in resilient theories, as well as of Morley sequences which are witnesses. As a by-product we obtain information on types co-dominated by generically stable types in dependent theories. For example, we prove that every Morley sequence in such a type is a witness.
We study the notions generic stability, regularity, homogeneous pregeometries, quasiminimality, and their mutual relations, in arbitrary first order theories. We prove that “infinite-dimensional homogeneous pregeometries” coincide with generically stable strongly …
We study the notions generic stability, regularity, homogeneous pregeometries, quasiminimality, and their mutual relations, in arbitrary first order theories. We prove that “infinite-dimensional homogeneous pregeometries” coincide with generically stable strongly regular types (p(x), x = x). We prove that in a theory without the strict order property, regular types are generically stable, and prove analogous results for quasiminimal structures. We prove that the “generic type” of a quasiminimal structure is “locally strongly regular”.
Abstract 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 …
Abstract 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]).
Abstract We prove that in a group without the independence property a nilpotent subgroup is always contained in a definable nilpotent subgroup of the same nilpotency class. The analogue for …
Abstract We prove that in a group without the independence property a nilpotent subgroup is always contained in a definable nilpotent subgroup of the same nilpotency class. The analogue for the soluble case is also shown when the subgroup is normal in the ambient group.
For G a group definable in some structure M, we define notions of "definable" compactification of G and "definable" action of G on a compact space X (definable G-flow), where …
For G a group definable in some structure M, we define notions of "definable" compactification of G and "definable" action of G on a compact space X (definable G-flow), where the latter is under a definability of types assumption on M. We describe the universal definable compactification of G as G⁎/(G⁎)M00 and the universal definable G-ambit as the type space SG(M). We also point out the existence and uniqueness of "universal minimal definable G-flows", and discuss issues of amenability and extreme amenability in this definable category, with a characterization of the latter. For the sake of completeness we also describe the universal (Bohr) compactification and universal G-ambit in model-theoretic terms, when G is a topological group (although it is essentially well-known).
Abstract We introduce the Boolean algebra of d -semialgebraic (more generally, d -definable) sets and prove that its Stone space is naturally isomorphic to the Ellis enveloping semigroup of the …
Abstract We introduce the Boolean algebra of d -semialgebraic (more generally, d -definable) sets and prove that its Stone space is naturally isomorphic to the Ellis enveloping semigroup of the Stone space of the Boolean algebra of semialgebraic (definable) sets. For definably connected o-minimal groups, we prove that this family agrees with the one of externally definable sets in the one-dimensional case. Nonetheless, we prove that in general these two families differ, even in the semialgebraic case over the real algebraic numbers. On the other hand, in the semialgebraic case we characterise real semialgebraic functions representing Boolean combinations of d -semialgebraic sets.
We study combinatorial properties of convex sets over arbitrary valued fields.We demonstrate analogs of some classical results for convex sets over the reals (for example, the fractional Helly theorem and …
We study combinatorial properties of convex sets over arbitrary valued fields.We demonstrate analogs of some classical results for convex sets over the reals (for example, the fractional Helly theorem and Bárány's theorem on points in many simplices), along with some additional properties not satisfied by convex sets over the reals, including finite breadth and VC dimension.These results are deduced from a simple combinatorial description of modules over the valuation ring in a spherically complete valued field.
For an infinite cardinal ${\it\kappa}$ , let $\text{ded}\,{\it\kappa}$ denote the supremum of the number of Dedekind cuts in linear orders of size ${\it\kappa}$ . It is known that ${\it\kappa}<\text{ded}\,{\it\kappa}\leqslant 2^{{\it\kappa}}$ …
For an infinite cardinal ${\it\kappa}$ , let $\text{ded}\,{\it\kappa}$ denote the supremum of the number of Dedekind cuts in linear orders of size ${\it\kappa}$ . It is known that ${\it\kappa}<\text{ded}\,{\it\kappa}\leqslant 2^{{\it\kappa}}$ for all ${\it\kappa}$ and that $\text{ded}\,{\it\kappa}<2^{{\it\kappa}}$ is consistent for any ${\it\kappa}$ of uncountable cofinality. We prove however that $2^{{\it\kappa}}\leqslant \text{ded}(\text{ded}(\text{ded}(\text{ded}\,{\it\kappa})))$ always holds. Using this result we calculate the Hanf numbers for the existence of two-cardinal models with arbitrarily large gaps and for the existence of arbitrarily large models omitting a type in the class of countable dependent first-order theories. Specifically, we show that these bounds are as large as in the class of all countable theories.
