ON THE NUMBER OF COUNTABLE MODELS OF A COUNTABLE NSOP1 THEORY WITHOUT WEIGHT ω

2019 ◽  
Vol 84 (3) ◽  
pp. 1168-1175
Author(s):  
BYUNGHAN KIM
Keyword(s):  
Image Position ◽  
Countable Models

AbstractIn this article, we prove that if a countable non-${\aleph _0}$-categorical NSOP1 theory with nonforking existence has finitely many countable models, then there is a finite tuple whose own preweight is ω. This result is an extension of a theorem of the author on any supersimple theory.

Refinements of Vaught's normal from theorem

1979 ◽  
Vol 44 (3) ◽  
pp. 289-306 ◽  
Author(s):  
Victor Harnik
Keyword(s):  
Normal Form ◽  
Boolean Combination ◽  
Form Theorem ◽  
Central Notion ◽  
The Matrix ◽  
Normal Form Theorem ◽  
Skolem Theorem ◽  
Image Position ◽  
Countable Models

The central notion of this paper is that of a (conjunctive) game-sentence, i.e., a sentence of the formwhere the indices ki, ji range over given countable sets and the matrix conjuncts are, say, open -formulas. Such game sentences were first considered, independently, by Svenonius [19], Moschovakis [13]—[15] and Vaught [20]. Other references are [1], [3]—[5], [10]—[12]. The following normal form theorem was proved by Vaught (and, in less general forms, by his predecessors).Theorem 0.1. Let L = L0(R). For every -sentence ϕ there is an L0-game sentence Θ such that ⊨′ ∃Rϕ ↔ Θ.(A word about the notations: L0(R) denotes the language obtained from L0 by adding to it the sequence R of logical symbols which do not belong to L0; “⊨′α” means that α is true in all countable models.)0.1 can be restated as follows.Theorem 0.1′. For every-sentence ϕ there is an L0-game sentence Θ such that ⊨ϕ → Θ and for any-sentence ϕ if ⊨ϕ → ϕ and L′ ⋂ L ⊆ L0, then ⊨ Θ → ϕ.(We sketch the proof of the equivalence between 0.1 and 0.1′.0.1 implies 0.1′. This is obvious once we realize that game sentences and their negations satisfy the downward Löwenheim-Skolem theorem and hence, ⊨′α is equivalent to ⊨α whenever α is a boolean combination of and game sentences.

scholarly journals COUNTABLE MODELS OF THE THEORIES OF BALDWIN–SHI HYPERGRAPHS AND THEIR REGULAR TYPES

2019 ◽  
Vol 84 (3) ◽  
pp. 1007-1019
Author(s):  
DANUL K. GUNATILLEKA
Keyword(s):  
Large Body ◽  
Countable Model ◽  
Stable Theory ◽  
Generic Structure ◽  
Elementary Chain ◽  
Regular Type ◽  
Image Position ◽  
Original Class ◽  
Countable Models

AbstractWe continue the study of the theories of Baldwin–Shi hypergraphs from [5]. Restricting our attention to when the rank δ is rational valued, we show that each countable model of the theory of a given Baldwin–Shi hypergraph is isomorphic to a generic structure built from some suitable subclass of the original class used in the construction. We introduce a notion of dimension for a model and show that there is a an elementary chain $\left\{ {\mathfrak{M}_\beta :\beta \leqslant \omega } \right\}$ of countable models of the theory of a fixed Baldwin–Shi hypergraph with $\mathfrak{M}_\beta \preccurlyeq \mathfrak{M}_\gamma $ if and only if the dimension of $\mathfrak{M}_\beta $ is at most the dimension of $\mathfrak{M}_\gamma $ and that each countable model is isomorphic to some $\mathfrak{M}_\beta $. We also study the regular types that appear in these theories and show that the dimension of a model is determined by a particular regular type. Further, drawing on a large body of work, we use these structures to give an example of a pseudofinite, ω-stable theory with a nonlocally modular regular type, answering a question of Pillay in [11].

