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.

Teks Anekdot dalam Cerita Lisan Yong Dollah Pewarisan Orang Melayu Sebagai Alternatif Pemilihan Bahan Ajar Bahasa Indonesia

GERAM ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 30-43
Author(s):  
Muhammad Mukhlis ◽  
Asnawi Asnawi
Keyword(s):  
Content Analysis ◽  
Research Method ◽  
The Other ◽  
Teaching Material ◽  
Generic Structure ◽  
Alternative Material ◽  
Descriptive Approach ◽  
Alternative Choice ◽  
Generic Structures ◽  
Bahasa Indonesia

This research is entitled as "Anecdotal Text in the Oral Story of Yong Dollah Inheritance of Malays as Alternative Choice for Indonesian Language Teaching Materials". It is inspired by the collection of Yong Dollah stories as the inheritance of Malays in Bengkalis Regency which contain of humor elements. In addition, the stories have the same characteristics with anecdotal text, so that it can be applied as teaching material for Indonesia Language subject in the school. This research method was content analysis of descriptive approach. This research was conducted during six months. The technique used to collect data were documentation and interview. The data of this study were the entire generic structure and language features of anecdotal texts contained in a collection of Yong Dollah stories which consisted of 11 stories. The result showed that as following. First, there are five texts contain of complete generic structures and six texts contain of incomplete generic structure which is coda part for data 2, 3, 5, 8, and 1. Second, about language features, there are four data contains of all language features of Anecdote text, but on the other side, there are seven incomplete language features in the texts. Third, the consideration of choosing Yong Dollah as alternative material for Indonesia Language subject refers to eight indicators that are conveyed based on teachers’ perception toward Anecdote text Yong Dolla. 55 % of number of teachers claim that these texts suitable to be implemented as teaching material, but 44% of them claim neutral, and 1% claim disagree on it.

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.

Stability theory and Algebra

1979 ◽  
Vol 44 (4) ◽  
pp. 599-608 ◽  
Author(s):  
John T. Baldwin
Keyword(s):  
Point Of View ◽  
Clear Understanding ◽  
Common Features ◽  
Finite Models ◽  
Complete Theories ◽  
Skolem Theorem ◽  
And Algebra ◽  
Infinite Power ◽  
Fixed Structure ◽  
Algebraically Closed Fields

There are two origins for first-order theories. One type of theory arises by generalizing the common features of a number of different structures, e.g. the theory of groups, and formulating a set of axioms to encode these common features. Here the set of axioms is well understood, frequently it is finite or at least recursive, but there usually is no clear understanding of all the logical consequences of these axioms. The second type of theory arises by considering the set, T = Th(A), of all sentences true in a fixed structure A,3 e.g. the theory of arithmetic (N, +, 0) or the theory of the field of complex numbers (alias: the theory of algebraically closed fields of characteristic zero). The second case gives little more insight as to the truth in A (i.e. membership in T) of a given sentence ∅. But it does guarantee that for a given sentence ∅, either ∅ or ¬∅ is in T, that is, that T is a complete theory. When does a theory T of the first type, i.e. with well-understood axioms, posses this completeness property? An obvious sufficient condition is that T be secretly of the second type, that it have only one model, or, in jargon, T is categorical. Unfortunately (or fortunately depending on your point of view) for any theory with an infinite model, the Löwenheim-Skolem theorem shows this to be impossible: The theory has a model in every infinite power. In the mid-50's Łoś and Vaught discovered that if a theory T with no finite models is categorical in some infinite power α (all models with cardinality α are isomorphic) then T is complete. We will be dealing below with countable complete theories and will assume, unless stated to the contrary, that each theory has no finite models.

scholarly journals STRUKTUR, DIKSI, DAN KONJUNGSI TEKS PROSEDUR KARYA SISWA KELAS VII SMP NEGERI 1 KOTA SOLOK

2020 ◽  
Vol 8 (3) ◽  
pp. 469
Author(s):  
Wildani Ulfa ◽  
Yulianti Rasyid
Keyword(s):  
Qualitative Method ◽  
Research Data ◽  
Seventh Grade ◽  
Generic Structure ◽  
Procedure Text ◽  
Many Sources ◽  
Generic Structures ◽  
Descriptive Qualitative

