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, ā¦
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.
We prove that for every Scott set $S$ there are $S$-saturated real closed fields and models of Presburger arithmetic.
We prove that for every Scott set $S$ there are $S$-saturated real closed fields and models of Presburger arithmetic.
We prove that for every Scott set $S$ there are $S$-saturated real closed fields and models of Presburger arithmetic.
We prove that for every Scott set $S$ there are $S$-saturated real closed fields and models of Presburger arithmetic.
We study the VapnikāChervonenkis (VC) density of definable families in certain stable first-order theories. In particular, we obtain uniform bounds on the VC density of definable families in finite U-rank ā¦
We study the VapnikāChervonenkis (VC) density of definable families in certain stable first-order theories. In particular, we obtain uniform bounds on the VC density of definable families in finite U-rank theories without the finite cover property, and we characterize those abelian groups for which there exist uniform bounds on the VC density of definable families.
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 ā¦
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.
For a vector field $\xi$ on $\mathbb{R}^2$ we construct, under certain assumptions on $\xi$, an ordered model-theoretic structure associated to the flow of $\xi$. We do this in such a ā¦
For a vector field $\xi$ on $\mathbb{R}^2$ we construct, under certain assumptions on $\xi$, an ordered model-theoretic structure associated to the flow of $\xi$. We do this in such a way that the set of all limit cycles of $\xi$ is represente
For a vector field F on the Euclidean plane we construct, under certain assumptions on F, an ordered model-theoretic structure associated to the flow of F. We do this in ā¦
For a vector field F on the Euclidean plane we construct, under certain assumptions on F, an ordered model-theoretic structure associated to the flow of F. We do this in such a way that the set of all limit cycles of F is represented by a definable set. This allows us to give two restatements of Dulac's Problem for F--that is, the question whether F has finitely many limit cycles--in model-theoretic terms, one involving the recently developed notion of thorn-rank and the other involving the notion of o-minimality.
Abstract A structure ( M , <, ā¦) is called quasi-o-minimal if in any structure elementarily equivalent to it the definable subsets are exactly the Boolean combinations of 0-definable subsets ā¦
Abstract A structure ( M , <, ā¦) is called quasi-o-minimal if in any structure elementarily equivalent to it the definable subsets are exactly the Boolean combinations of 0-definable subsets and intervals. We give a series of natural examples of quasi-o-minimal structures which are not o-minimal; one of them is the ordered group of integers. We develop a technique to investigate quasi-o-minimality and use it to study quasi-o-minimal ordered groups (possibly with extra structure). Main results: any quasi-o-minimal ordered group is abelian; any quasi-o-minimal ordered ring is a real closed field, or has zero multiplication; every quasi-o-minimal divisible ordered group is o-minimal; every quasi-o-minimal archimedian densely ordered group is divisible. We show that a counterpart of quasi-o-minimality in stability theory is the notion of theory of U -rank 1.
In this paper we formulate a notion similar to o -minimality but appropriate for the p -adics. The paper is in a sense a sequel to [11] and [5]. In ā¦
In this paper we formulate a notion similar to o -minimality but appropriate for the p -adics. The paper is in a sense a sequel to [11] and [5]. In [11] a notion of minimality was formulated, as follows. Suppose that L, L + are first-order languages and + is an L + -structure whose reduct to L is . Then + is said to be -minimal if, for every N + elementarily equivalent to + , every parameterdefinable subset of its domain N + is definable with parameters by a quantifier-free L -formula. Observe that if L has a single binary relation which in is interpreted by a total order on M , then we have just the notion of strong o-minimality , from [13]; and by a theorem from [6], strong o -minimality is equivalent to o -minimality. If L has no relations, functions, or constants (other than equality) then the notion is just strong minimality . In [11], -minimality is investigated for a number of structures . In particular, the C-relation of [1] was considered, in place of the total order in the definition of strong o -minimality. The C -relation is essentially the ternary relation which naturally holds on the maximal chains of a sufficiently nice tree; see [1], [11] or [5] for more detail, and for axioms. Much of the motivation came from the observation that a C -relation on a field F which is preserved by the affine group AGL(1, F ) (consisting of permutations ( a,b ) : x ā¦ ax + b , where a ā F \ {0} and b ā F ) is the same as a non-trivial valuation: to get a C -relation from a valuation Ī½, put C ( x;y,z ) if and only if Ī½( y ā x ) < Ī½( y ā z ).
