Examples in the theory of existential completeness

1978 ◽  
Vol 43 (4) ◽  
pp. 650-658
Author(s):  
Joram Hirschfeld
Keyword(s):  
Saturated Model ◽  
Prime Model ◽  
Model Companion ◽  
Generic Models ◽  
Universal Formula ◽  
Categorical Model ◽  
Existential Formula ◽  
Elementary Classes ◽  
Class C ◽  
Countable Theory

Since A. Robinson introduced the classes of existentially complete and generic models, conditions which were interesting for elementary classes were considered for these classes. In [6] H. Simmons showed that with the natural definitions there are prime and saturated existentially complete models and these are very similar to their elementary counterparts which were introduced by Vaught [2, 2.3]. As Example 6 will show, there is a limit to the similarity—there are theories which have exactly two existentially complete models.In [6] H. Simmons considers the following list of properties, shows that each property implies the next one and asks whether any of them implies the previous one:1.1. T is ℵ0-categorical.1.2. T has an ℵ0-categorical model companion.1.3. ∣E∣ = 1.1.4. ∣E∣ < .1.5. T has a countable ∃-saturated model.1.6. T has a ∃-prime model.1.7. Each universal formula is implied by a ∃-atomic existential formula.[The reader is referred to [1], [3], [4] and [6] for the definitions and background.We only mention that T is always a countable theory. All the models under discussion are countable. Thus E is the class of countable existentially complete models and F and G, respectively, are the classes of countable finite and infinite generic models. For every class C,∣C∣ is the number of (countable) models in C.]

Recursively presentable prime models

1974 ◽  
Vol 39 (2) ◽  
pp. 305-309 ◽  
Author(s):  
Leo Harrington
Keyword(s):  
Decision Procedure ◽  
Positive Answer ◽  
Prime Model ◽  
Property A ◽  
Categorical Theory ◽  
Decidable Theory ◽  
Free Variables ◽  
Closed Fields ◽  
Countable Theory ◽  
Algebraically Closed Fields

It is well known that a decidable theory possesses a recursively presentable model. If a decidable theory also possesses a prime model, it is natural to ask if the prime model has a recursive presentation. This has been answered affirmatively for algebraically closed fields [5], and for real closed fields, Hensel fields and other fields [3]. This paper gives a positive answer for the theory of differentially closed fields, and for any decidable ℵ1-categorical theory.The language of a theory T is denoted by L(T). All languages will be presumed countable. An x-type of T is a set of formulas with free variables x, which is consistent with T and which is maximal in this property. A formula with free variables x is complete if there is exactly one x-type containing it. A type is principal if it contains a complete formula. A countable model of T is prime if it realizes only principal types. Vaught has shown that a complete countable theory can have at most one prime model up to isomorphism.If T is a decidable theory, then the decision procedure for T equips L(T) with an effective counting. Thus the formulas of L(T) correspond to integers. The integer a formula φ(x) corresponds to is generally called the Gödel number of φ(x) and is denoted by ⌜φ(x)⌝. The usual recursion theoretic notions defined on the set of integers can be transferred to L(T). In particular a type Γ is recursive with index e if {⌜φ⌝.; φ ∈ Γ} is a recursive set of integers with index e.

scholarly journals INDEPENDENCE IN GENERIC INCIDENCE STRUCTURES

2019 ◽  
Vol 84 (02) ◽  
pp. 750-780
Author(s):  
GABRIEL CONANT ◽  
ALEX KRUCKMAN
Keyword(s):  
Algebraic Independence ◽  
Incidence Structure ◽  
Projective Planes ◽  
Saturated Model ◽  
Prime Model ◽  
Incidence Structures ◽  
Finite Projective Planes ◽  
Existentially Closed ◽  
Image Position ◽  
Open Question