Generalized quantifiers and elementary extensions of countable models

1977 ◽  
Vol 42 (3) ◽  
pp. 341-348 ◽  
Author(s):  
Małgorzata Dubiel
Keyword(s):  
Countable Model ◽  
Generalized Quantifiers ◽  
Natural Question ◽  
Elementary Extension ◽  
First Order ◽  
Transitive Model ◽  
Order Language ◽  
Image Position ◽  
Countable Models

Let L be a countable first-order language and L(Q) be obtained by adjoining an additional quantifier Q. Q is a generalization of the quantifier “there exists uncountably many x such that…” which was introduced by Mostowski in [4]. The logic of this latter quantifier was formalized by Keisler in [2]. Krivine and McAloon [3] considered quantifiers satisfying some but not all of Keisler's axioms. They called a formula φ(x) countable-like iffor every ψ. In Keisler's logic, φ(x) being countable-like is the same as ℳ⊨┐Qxφ(x). The main theorem of [3] states that any countable model ℳ of L[Q] has an elementary extension N, which preserves countable-like formulas but no others, such that the only sets definable in both N and M are those defined by formulas countable-like in M. Suppose C(x) in M is linearly ordered and noncountable-like but with countable-like proper segments. Then in N, C will have new elements greater than all “old” elements but no least new element — otherwise it will be definable in both models. The natural question is whether it is possible to use generalized quantifiers to extend models elementarily in such a way that a noncountable-like formula C will have a minimal new element. There are models and formulas for which it is not possible. For example let M be obtained from a minimal transitive model of ZFC by letting Qxφ(x) mean “there are arbitrarily large ordinals satisfying φ”.

Bad models in nice neighborhoods

1986 ◽  
Vol 51 (4) ◽  
pp. 1043-1055 ◽  
Author(s):  
Terry Millar
Keyword(s):  
Linear Order ◽  
Homogeneous Model ◽  
Order Type ◽  
Saturated Model ◽  
Countable Number ◽  
Decidable Theory ◽  
Type Spectrum ◽  
Image Position ◽  
Countable Models

This paper contains an example of a decidable theory which has1) only a countable number of countable models (up to isomorphism);2) a decidable saturated model; and3) a countable homogeneous model that is not decidable.By the results in [1] and [2], this can happen if and only if the set of types realized by the homogeneous model (the type spectrum of the model) is not .If Γ and Σ are types of a theory T, define Γ ◁ Σ to mean that any model of T realizing Γ must realize Σ. In [3] a decidable theory is constructed that has only countably many countable models, only recursive types, but whose countable saturated model is not decidable. This is easy to do if the restriction on the number of countable models is lifted; the difficulty arises because the set of types must be recursively complex, and yet sufficiently related to control the number of countable models. In [3] the desired theory T is such thatis a linear order with order type ω*. Also, the set of complete types of T is not . The last feature ensures that the countable saturated model is not decidable; the first feature allows the number of countable models to be controlled.

scholarly journals ALMOST GALOIS ω-STABLE CLASSES

2015 ◽  
Vol 80 (3) ◽  
pp. 763-784 ◽  
Author(s):  
JOHN T. BALDWIN ◽  
PAUL B. LARSON ◽  
SAHARON SHELAH
Keyword(s):  
Elementary Class ◽  
Joint Embedding ◽  
Image Position ◽  
Abstract Elementary Class ◽  
Countable Models

AbstractTheorem. Suppose that k = (K, $$\prec_k$$) is an ℵ0-presentable abstract elementary class with Löwenheim–Skolem number ℵ0, satisfying the joint embedding and amalgamation properties in ℵ0. If K has only countably many models in ℵ1, then all are small. If, in addition, k is almost Galois ω-stable then k is Galois ω-stable. Suppose that k = (K, $$\prec_k$$) is an ℵ0-presented almost Galois ω-stable AEC satisfying amalgamation for countable models, and having a model of cardinality ℵ1. The assertion that K is ℵ1-categorical is then absolute.

On Cantor-Bendixson spectra containing (1,1). II

1985 ◽  
Vol 50 (3) ◽  
pp. 611-618 ◽  
Author(s):  
Annalisa Marcja ◽  
Carlo Toffalori
Keyword(s):  
Boolean Algebras ◽  
Stable Theory ◽  
Definable Subset ◽  
Limit Ordinal ◽  
Image Position ◽  
Countable Models