This book is a modern introduction to model theory which stresses applications to algebra throughout the text. The first half of the book includes classical material on model construction techniques, ā¦
This book is a modern introduction to model theory which stresses applications to algebra throughout the text. The first half of the book includes classical material on model construction techniques, type spaces, prime models, saturated models, countable models, and indiscernibles and their applications. The author also includes an introduction to stability theory beginning with Morley's Categoricity Theorem and concentrating on omega-stable theories. One significant aspect of this text is the inclusion of chapters on important topics not covered in other introductory texts, such as omega-stable groups and the geometry of strongly minimal sets. The author then goes on to illustrate how these ingredients are used in Hrushovski's applications to diophantine geometry. David Marker is Professor of Mathematics at the University of Illinois at Chicago. His main area of research involves mathematical logic and model theory, and their applications to algebra and geometry. This book was developed from a series of lectures given by the author at the Mathematical Sciences Research Institute in 1998.
We show that a class of subsets of a structure uniformly definable by a first-order formula is a Vapnik-Chervonenkis class if and only if the formula does not have the ā¦
We show that a class of subsets of a structure uniformly definable by a first-order formula is a Vapnik-Chervonenkis class if and only if the formula does not have the independence property. Via this connection we obtain several new examples of Vapnik-Chervonenkis classes, including sets of positivity of finitely subanalytic functions.
Abstract Morley rank and Lascar rank are equal on generic types of definable groups in differentially closed fields with finitely many commuting derivations.
Abstract Morley rank and Lascar rank are equal on generic types of definable groups in differentially closed fields with finitely many commuting derivations.
Abstract We give a valuation theoretic characterization for a real closed field to be recursively saturated. This builds on work in [9], where the authors gave such a characterization for ā¦
Abstract We give a valuation theoretic characterization for a real closed field to be recursively saturated. This builds on work in [9], where the authors gave such a characterization for Īŗ -saturation, for a cardinal $\kappa \ge \aleph _0 $ . Our result extends the characterization of Harnik and Ressayre [7] for a divisible ordered abelian group to be recursively saturated.
Introduction.Dedekind's pigeon-hole principle, also known as the box argument or the chest of drawers argument (Schubfachprinzip) can be described, rather vaguely, as follows.If sufficiently many objects are distributed over not ā¦
Introduction.Dedekind's pigeon-hole principle, also known as the box argument or the chest of drawers argument (Schubfachprinzip) can be described, rather vaguely, as follows.If sufficiently many objects are distributed over not too many classes, then at least one class contains many of these objects.In 1930 F. P. Ramsey [12] discovered a remarkable extension of this principle which, in its simplest form, can be stated as follows.Let S be the set of all positive integers and suppose that all unordered pairs of distinct elements of S are distributed over two classes.Then there exists an infinite subset A of S such that all pairs of elements of A belong to the same class.As is well known, Dedekind's principle is the central step in many investigations.Similarly, Ramsey's theorem has proved itself a useful and versatile tool in mathematical arguments of most diverse character.The object of the present paper is to investigate a number of analogues and extensions of Ramsey's theorem.We shall replace the sets S and A by sets of a more general kind and the unordered pairs, as is the case already in the theorem proved by Ramsey, by systems of any fixed number r of elements of S. Instead of an unordered set S we consider an ordered set of a given order type, and we stipulate that the set A is to be of a prescribed order type.Instead of two classes we admit any finite or infinite number of classes.Further extension will be explained in Ā§ Ā§2, 8 and 9.The investigation centres round what we call partition relations connecting given cardinal numbers or order types and in each given case the problem arises of deciding whether a particular partition relation is true or false.It appears that a large number of seemingly unrelated arguments in set theory are, in fact, concerned with just such a problem.It might therefore be of interest to study such relations for their own sake and to build up a partition calculus which might serve as a new and unifying principle in set theory.In some cases we have been able to find best possible partition relations, in one sense or another.In other cases the methods