AbstractWe study the theory Tm,n of existentially closed incidence structures omitting the complete incidence structure Km,n, which can also be viewed as existentially closed Km,n-free bipartite graphs. In the case m = n = 2, this is the theory of existentially closed projective planes. We give an $\forall \exists$-axiomatization of Tm,n, show that Tm,n does not have a countable saturated model when m, n ≥ 2, and show that the existence of a prime model for T2,2 is equivalent to a longstanding open question about finite projective planes. Finally, we analyze model theoretic notions of complexity for Tm,n. We show that Tm,n is NSOP1, but not simple when m, n ≥ 2, and we show that Tm,n has weak elimination of imaginaries but not full elimination of imaginaries. These results rely on combinatorial characterizations of various notions of independence, including algebraic independence, Kim independence, and forking independence.

A counterexample in the theory of model companions

1975 ◽  
Vol 40 (1) ◽  
pp. 31-34 ◽  
Author(s):  
D. Saracino
Keyword(s):  
Complete Model ◽  
Complete Structure ◽  
Model Companion ◽  
Categorical Theory ◽  
Generic Structure ◽  
Finite Models ◽  
Infinite Power ◽  
Countable Theory ◽  
Generic Structures

In [7] we proved that (I) if T is a countable ℵ0-categorical theory without finite models then T has a model companion; and several people have observed that (II) if T is a countable theory without finite models which is ℵ1-categorical and forcingcomplete for infinite forcing (i.e., T= TF) then T is model-complete. It is natural to ask (1) whether in (I) we can replace ℵ0 by ℵ1; (2) whether in (II) we can replace TF by Tf; and (3) in connection with (II), whether the categoricity of the class of infinitely generic structures for a theory K in some or all infinite powers implies the existence of a model companion for K. The purpose of this note is to provide negative answers to (1), (2), and (3). Specifically, we will prove:Theorem. There exists a countable theory T such that(i) T has no finite models and is ℵ-categorical;(ii) T is forcing-complete for finite forcing, i.e., T = Tf;(iii) T has no model companion (i.e., in light of (ii), T is not model-complete);(iv) the class of infinitely generic structures for T is categorical in every infinite power;(v) every uncountable existentially complete structure for T is infinitely generic;(vi) there is, up to isomorphism, precisely one countable existentially complete model of Tf, and there are no uncountable e.c. models of Tf (in particular, there is just one countable finitely generic structure and there are no uncountable ones);(vii) there are precisely ℵ0isomorphism types of countable existentially complete structures for T.

Existentially complete torsion-free nilpotent groups

1978 ◽  
Vol 43 (1) ◽  
pp. 126-134 ◽  
Author(s):  
D. Saracino
Keyword(s):  
Free Groups ◽  
Nilpotent Groups ◽  
Generic Model ◽  
Model Companion ◽  
Generic Models ◽  
Direct Sum ◽  
Carry Over ◽  
Torsion Free

This paper continues the study of existentially complete nilpotent groups initiated in [6]. Following [6], we let Kn denote the theory of groups nilpotent of class ≤ n and let Kn+ denote the theory of torsion-free groups nilpotent of class ≤ n. The principal results of [6] were that for n ≥ 2, neither Kn nor Kn+ has a model companion, and the classes E, F, and G of existentially complete, finitely generic and infinitely generic models of Kn are all distinct. The question of the relationships between these classes in the context of Kn was left open, however, and the proof of their distinctness for Kn+ obviously did not carry over to Kn+, because it made strong use of torsion elements.In this paper we establish the relationships between E, F, and G for K2+. We show that all three classes are distinct. We also show that there is only one countable finitely generic model, and only one countable infinitely generic model, and that all the countable existentially complete models can be arranged in a sequence N1 ⊆ N2 ⊆ N3 ⊆ … ⊆ Nω, where Z(Nn) is the direct sum of n copies of Q. Another result is that the finite and infinite forcing companions of K2+ differ by an ∀∃∀ sentence. Finally, we show that there exist finitely generic models of K2+ in all infinite cardinalities.