Abstract We consider definable topological dynamics for NIP groups admitting certain decompositions in terms of specific classes of definably amenable groups. For such a group, we find a description of …
Abstract We consider definable topological dynamics for NIP groups admitting certain decompositions in terms of specific classes of definably amenable groups. For such a group, we find a description of the Ellis group of its universal definable flow. This description shows that the Ellis group is of bounded size. Under additional assumptions, it is shown to be independent of the model, proving a conjecture proposed by Newelski. Finally we apply the results to new classes of groups definable in o-minimal structures, generalizing all of the previous results for this setting.
The article gives an overview of the development of model theory in Kazakhstan over the past 60 years from the standpoint of the tasks that stood at that time.We consider …
The article gives an overview of the development of model theory in Kazakhstan over the past 60 years from the standpoint of the tasks that stood at that time.We consider directions that naturally arose from the definition of truth in a model and under the influence of Doctor of Physical and Mathematical Sciences, Academician A. Taimanov, initiator of the growth of model theory in Kazakhstan.
Abstract We show basic facts about dp-minimal ordered structures. The main results are: dp-minimal groups are abelian-by-finite-exponent, in a divisible ordered dp-minimal group, any infinite set has nonempty interior, and …
Abstract We show basic facts about dp-minimal ordered structures. The main results are: dp-minimal groups are abelian-by-finite-exponent, in a divisible ordered dp-minimal group, any infinite set has nonempty interior, and any theory of pure tree is dp-minimal.
Let $T$ be a (first order complete) dependent theory, ${\mathfrak{C}}$ a $\barκ$-saturated model of $T$ and $G$ a definable subgroup which is abelian. Among subgroups of bounded index which are …
Let $T$ be a (first order complete) dependent theory, ${\mathfrak{C}}$ a $\barκ$-saturated model of $T$ and $G$ a definable subgroup which is abelian. Among subgroups of bounded index which are the union of $
Does the class of linear orders have (one of the variants of) the so called $(\lambda,\kappa)$-limit model? It is necessarily unique, and naturally assuming some instances of G.C.H. we get …
Does the class of linear orders have (one of the variants of) the so called $(\lambda,\kappa)$-limit model? It is necessarily unique, and naturally assuming some instances of G.C.H. we get some positive results. More generally, letting $T$ be a complete first order theory and for simplicity assume G.C.H., for regular $\lambda > \kappa > |T|$ does $T$ have (variants of) a $(\lambda,\kappa)$-limit models, except for stable $T$? For some, yes, the theory of dense linear order, for some, no. Moreover, for independent $T$ we get negative results. We deal more with linear orders.
Cette note est une suite à mon article Théories instables , this Journal, vol. 46 (1981), pp. 513–522. Quand il en a eu pris connaissance, Saharon Shelah–et c'est remplir un …
Cette note est une suite à mon article Théories instables , this Journal, vol. 46 (1981), pp. 513–522. Quand il en a eu pris connaissance, Saharon Shelah–et c'est remplir un devoir agréable que de lui témoigner ici ma reconnaissance pour l'intérêt qu'il porte à mes travaux–m'a fait savoir qu'il pouvait en simplifier les résultats, et a publié le Théorème 1 de ce post-scriptum, pour le cas des cohéritiers, dans les “added in proof” de son Simple unstable théories, Annals of Mathematical Logic , vol. 19 (1980), pp. 177–203, plusieurs mois avant que ne paraisse l'article qu'il améliore. On verra ici que ce théorème se généralise sans problème aux fils spéciaux, et que son corollaire, le Théorème 2, représente un net progrès sur mon résultat original, puisqu'il est à la fois plus général, plus précis, et obtenu à moindre frais, le lemme sur les ultrafiltres devenant inutile. Quelques rappels pour commencer: je considère une théorie complète T , sans propriété d'indépendance; cela signifie que toute suite … a i … indicernable dans l'ordre est insécable , c'est-à-dire qu'il ne peut exister de formule f ( x, ȳ ) et de paramètres tels que f ( a i , ) soit vraie cofinalement, et fausse cofinalement, dans la suite; cela signifie encore que cette suite s est découpée par la formule f ( x , ) en un nombre fini de segments S 1 , …, S k , la formule prenant la même valeur de vérité dans chaque segment, et des valeurs opposées dans des segments consécutifs; ce nombre k de segments sera appelé nombre d'alternance de la formule f ( x , ) sur la suite s . On remarquera que, par compacité, ce nombre d'alternance est majoré en fonction seulement de f ( x, ȳ ).