ABSTRACT The research aims to describe generic structure and language features of procedure texts by seventh grade students in SMP Negeri 1 Kota Solok. The method that is used is descriptive qualitative method. The data of the research are procedure texts collected in many sources such as the documents of 32 text procedure written by the students. The instrument of this research is the researcher her self. Data were analyzed by describing, analyzing, and discussing the data toward the theory. Based on the results of the study, there are two things that can be concluded. First, the seventh grade students in SMP Negeri 1 Kota Solok have used the five generic structure of procedure text in writing. The generic structures of of procedure text are title, purpose, tool or meterial, and closing. It is proven by 32 procedure texts that have been analyzed, there are all texts have titles, tools, and materials, as well working steps, only 8 texts that have a purpose. Students have not been able to write a good closing in making the procedure text. This is evident from 32 texts analyzed only 6 texts that have a cover. Second, in terms of language. The happiness analyzed is three, namely diction, conjunction, and imperative sentences. Of the 32 research data found 4,099 the number of diction consisting of 3,949 standard diction and 150 diction which are not standard. From 32 research data found 436 number of conjunctions consisting of 420 uses of precision conjunctions and 16 improper conjunctions.  Kata Kunci: Struktur Teks, Diksi Teks, Konjungsi Teks

scholarly journals Interpolative fusions

2020 ◽  
pp. 2150010
Author(s):  
Alex Kruckman ◽  
Chieu-Minh Tran ◽  
Erik Walsberg
Keyword(s):  
Sufficient Conditions ◽  
Model Companion ◽  
First Order ◽  
Generic Structures

We define the interpolative fusion [Formula: see text] of a family [Formula: see text] of first-order theories over a common reduct [Formula: see text], a notion that generalizes many examples of random or generic structures in the model-theoretic literature. When each [Formula: see text] is model-complete, [Formula: see text] coincides with the model companion of [Formula: see text]. By obtaining sufficient conditions for the existence of [Formula: see text], we develop new tools to show that theories of interest have model companions.

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.]

scholarly journals An Analysis of Students' Reading Ability in Identifying Generic Structure of News Item Text

2019 ◽  
Vol 3 (2) ◽  
pp. 27-31
Author(s):  
Widya Juli Astria
Keyword(s):  
Reading Ability ◽  
Multiple Choice ◽  
Moderate Level ◽  
News Item ◽  
Multiple Choice Questions ◽  
Generic Structure ◽  
Generic Structures ◽  
Descriptive Method

This research aims to determine the ability of students to identify the generic structure of the news item text. This research uses a descriptive method. Data is taken through examinations in the form of multiple choice questions. The results showed that the ability of students to identify newsworthy events, event backgrounds and sources of news item texts was at a moderate level. This is evidenced by the percentage of student scores for each of the generic structures is 58.46%, 69, 23% and 61, 54%. Therefore, teachers are advised to discuss more and provide training on generic structure of the news item text. Then, students are advised to study harder to understand and do generic structure exercises from news item text. Furthermore, the next researcher is expected to conduct research related to the problems faced by students in identifying the generic structure of the news item text.

scholarly journals THE ANALYZES OF GENERIC STRUCTURES OF READING EXTS OF TEST OF THE NATIONAL EXAMINATIO

2017 ◽  
Author(s):  
Seminar Nasional Multidisiplin Ilmu 2017
Keyword(s):  
Social Function ◽  
Reading Test ◽  
Generic Structure ◽  
The Social ◽  
Reading Text ◽  
Qualitative Descriptive ◽  
Procedure Text ◽  
Generic Structures ◽  
Descriptive Method ◽  
Report Text

This study is aimed at describing the implementation of genres in reading text of the UN. In relation to the focus, objectives of the study are. In the research, qualitative descriptive method was conducted by the researcher. In analyzing the genre of reading passage, the writer also analyzed the characteristics of each genre in the term of social function, generic structure, and grammatical feature of each reading passage. The writer used the theory of Mark Anderson &amp; Kathy Anderson to know the characteristics of genre. From the result of data analysis, the writer found that the reading test presents the social function and in the side of generic structure, conclusion is not distributed in all report text and also coda is not distributed in all narrative text. For grammatical features, the reading test does not present the use of present tense and the use of adverb in the procedure text in all reading passages.

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.

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’.

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