Almost strongly minimal theories. II

1972 ◽  
Vol 37 (4) ◽  
pp. 657-660 ◽  
Author(s):  
John T. Baldwin
Keyword(s):  
Elementary Theory ◽  
Intersection Property ◽  
Stone Space ◽  
Sufficient Condition ◽  
Categorical Theory ◽  
Elementary Submodels ◽  
Minimal Theory ◽  
Elementary Submodel

The notion of an almost strongly minimal theory was introduced in [1]. Such a theory is a particularly simple sort of ℵ1-categorical theory. In [1] we characterized this simplicity in terms of the Stone space of models of T. Here, we characterize almost strongly minimal theories which are not ℵ0-categorical in terms of D. M. R. Park's notion [4] of a theory with the strong elementary intersection property. In addition we prove a useful sufficient condition for an elementary theory to be an almost strongly minimal theory. Our notation is from [1] but this paper is independent of the results proved there. We do assume familiarity with §1 and §2 of [2].In [4], Park defines a theory T to have the strong elementary intersection property (s.e.i.p.) if for each model C of T and each pair of elementary submodels of C either is an elementary submodel of C. T has the nontrivial strong elementary intersection property (n.s.e.i.p.) if for each triple C, as above Park proves the following two statements equivalent:

scholarly journals On biorthogonal systems and Maxur's intersection property

2004 ◽  
Vol 69 (1) ◽  
pp. 107-111 ◽  
Author(s):  
Jan Rychtář
Keyword(s):  
Banach Spaces ◽  
Intersection Property ◽  
Sufficient Condition ◽  
Biorthogonal Systems ◽  
Markushevich Basis ◽  
Mazur's Intersection Property

We give a characterisation of Banach spaces X containing a subspace with a shrinking Markushevich basis {xγ, fγ}γ∈Γ. This gives a sufficient condition for X to have a renorming with Mazur's intersection property.

Unique decomposition in classifiable theories

2002 ◽  
Vol 67 (1) ◽  
pp. 61-68
Author(s):  
Bradd Hart ◽  
Ehud Hrushovski ◽  
Michael C. Laskowski
Keyword(s):  
Stability Theory ◽  
Saturated Model ◽  
Prime Model ◽  
Order Property ◽  
Elementary Submodels ◽  
Saturated Models ◽  
Unique Decomposition ◽  
Elementary Submodel

By a classifiable theory we shall mean a theory which is superstable, without the dimensional order property, which has prime models over pairs. In order to define what we mean by unique decomposition, we remind the reader of several definitions and results. We adopt the usual conventions of stability theory and work inside a large saturated model of a fixed classifiable theory T; for instance, if we write M ⊆ N for models of T, M and N we are thinking of these models as elementary submodels of this fixed saturated models; so, in particular, M is an elementary submodel of N. Although the results will not depend on it, we will assume that T is countable to ease notation.We do adopt one piece of notation which is not completely standard: if T is classifiable, M0 ⊆ Mi for i = 1, 2 are models of T and M1 is independent from M2 over M0 then we write M1M2 for the prime model over M1 ∪ M2.

Non Σn axiomatizable almost strongly minimal theories

1989 ◽  
Vol 54 (3) ◽  
pp. 921-927 ◽  
Author(s):  
David Marker
Keyword(s):  
Natural Number ◽  
Algebraic Closure ◽  
Predicate Symbol ◽  
Elementary Extension ◽  
Minimal Set ◽  
Categorical Theory ◽  
Minimal Theory ◽  
Axiomatizable Theory ◽  
Definition Of ◽  
Infinite Power