I am grateful to Peter Freyd for his generous introductory comments, and I am grateful also to the [Selection] Committee for extending the invitation to speak to you.Last month a …
I am grateful to Peter Freyd for his generous introductory comments, and I am grateful also to the [Selection] Committee for extending the invitation to speak to you.Last month a colleague with whom I attempted to discuss today's remarks suggested drily that as half of the team responsible for [CN2] I have probably already said more than enough about ultrafllters, and that if I insist on pursuing the matter further today I could do so most gracefully and efficiently simply by offering a complimentary copy of [CN2] to each of you.Eschewing that advice I shall in the hour allotted to me attempt to achieve the following three goals.(A) To acquaint you with what I think are some of the most basic, fundamental facts about ultrafllters on a discrete topological space; this material is sufficiently simple and elegant that it can be absorbed comfortably into a first-year graduate course in general topology.(B) To give some partial results, less definitive and less conclusive than the optimal theorems available, concerning the existence of particular ultrafllters with special properties; I hope that the results chosen in this connection have the complementary virtues that they are sufficiently powerful to handle most of the situations treated by the more powerful results which we shall ignore, and that their proofs are significantly simpler than those of the more general results.(C) To record some results about ultrafllters which came to my attention after the publication of [CN2]; I have chosen today to emphasize three relatively new results which are not formally concerned with ultrafllters and which indeed make no mention of ultrafllters in their statements, but which nevertheless have been given proofs in which ultrafllters play an important catalytic role.My hope is that (even) those of you not professionally inclined toward topology or set theory will find something potentially useful, or amusing, among the basic results given in (A).The theorems selected for inclusion in (B) are given not only because of the intrinsic beauty and elegance of their proofs, but also because they serve to indicate the principal sorts of questions
We try to understand complete types over a somewhat saturated model of a complete first-order theory which is dependent (previously called NIP), by "decomposition theorems for such types". Our thesis …
We try to understand complete types over a somewhat saturated model of a complete first-order theory which is dependent (previously called NIP), by "decomposition theorems for such types". Our thesis is that the picture of dependent theory is the combination of the one for stable theories and the one for the theory of dense linear order or trees (and first, we should try to understand the quite saturated case). As a measure of our progress, we give several applications considering some test questions; in particular, we try to prove the generic pair conjecture and do it for measurable cardinals.
For a dependent theory <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper C Subscript upper T"> <mml:semantics> <mml:msub> …
For a dependent theory <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper C Subscript upper T"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">C</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>T</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">{\mathfrak {C}}_{T}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for every type definable group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the intersection of type definable subgroups with bounded index is a type definable subgroup with bounded index.
Resumo Ni montras propecon de el j eteco de la kvantoro (∃y ∈ M ) pri la (sufi c e) belaj paroj de modeloj de una O -plimalpova teorio. G …
Resumo Ni montras propecon de el j eteco de la kvantoro (∃y ∈ M ) pri la (sufi c e) belaj paroj de modeloj de una O -plimalpova teorio. G i havas korolaron ke, se ni aldonas malkavajn unarajn predikatojn a la lingvo de kelka O -plimalpova strukturo, ni ricevas malforte O -plimalpovan strukturon. Tui c i rezultato estis en speciala kaso pruvita de [5], kaj la g ia g eneralize c o estis anoncita en [1].
In this paper we give characterizations of the stable and ℵ 0 ‐stable theories, in terms of an external property called representation. In the sense of the representation property, the …
In this paper we give characterizations of the stable and ℵ 0 ‐stable theories, in terms of an external property called representation. In the sense of the representation property, the mentioned classes of first‐order theories can be regarded as “not very complicated”.