available to the authors do not seem to lead anywhere near the ultimate Part of this paper was material from an address delivered by P. Erdƶs under the title Combinatorial problems in set theory before the New York meeting of the Society on October 24, 1953, by invitation of the Committee to Select Hour Speakers for Eastern Sectional Meetings; received by the editors May 17, 1955.1956] Ā«-> (ft, ā¢ā¢ā¢)*; <*-Ā» (*o, ā¢ā¢ā¢)*ā¢ THEOREM 16.If a-*(j8 0 , ft, ā¢ ā¢ ā¢ )ĆÆ +4 ; ftr-KYo, Ti, ā¢ ā¢ ā¢ )ĆÆ, then Ā«-Ā» (TO, TI, ā¢ ā¢ ā¢ , ft, ft, ā¢ * ā¢ )*+*ā¢In this proposition the types a, ft, y v may be replaced by cardinals.In formulating the last theorem we use an obvious extension of the symbol (2).PROOF.We consider the case of types.Let 5 = a, [S] r = Zfr < *]*ox + Ā£[0 < * < 1 + *]*,.Put Ko = ]C [X </]JSTox.Then, by hypothesis, there are A CS\ P<1 +k
Let<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"><mml:semantics><mml:mi>L</mml:mi><mml:annotation encoding="application/x-tex">L</mml:annotation></mml:semantics></mml:math></inline-formula>contain only the equality symbol and let<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript plus"><mml:semantics><mml:msup><mml:mi>L</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:annotation encoding="application/x-tex">L^+</mml:annotation></mml:semantics></mml:math></inline-formula>be an arbitrary finite symmetric relational language containing<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"><mml:semantics><mml:mi>L</mml:mi><mml:annotation ā¦
Let<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"><mml:semantics><mml:mi>L</mml:mi><mml:annotation encoding="application/x-tex">L</mml:annotation></mml:semantics></mml:math></inline-formula>contain only the equality symbol and let<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript plus"><mml:semantics><mml:msup><mml:mi>L</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:annotation encoding="application/x-tex">L^+</mml:annotation></mml:semantics></mml:math></inline-formula>be an arbitrary finite symmetric relational language containing<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"><mml:semantics><mml:mi>L</mml:mi><mml:annotation encoding="application/x-tex">L</mml:annotation></mml:semantics></mml:math></inline-formula>. Suppose probabilities are defined on finite<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript plus"><mml:semantics><mml:msup><mml:mi>L</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:annotation encoding="application/x-tex">L^+</mml:annotation></mml:semantics></mml:math></inline-formula>structures with āedge probabilityā<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n Superscript negative alpha"><mml:semantics><mml:msup><mml:mi>n</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>ā</mml:mo><mml:mi>Ī±</mml:mi></mml:mrow></mml:msup><mml:annotation encoding="application/x-tex">n^{-\alpha }</mml:annotation></mml:semantics></mml:math></inline-formula>. By<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Superscript alpha"><mml:semantics><mml:msup><mml:mi>T</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>Ī±</mml:mi></mml:mrow></mml:msup><mml:annotation encoding="application/x-tex">T^{\alpha }</mml:annotation></mml:semantics></mml:math></inline-formula>, the almost sure theory of random<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript plus"><mml:semantics><mml:msup><mml:mi>L</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:annotation encoding="application/x-tex">L^+</mml:annotation></mml:semantics></mml:math></inline-formula>-structures we mean the collection of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript plus"><mml:semantics><mml:msup><mml:mi>L</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:annotation encoding="application/x-tex">L^+</mml:annotation></mml:semantics></mml:math></inline-formula>-sentences which have limit probability 1.<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Subscript alpha"><mml:semantics><mml:msub><mml:mi>T</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>Ī±</mml:mi></mml:mrow></mml:msub><mml:annotation encoding="application/x-tex">T_{\alpha }</mml:annotation></mml:semantics></mml:math></inline-formula>denotes the theory of the generic structures for<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper K Subscript alpha"><mml:semantics><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">K</mml:mi></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>Ī±</mml:mi></mml:mrow></mml:msub><mml:annotation encoding="application/x-tex">\mathbb {K}_{\alpha }</mml:annotation></mml:semantics></mml:math></inline-formula>(the collection of finite graphs<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>with<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="delta Subscript alpha Baseline left-parenthesis upper G right-parenthesis equals StartAbsoluteValue upper G EndAbsoluteValue minus alpha dot StartAbsoluteValue edges of upper G EndAbsoluteValue"><mml:semantics><mml:mrow><mml:msub><mml:mi>Ī“</mml:mi><mml:mi>Ī±</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>G</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mi>G</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mo>ā</mml:mo><mml:mi>Ī±</mml:mi><mml:mo>ā </mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mrow><mml:mtext>edges of </mml:mtext><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>G</mml:mi></mml:mrow></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">|</mml:mo></mml:mrow></mml:mrow><mml:annotation encoding="application/x-tex">\delta _\alpha (G) = |G| - \alpha \cdot |\text {edges of $G$}|</mml:annotation></mml:semantics></mml:math></inline-formula>hereditarily nonnegative). <bold>Theorem.