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.

Number of countable models

1978 ◽  
Vol 43 (3) ◽  
pp. 492-496 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Finite Number ◽  
Complete Theory ◽  
Prime Model ◽  
Categorical Theory ◽  
Definable Subset ◽  
Algebraic Elements ◽  
Minimal Prime ◽  
Countable Theory ◽  
The Universe ◽  
Countable Models

We prove that a countable complete theory whose prime model has an infinite definable subset, all of whose elements are named, has at least four countable models up to isomorphism. The motivation for this is the conjecture that a countable theory with a minimal model has infinitely many countable models. In this connection we first prove that a minimal prime model A has an expansion by a finite number of constants A′ such that the set of algebraic elements of A′ contains an infinite definable subset.We note that our main conjecture strengthens the Baldwin–Lachlan theorem. We also note that due to Vaught's result that a countable theory cannot have exactly two countable models, the weakest possible nontrivial result for a non-ℵ0-categorical theory is that it has at least four countable models.§1. Notation and preliminaries. Our notation follows Chang and Keisler [1], except that we denote models by A, B, etc. We use the same symbol to refer to the universe of a model. Models we refer to are always in a countable language. For T a countable complete theory we let n(T) be the number of countable models of T up to isomorphism. ∃n means ‘there are exactly n’.

Automorphisms moving all non-algebraic points and an application to NF

1998 ◽  
Vol 63 (3) ◽  
pp. 815-830 ◽  
Author(s):  
Friederike Körner
Keyword(s):  
Set Theory ◽  
Sufficient Conditions ◽  
Saturated Model ◽  
Natural Numbers ◽  
Algebraic Point ◽  
Complete Theories ◽  
Necessary And Sufficient ◽  
Countable Theory ◽  
Recursively Saturated ◽  
Recursively Saturated Model

AbstractSection 1 is devoted to the study of countable recursively saturated models with an automorphism moving every non-algebraic point. We show that every countable theory has such a model and exhibit necessary and sufficient conditions for the existence of automorphisms moving all non-algebraic points. Furthermore we show that there are many complete theories with the property that every countable recursively saturated model has such an automorphism.In Section 2 we apply our main theorem from Section 1 to models of Quine's set theory New Foundations (NF) to answer an old consistency question. If NF is consistent, then it has a model in which the standard natural numbers are a definable subclass ℕ of the model's set of internal natural numbers Nn. In addition, in this model the class of wellfounded sets is exactly .

Surgical Management of Class C and D Glomus Jugulare Tumors: Fisch Infratemporal Fossa Type A Approach

2020 ◽  
Author(s):  
Yin Xia ◽  
Yubin Xue ◽  
Ting Ye ◽  
Xiaopeng Qu ◽  
Xukun Yan ◽  
...  
Keyword(s):  
Surgical Management ◽  
Infratemporal Fossa ◽  
Type A ◽  
Glomus Jugulare ◽  
Class C

USING THE TECHNOLOGY OF DEVELOPING CRITICAL THINKING AT UKRAINIAN LESSONS IN ELEMENTARY CLASSES

2018 ◽  
Vol 2 (21) ◽  
pp. 51-56
Author(s):  
S.M. Dubiaha ◽  
◽  
Y.O. Saienko ◽  
V.I. Dubiaha ◽  
◽  
...  
Keyword(s):  
Critical Thinking ◽  
Elementary Classes

A Constant-Current-Controlled Class-C Voltage-Controlled Oscillator using Self-Adjusting Replica Bias Circuit

2015 ◽  
Vol E98.C (6) ◽  
pp. 471-479
Author(s):  
Teerachot SIRIBURANON ◽  
Wei DENG ◽  
Kenichi OKADA ◽  
Akira MATSUZAWA
Keyword(s):  
Constant Current ◽  
Voltage Controlled Oscillator ◽  
Class C
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