Baizhanov and Baldwin [1] introduce the notions of benign and weakly benign sets to investigate the preservation of stability by naming arbitrary subsets of a stable structure. They connect the …
Baizhanov and Baldwin [1] introduce the notions of benign and weakly benign sets to investigate the preservation of stability by naming arbitrary subsets of a stable structure. They connect the notion with work of Baldwin, Benedikt, Bouscaren, Casanovas, Poizat, and Ziegler. Stimulated by [1], we investigate here the existence of benign or weakly benign sets. Definition 0.1. (1) The set A is benign in M if for every α, β ∊ M if p = tp(α/ A ) = tp(β/ A ) then tp * (α/ A ) = tp * (β/ A ) where the *-type is the type in the language L* with a new predicate P denoting A . (2) The set A is weakly benign in M if for every α,β ∊ M if p = stp(α/ A ) = stp(β/ A ) then tp * (α/ A ) = tp * (β/A) where the *-type is the type in language with a new predicate P denoting A . Conjecture 0.2 (too optimistic). If M is a model of stable theory T and A ⊆ M then A is benign . Shelah observed, after learning of the Baizhanov-Baldwin reductions of the problem to equivalence relations, the following counterexample. Lemma 0.3. There is an ω-stable rank 2 theory T with ndop which has a model M and set A such that A is not benign in M .
Our thesis is that for the family of classes of the form EC(T),T a com- plete first order theory with the dependence property (which is just the negation of the …
Our thesis is that for the family of classes of the form EC(T),T a com- plete first order theory with the dependence property (which is just the negation of the independence property) there is a substantial theory which means: a substantial body of basic results for all such classes and some complimentary results for the first order theories with the independence property, as for the family of stable (and the family of simple) first order theories. We examine some properties.
We further investigate the class of models of a strongly dependent (first order complete) theory T, continuing math.LO/0406440. If |A|+|T| =beth_{|T|^+}(mu) then some J subseteq I of cardinality mu^+ is …
We further investigate the class of models of a strongly dependent (first order complete) theory T, continuing math.LO/0406440. If |A|+|T| =beth_{|T|^+}(mu) then some J subseteq I of cardinality mu^+ is an indiscernible sequence over A .
L'absence ou la présence de la propriété d'indépendance, introduite par Shelah, est sans aucun doute une mesure significative de la complexité, du point de vue de la théorie des modèles, …
L'absence ou la présence de la propriété d'indépendance, introduite par Shelah, est sans aucun doute une mesure significative de la complexité, du point de vue de la théorie des modèles, d'une théorie instable. Il importe donc de distinguer les théories qui ont cette propriété de celles qui ne l'ont pas; le critère de Keisler [1] et de Shelah [5], qui consiste à compter le nombre de types qu'on peut obtenir sur un ensemble de paramètres de cardinal λ , n'emporte la décision que si l'on nie assez brutalement l'hypothèse du continu généralisée. Je propose ici de compter, étant donnée une partie X , de cardinal λ , d'un espace de types S i ( M ), quel peut être le cardinal de l'adhérence de X ; ce point de vue équivaut au précédent si la théorie T est stable (voir la remarque après le Théorème 7); et si T n'a pas la propriété d'indépendance ce cardinal est au plus 2 λ , tandis que si T a la propriété d'indépendance il peut atteindre 2 2 λ (Théorème 7). Cela donne, indépendamment d'hypothèses de théorie des ensembles, une mesure quantitative de la plus grande complexité des espaces de types s'il y a propriété d'indépendance; et sous la forme du Théorème 8, cela devient un test très maniable dans les situations concrètes, car l'expérience m'a prouvé que les cohéritiers d'un type sont toujours faciles à déterminer. Dans une première section, je considère un type complet p au dessus d'un modèle M d'une théorie complète T , et parmi les fils de p sur une extension élémentaire de M j'en distingue certains que j'appelle fils spéciaux de p : cette notion est naturelle et utile pour la suite, car le critère que j'ai décrit revient à prouver l'abondance de certains fils spéciaux d'un même type; on notera au passage le Théorème 3. Le théorème de caractérisation de la propriété d'indépendance est prouvé dans une deuxième section, grâce à un résultat de combinatoire (Théorème 6); j'y montre aussi, à titre d'illustration, qu'un ordre total n'a jamais la propriété d'indépendance.