Recall that a theory is said to be almost strongly minimal if in every model every element is in the algebraic closure of a strongly minimal set. In 1970 Hodges and Macintyre conjectured that there is a natural number n such that every ℵ0-categorical almost strongly minimal theory is Σn axiomatizable. Recently Ahlbrandt and Baldwin [A-B] proved that if T is ℵ0-categorical and almost strongly minimal, then T is Σn axiomatizable for some n. This result also follows from Ahlbrandt and Ziegler's results on quasifinite axiomatizability [A-Z]. In this paper we will refute Hodges and Macintyre's conjecture by showing that for each n there is an ℵ0-categorical almost strongly minimal theory which is not Σn axiomatizable.Before we begin we should note that in all these examples the complexity of the theory arises from the complexity of the definition of the strongly minimal set. It is still open whether the conjecture is true if we allow a predicate symbol for the strongly minimal set.We will prove the following result.Theorem. For every n there is an almost strongly minimal ℵ0-categorical theory T with models M and N such that N is Σn elementary but not Σn + 1 elementary.To show that these theories yield counterexamples to the conjecture we apply the following result of Chang [C].Theorem. If T is a Σn axiomatizable theory categorical in some infinite power, M and N are models of T and N is a Σn elementary extension of M, then N is an elementary extension of M.

The real line in elementary submodels of set theory

2000 ◽  
Vol 65 (2) ◽  
pp. 683-691 ◽  
Author(s):  
Kenneth Kunen ◽  
Franklin D. Tall
Keyword(s):  
Set Theory ◽  
Transitive Closure ◽  
Topological Spaces ◽  
Linear Orders ◽  
Combinatorial Objects ◽  
Elementary Submodels ◽  
Standard Tool ◽  
Elementary Submodel ◽  
Skolem Theorem ◽  
The Universe

The use of elementary submodels has become a standard tool in set-theoretic topology and infinitary combinatorics. Thus, in studying some combinatorial objects, one embeds them in a set, M, which is an elementary submodel of the universe, V (that is, (M; Є) ≺ (V; Є)). Applying the downward Löwenheim-Skolem Theorem, one can bound the cardinality of M. This tool enables one to capture various complicated closure arguments within the simple “≺”.However, in this paper, as in the paper [JT], we study the tool for its own sake. [JT] discussed various general properties of topological spaces in elementary submodels. In this paper, we specialize this consideration to the space of real numbers, ℝ. Our models M are not in general transitive. We will always have ℝ Є M, but not usually ℝ ⊆ M. We plan to study properties of the ℝ ⋂ M's. In particular, as M varies, we wish to study whether any two of these ℝ ⋂ M's are isomorphic as topological spaces, linear orders, or fields.As usual, it takes some sleight-of-hand to formalize these notions within the standard axioms of set theory (ZFC), since within ZFC, one cannot actually define the notion (M;Є) ≺ (V;Є). Instead, one proves theorems about M such that (M;Є) ≺ (H(θ);Є), where θ is a “large enough” cardinal; here, H(θ) is the collection of all sets whose transitive closure has size less than θ.

The ⊰-order on submodels

1976 ◽  
Vol 41 (1) ◽  
pp. 215-221
Author(s):  
Leo Marcus
Keyword(s):  
Large Class ◽  
Linear Order ◽  
Partial Ordering ◽  
Natural Order ◽  
Prime Model ◽  
Natural Question ◽  
Categorical Theory ◽  
Elementary Submodels ◽  
Homogeneous Models ◽  
Element Theorem

The relation M1 ⊰ M2, where M1 and M2 are elementary submodels of a given model M, defines a partial ordering ⊰(M). A natural question is: What are the general properties of this ordering and what special properties can it have for given M? Here we give an example of a prime model M for which ⊰(M) is a dense linear order, in particular ⊰(M) is isomorphic to the natural order of (− ∞, ∞], the reals with a last element (Theorem 1). In fact, this is the only way for ⊰(M) to be a linear order when M is prime (Theorem 3).We then remark that if N is a model of an ℵ0-categorical theory then ⊰(N) cannot be a linear order, but we give an example where ⊰(N) is a dense partial order, N a model of an ℵ0-categorical theory.Independently, and about the same time, Benda [1] found nonprime models M where ⊰(M) is isomorphic to any one of a large class of orders, including (− ∞, ∞]. His methods are very similar to ours.Notation. We assume the reader is familar with Vaught [2] where he will find the definitions and standard results pertaining to prime models, homogeneous models, and ℵ0-categorical theories.