</bold><inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Superscript alpha"><mml:semantics><mml:msup><mml:mi>T</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>Ī±</mml:mi></mml:mrow></mml:msup><mml:annotation encoding="application/x-tex">T^{\alpha }</mml:annotation></mml:semantics></mml:math></inline-formula>, the almost sure theory of random<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript plus"><mml:semantics><mml:msup><mml:mi>L</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:annotation encoding="application/x-tex">L^+</mml:annotation></mml:semantics></mml:math></inline-formula>-structures, is the same as the theory<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Subscript alpha"><mml:semantics><mml:msub><mml:mi>T</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>Ī±</mml:mi></mml:mrow></mml:msub><mml:annotation encoding="application/x-tex">T_{\alpha }</mml:annotation></mml:semantics></mml:math></inline-formula>of the<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper K Subscript alpha"><mml:semantics><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">K</mml:mi></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>Ī±</mml:mi></mml:mrow></mml:msub><mml:annotation encoding="application/x-tex">\mathbb {K}_{\alpha }</mml:annotation></mml:semantics></mml:math></inline-formula>-generic model. This theory is complete, stable, and nearly model complete. Moreover, it has the finite model property and has only infinite models so is not finitely axiomatizable.
In this paper we extend two theorems from [2] on p-adic subanalytic sets, where p is a fixed prime number, Qp is the field of p-adic numbers and Zp is ā¦
In this paper we extend two theorems from [2] on p-adic subanalytic sets, where p is a fixed prime number, Qp is the field of p-adic numbers and Zp is the ring of p-adic integers. One of these theorems [2, 3.32] says that each subanalytic subset of Zp is semialgebraic. This is extended here as follows.
RESUME. Lāanalyse de Delon des types dans les corps values devient particulierement transparente dans le langage des corps values auquel on ajoute une āŖ application coeļ¬cient ā«. On obtient ainsi ā¦
RESUME. Lāanalyse de Delon des types dans les corps values devient particulierement transparente dans le langage des corps values auquel on ajoute
une āŖ application coeļ¬cient ā«. On obtient ainsi une preuve du transfert a la Ax-Kochen-Ershov de la propriete dāindependance en egale caracteristique
zero. /
ABSTRACT. Delonās analysis of types in valued
We prove that no complete theory of ordered abelian groups has the independence property, thus answering a question by B. Poizat. The main tool is a result contained in the ā¦
We prove that no complete theory of ordered abelian groups has the independence property, thus answering a question by B. Poizat. The main tool is a result contained in the doctoral dissertation of Yuri Gurevich and also in P. H. Schmittās <italic>Elementary properties of ordered abelian groups</italic>, which basically transforms statements on ordered abelian groups into statements on coloured chains. We also prove that every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-type in the theory of coloured chains has at most <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 Superscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{2^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> coheirs, thereby strengthening a result by B. Poizat.
We study the notion of dp-minimality, beginning by providing several essential facts about dp-minimality, establishing several equivalent definitions for dp-minimality, and comparing dp-minimality to other minimality notions. The majority of ā¦
We study the notion of dp-minimality, beginning by providing several essential facts about dp-minimality, establishing several equivalent definitions for dp-minimality, and comparing dp-minimality to other minimality notions. The majority of the rest of the paper is dedicated to examples. We establish via a simple proof that any weakly o-minimal theory is dp-minimal and then give an example of a weakly o-minimal group not obtained by adding traces of externally definable sets. Next we give an example of a divisible ordered Abelian group which is dp-minimal and not weakly o-minimal. Finally we establish that the field of p-adic numbers is dp-minimal.