Let T be a (countable, complete, quantifier eliminable) ω-stable theory; an analysis of T, and consequently a classification of ω-stable theories, can be done by looking at the Boolean algebras B(M) of definable subsets of its countable models M (as usual, we often confuse a definable subset of M with the class of formulas defining it). If M ⊨ T, ∣M∣ = ℵ0, then, for every LM-formula ϕ(v) and for every ordinal α, we define a relation(CB = Cantor-Bendixson, of course) by induction on α:CB-rank ϕ(v) ≥ 0 if ϕ(M) ≠ ∅CB-rank ϕ(v) ≥ λ for λ a limit ordinal, if CB-rank ϕ(v) ≥ for all v < λ;CB-rank ϕ(v)≥ α + 1 if, for all n ∈ ω,(*) there are LM-formulas ϕ0(v), …, ϕn − 1(v) such thatIt is well known that the ω-stability of T implies that, for every consistent LM-formula ϕ(v), there is exactly one ordinal α < ω1 such that CB-rank ϕ(v) ≥ α and CB-rank ϕ(v)≱α + 1. Therefore we define:CB-rank ϕ(v) = αCB-degree ϕ(v) = d if d is the maximal n ∈ ω satisfying (*); andCB-type ϕ(v) = (α, d).

On Scott and Karp trees of uncountable models

1990 ◽  
Vol 55 (3) ◽  
pp. 897-908 ◽  
Author(s):  
Tapani Hyttinen ◽  
Jouko Väänänen
Keyword(s):  
Independent Interest ◽  
Relational Models ◽  
First Order ◽  
Countable Ordinal ◽  
Order Language ◽  
Similar Relation ◽  
Image Position ◽  
Countable Models

AbstractLet 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 thati.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.

scholarly journals THE BOREL COMPLEXITY OF ISOMORPHISM FOR O-MINIMAL THEORIES

2017 ◽  
Vol 82 (2) ◽  
pp. 453-473 ◽  
Author(s):  
RICHARD RAST ◽  
DAVENDER SINGH SAHOTA
Keyword(s):  
Lower Bounds ◽  
General Model ◽  
Isomorphism Problem ◽  
Linear Orders ◽  
Minimal Theory ◽  
Borel Complexity ◽  
Image Position ◽  
Countable Models

AbstractGiven a countable o-minimal theory T, we characterize the Borel complexity of isomorphism for countable models of T up to two model-theoretic invariants. If T admits a nonsimple type, then it is shown to be Borel complete by embedding the isomorphism problem for linear orders into the isomorphism problem for models of T. This is done by constructing models with specific linear orders in the tail of the Archimedean ladder of a suitable nonsimple type.If the theory admits no nonsimple types, then we use Mayer’s characterization of isomorphism for such theories to compute invariants for countable models. If the theory is small, then the invariant is real-valued, and therefore its isomorphism relation is smooth. If not, the invariant corresponds to a countable set of reals, and therefore the isomorphism relation is Borel equivalent to F2.Combining these two results, we conclude that $\left( {{\rm{Mod}}\left( T \right), \cong } \right)$ is either maximally complicated or maximally uncomplicated (subject to completely general model-theoretic lower bounds based on the number of types and the number of countable models).

Simple Identification of Electron Diffraction Ring Patterns

1969 ◽  
Vol 27 ◽  
pp. 160-161
Author(s):  
Carolyn Nohr ◽  
Ann Ayres
Keyword(s):  
Electron Microscope ◽  
Electron Diffraction ◽  
Diffraction Ring ◽  
Image Position

Texts on electron diffraction recommend that the camera constant of the electron microscope be determine d by calibration with a standard crystalline specimen, using the equation

Atomic orientation effects in proton stopping at low velocity

1983 ◽  
Vol 41 ◽  
pp. 708-709
Author(s):  
Kin Lam
Keyword(s):  
Polycrystalline Material ◽  
Charged Particles ◽  
Target Material ◽  
Stopping Power ◽  
Low Velocity ◽  
Electronic Density ◽  
Interatomic Distances ◽  
Frame Work ◽  
Image Position ◽  
Atomic Orientation

The energy of moving ions in solid is dependent on the electronic density as well as the atomic structural properties of the target material. These factors contribute to the observable effects in polycrystalline material using the scanning ion microscope. Here we outline a method to investigate the dependence of low velocity proton stopping on interatomic distances and orientations.The interaction of charged particles with atoms in the frame work of the Fermi gas model was proposed by Lindhard. For a system of atoms, the electronic Lindhard stopping power can be generalized to the formwhere the stopping power function is defined as

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