New Characterizations of the Reflexivity in Terms of the Set of Norm Attaining Functionals

1998 ◽  
Vol 41 (3) ◽  
pp. 279-289 ◽  
Author(s):  
Mariá D. Acosta ◽  
Manuel Ruiz Galán
Keyword(s):  
Banach Space ◽  
Positive Result ◽  
Unit Ball ◽  
Separable Banach Space ◽  
Intersection Property ◽  
Sufficient Condition ◽  
Empty Interior ◽  
Norm Topology ◽  
Fréchet Differentiable ◽  
Norm Attaining

AbstractAs a consequence of results due to Bourgain and Stegall, on a separable Banach space whose unit ball is not dentable, the set of norm attaining functionals has empty interior (in the norm topology). First we show that any Banach space can be renormed to fail this property. Then, our main positive result can be stated as follows: if a separable Banach space X is very smooth or its bidual satisfies the w*-Mazur intersection property, then either X is reflexive or the set of norm attaining functionals has empty interior, hence the same result holds if X has the Mazur intersection property and so, if the norm of X is Fréchet differentiable. However, we prove that smoothness is not a sufficient condition for the same conclusion.

Finite inseparability of some theories of cylindrification algebras

1969 ◽  
Vol 34 (2) ◽  
pp. 171-176 ◽  
Author(s):  
Stephen D. Comer
Keyword(s):  
Elementary Theory ◽  
The Other ◽  
Sufficient Condition ◽  
The One

An elementary theory T in a language L is (strongly) finitely inseparable if the set of logically valid sentences of L and the set of T-finitely refutable sentences are recursively inseparable. In §1 we establish a sufficient condition for the elementary theory of a class of BA's with operators to be finitely inseparable. This is done using the methods developed independently by M. Rabin and D. Scott (see [6]) on the one hand and by Ershov on the other (see [2]).

scholarly journals THE STABILITY SPECTRUM FOR CLASSES OF ATOMIC MODELS

2012 ◽  
Vol 12 (01) ◽  
pp. 1250001 ◽  
Author(s):  
JOHN T. BALDWIN ◽  
SAHARON SHELAH
Keyword(s):  
Positive Integer ◽  
Stone Space ◽  
Sufficient Condition ◽  
Stable Case ◽  
Stability Spectrum ◽  
Atomic Models ◽  
Unstable Case ◽  
The Stability

We prove two results on the stability spectrum for Lω1,ω. Here [Formula: see text] denotes an appropriate notion (at or mod) of Stone space of m-types over M. (1) Theorem for unstable case: Suppose that for some positive integer m and for every α < δ(T), there is an M ∈ K with [Formula: see text]. Then for every λ ≥ |T|, there is an M with [Formula: see text]. (2) Theorem for strictly stable case: Suppose that for every α < δ (T), there is Mα ∈ K such that λα = |Mα| ≥ ℶα and [Formula: see text]. Then for any μ with μℵ0 > μ, K is not i-stable in μ. These results provide a new kind of sufficient condition for the unstable case and shed some light on the spectrum of strictly stable theories in this context. The methods avoid the use of compactness in the theory under study. In this paper, we expound the construction of tree indiscernibles for sentences of Lω1,ω. Further we provide some context for a number of variants on the Ehrenfeucht–Mostowski construction.

The Current Situation in Chemical Stabilization of Biological Materials

1972 ◽  
Vol 30 ◽  
pp. 132-133
Author(s):  
John H. Luft
Keyword(s):  
Electron Microscopy ◽  
Electron Microscope ◽  
Osmium Tetroxide ◽  
Molecular Level ◽  
Cross Linking ◽  
Chemical Stabilization ◽  
Specimen Preparation ◽  
Sufficient Condition ◽  
Radio Telescopes

With information processing devices such as radio telescopes, microscopes or hi-fi systems, the quality of the output often is limited by distortion or noise introduced at the input stage of the device. This analogy can be extended usefully to specimen preparation for the electron microscope; fixation, which initiates the processing sequence, is the single most important step and, unfortunately, is the least well understood. Although there is an abundance of fixation mixtures recommended in the light microscopy literature, osmium tetroxide and glutaraldehyde are favored for electron microscopy. These fixatives react vigorously with proteins at the molecular level. There is clear evidence for the cross-linking of proteins both by osmium tetroxide and glutaraldehyde and cross-linking may be a necessary if not sufficient condition to define fixatives as a class.

Sign in / Sign up

Export Citation Format

Share Document