Abstract We study dp-minimal and strongly dependent theories and investigate connections between these notions and weight.
Abstract We study dp-minimal and strongly dependent theories and investigate connections between these notions and weight.
We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating ā¦
We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating definably compact groups $G$ in saturated $o$-minimal structures to compact Lie groups. We also prove some other structural results about such $G$, for example the existence of a left invariant finitely additive probability measure on definable subsets of $G$. We finally introduce the new notion of "compact domination" (domination of a definable set by a compact space) and raise some new conjectures in the $o$-minimal case.
We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are ā¦
We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are (i) if p = \mathrm{tp}(b/A) does not fork over A then the Lascar strong type of b over A coincides with the compact strong type of b over A and any global nonforking extension of p is Borel definable over \mathrm{bdd}(A) , (ii) analogous statements for Keisler measures and definable groups, including the fact that G^{000} = G^{00} for G definably amenable, (iii) definitions, characterizations and properties of āgenerically stableā types and groups, (iv) uniqueness of invariant (under the group action) Keisler measures on groups with finitely satisfiable generics, (v) a proof of the compact domination conjecture for (definably compact) commutative groups in o -minimal expansions of real closed fields.
Abstract No theory of a partially ordered set of finite width has the independence property, generalizing Poizat's corresponding result for linearly ordered sets. In fact, a question of Poizat concerning ā¦
Abstract No theory of a partially ordered set of finite width has the independence property, generalizing Poizat's corresponding result for linearly ordered sets. In fact, a question of Poizat concerning linearly ordered sets is answered by showing, moreover, that no theory of a partially ordered set of finite width has the multi-order property. It then follows that a distributive lattice is not finite-dimensional iff its theory has the independence property iff its theory has the multi-order property.
Abstract We prove the elimination of Magidor-Malitz quantifiers for R -modules relative to certain -core sentences and positive primitive formulas. For complete extensions of the elementary theory of R-modules it ā¦
Abstract We prove the elimination of Magidor-Malitz quantifiers for R -modules relative to certain -core sentences and positive primitive formulas. For complete extensions of the elementary theory of R-modules it follows that all Ramsey quantifiers (āµ 0 -interpretation) are eliminable. By a result of Baldwin and Kueker [1] this implies that there is no R -module having the finite cover property.
Using relativizations of results of Goncharov and Peretyatākin on decidable homogeneous models, we prove that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is ā¦
Using relativizations of results of Goncharov and Peretyatākin on decidable homogeneous models, we prove that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-saturated for some Scott set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an enumeration of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has a presentation recursive in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Applying this result we are able to classify degrees coding (i) the reducts of models of PA to addition or multiplication, (ii) internally finite initial segments and (iii) nonstandard residue fields. We also use our results to simplify Solovayās characterization of degrees coding nonstandard models of Th(<bold>N</bold>).
Let K be a field of characteristic 0 and let n be a natural number.Let Ī be a subgroup of the multiplicative group (K * ) n of finite rank ā¦
Let K be a field of characteristic 0 and let n be a natural number.Let Ī be a subgroup of the multiplicative group (K * ) n of finite rank r.Given a 1 , . . ., a n ā K * write A(a 1 , . . ., a n , Ī) for the number of solutionsWe derive an explicit upper bound for A(a 1 , . . ., a n , Ī) which depends only on the dimension n and on the rank r.
A linearly ordered structure is weakly o-minimal if all of its definable sets in one variable are the union of finitely many convex sets in the structure. Weakly o-minimal structures ā¦
A linearly ordered structure is weakly o-minimal if all of its definable sets in one variable are the union of finitely many convex sets in the structure. Weakly o-minimal structures were introduced by Dickmann, and they arise in several contexts. We here prove several fundamental results about weakly o-minimal structures. Foremost among these, we show that every weakly o-minimal ordered field is real closed. We also develop a substantial theory of definable sets in weakly o-minimal structures, patterned, as much as possible, after that for o-minimal structures.
A set J of Turing degrees is called an ideal if (1) J ā ā , (2) for any pair of degrees Ć£, , if Ć£, Ļµ J , then Ć£ ā¦
A set J of Turing degrees is called an ideal if (1) J ā ā , (2) for any pair of degrees Ć£, , if Ć£, Ļµ J , then Ć£ ā Ļµ J , and (3) for any ā Ļµ J and any , if < ā, then Ļµ J . A set J of degrees is said to be closed if for any theory T with a set of axioms of degree in J, T has a completion of degree in J . Closed ideals of degrees arise naturally in the following way. If is a recursively saturated structure, let I ( ) = { for some Ä Ļµ }. Let D ( ) = { : is recursive in d -saturated}. (Recursive in d -saturation is defined like recursive saturation except that the sets of formulas considered are recursive in d .) These two sets of degrees were investigated in [2]. It was shown that if is a recursively saturated model of P , Pr = Th(Ļ, +), or Prā² = Th( Z , +, 1), then I ( ) = D ( ), and this set is a closed ideal. Any closed ideal J can be represented as I ( ) = D ( ) for some recursively saturated model of Prā². For sets J of power at most āµ 1 , Prā² can be replaced by P . Assuming CH, all closed ideals have power at most āµ 1 , but if CH fails, there are closed ideals of power greater than āµ 1 , and it is not known whether these can be represented as I ( ) = D ( ) for a recursively saturated model of P . In the present paper, it will first be shown that information about representation of closed ideals provides new information about an old problem of MacDowell and Specker [6] and extends an old result of Scott [8] in a natural way. It will also be shown that the representation results from [2] answer a problem of Friedman [1]. This part of the paper is aimed at convincing the reader that representation problems are worth investigating.
We formulate the measure analogue of generically stable types in first order theories with $NIP$ (without the independence property), giving several characterizations, answering some questions from an earlier paper by ā¦
We formulate the measure analogue of generically stable types in first order theories with $NIP$ (without the independence property), giving several characterizations, answering some questions from an earlier paper by Hrushovski and Pillay, and giving another treatment of uniqueness results from the same paper. We introduce a notion of "generic compact domination", relating it to stationarity of the Keisler measures, and also giving definable group versions. We also prove the "approximate definability" of arbitrary Borel probability measures on definable sets in the real and $p$-adic fields.
If is a countable recursively saturated structure and T is a recursively axiomatizable theory that is consistent with Th( ), then it is well known that can be expanded to ā¦
If is a countable recursively saturated structure and T is a recursively axiomatizable theory that is consistent with Th( ), then it is well known that can be expanded to a recursively saturated model of T [7, p. 186]. This is what has made recursively saturated models useful in model theory. Recursive saturation is the weakest notion of saturation for which this expandability result holds. In fact, if is a countable model of Pr = Th(Ļ, +), then can be expanded to a model of first order Peano arithmetic P just in case is recursively saturated (see [3]). In this paper we investigate two natural sets of Turing degrees that tell a good deal about the expandability of a given structure. If is a recursively saturated structure, I ( ) consists of the degrees of sets that are recursive in complete types realized in . The second set of degrees, D ( ), consists of the degrees of sets S such that is recursive in S -saturated. In general, I ( ) ā D ( ). Moreover, I ( ) is obviously an āidealā of degrees. For countable structures , D ( ) is āclosedā in the following sense: For any class C ā 2 Ļ , if C is co-r.e. in S for some set S such that , then there is some Ļ ā C such that . For uncountable structures , we do not know whether D ( ) must be closed.
For random graph theorists (see, e.g., Bollobas [1] for general reference) p any constant is not the only, not even the most interesting case. Rather, they consider p = p(n), ā¦
For random graph theorists (see, e.g., Bollobas [1] for general reference) p any constant is not the only, not even the most interesting case. Rather, they consider p = p(n), a function approaching zero. In their seminal paper, Erd6s and Renyi [5] showed that for many interesting A there is a function p(n), which they called a threshold function, so that if r(n) < p(n) then f(n, r(n), A) -+ 0 while if p(n) < r(n) then f(n, r(n), A) 1-+ . (Notation: p < r means lim p/r = 0. All limits are as n approaches infinity.) Let us say p = p(n) satisfies the Zero-One Law if for all A in GRA, limf(n, p, A) = 0 or 1. We shall partially characterize those p = p(n) which satisfy the Zero-One Law.