Omitting models

1977 ◽  
Vol 42 (1) ◽  
pp. 29-32
Author(s):  
Ernest Snapper
Keyword(s):  
Homogeneous Model ◽  
Countable Model ◽  
Complete Theory ◽  
Finite Set ◽  
Homogeneous Models ◽  
Definition Of ◽  
General Aspects ◽  
The Universe ◽  
Main Definition ◽  
Countable Models

The purpose of this paper is to introduce the notion of “omitting models” and to derive a very natural theorem concerning it (Theorem 1). A corollary of this theorem is the remarkable theorem of Vaught [3] which states that a countable complete theory cannot have precisely two nonisomorphic countable models. In fact, we show that our theorem implies Rosenstein's theorem [2] which, in turn, implies Vaught's theorem.T stands for a countable complete theory whose (countable) language is denoted by L. Following [1], a countably homogeneous model of T is a countable model of T with the property that, for any two n-tuples a1, …, an and b1,…,bn of the universe of whose types are the same, there is an automorphism of which maps ai, on bi, for i = 1, …, n [1, p. 129 and Proposition 3.2.9, p. 131]. “Homogeneous model” always means “countably homogeneous model.” “Type of T” always stands for “n-type of T” where n ≥ s 0, i.e., for the type of some n-tuple of individuals of the universe of some model of T. We often use that two homogeneous models which realize the same types are isomorphic [1, Proposition 3.2.9, p. 131].It is well known that every type of T is realized by at least one countable model of T. The main definition of this paper is:Definition 1. A set of countable models of T is omissible or “may be omitted” if every type of T is realized by at least one countable model of T which is not isomorphic to a model in the set.The main theorem of the paper is:Theorem 1. If a countable complete theory is not ω-categorical, every finite set of its homogeneous models may be omitted.The theorem is proved in §1 and in §2 it is shown how Vaught's and Rosenstein's theorems follow from it. §3 discusses some general aspects of omitting models.

On the number of countable homogeneous models

1983 ◽  
Vol 48 (3) ◽  
pp. 539-541 ◽  
Author(s):  
Libo Lo
Keyword(s):  
Homogeneous Model ◽  
Countable Model ◽  
The Other ◽  
Reduced Model ◽  
Elementary Extension ◽  
Other Hand ◽  
Homogeneous Models ◽  
Countable Theory ◽  
Countable Models

The number of homogeneous models has been studied in [1] and other papers. But the number of countable homogeneous models of a countable theory T is not determined when dropping the GCH. Morley in [2] proves that if a countable theory T has more than ℵ1 nonisomorphic countable models, then it has such models. He conjectures that if a countable theory T has more than ℵ0 nonisomorphic countable models, then it has such models. In this paper we show that if a countable theory T has more than ℵ0 nonisomorphic countable homogeneous models, then it has such models.We adopt the conventions in [1]–[3]. Throughout the paper T is a theory and the language of T is denoted by L which is countable.Lemma 1. If a theory T has more than ℵ0types, then T hasnonisomorphic countable homogeneous models.Proof. Suppose that T has more than ℵ0 types. From [2, Corollary 2.4] T has types. Let σ be a Ttype with n variables, and T′ = T ⋃ {σ(c1, …, cn)}, where c1, …, cn are new constants. T′ is consistent and has a countable model (, a1, …, an). From [3, Theorem 3.2.8] the reduced model has a countable homogeneous elementary extension . σ is realized in . This shows that every type σ is realized in at least one countable homogeneous model of T. But each countable model can realize at most ℵ0 types. Hence T has at least countable homogeneous models. On the other hand, a countable theory can have at most nonisomorphic countable models. Hence the number of nonisomorphic countable homogeneous models of T is .In the following, we shall use the languages Lα (α = 0, 1, 2) defined in [2]. We give a brief description of them. For a countable theory T, let K be the class of all models of T. L = L0 is countable.

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’.

Typical ambiguity and the axiom of choice

1984 ◽  
Vol 49 (4) ◽  
pp. 1074-1078 ◽  
Author(s):  
Marcel Crabbé
Keyword(s):  
Positive Solution ◽  
Type Theory ◽  
Axiom Of Choice ◽  
Famous Result ◽  
Different Types ◽  
Finite Set ◽  
Definition Of ◽  
The Axiom Of Choice ◽  
Cardinal Arithmetic ◽  
The Universe

E. Specker has proved that the axiom of choice (AC) is false in NF [6]. Since AC is stratified, one can, according to another famous result of Specker [7], prove directly ¬AC in type theory (TT) plus some finite set of ambiguity axioms, i.e. sentences of the form φ ↔ φ+, where φ+ results from φ by adding one to its type indices.We shall in §2 of this paper give a disproof of AC directly in TT plus some axioms of ambiguity. The argument will be split into two parts. The first one (contained in Proposition 2) concerns cardinal arithmetic and has nothing to do with typical ambiguity. Though carried out in TT, it could have been done in other set theories such as Zermelo's Z or ZF. The second part is an application of this to the cardinals of the universes at different types. This is made possible through the introduction of an appropriate definition of 2α in §1 enabling one to express shifting sentences as “typed properties” of the universe, in Boffa's sense. The disproof of AC is then completed in TT plus two extra ambiguity axioms. In §3, we show that this is in a sense the best possible result: that means that every single ambiguity axiom is consistent with TT plus AC, thus giving a positive solution to a conjecture of Specker [7, p. 119].

The degree spectra of homogeneous models

2008 ◽  
Vol 73 (3) ◽  
pp. 1009-1028 ◽  
Author(s):  
Karen Lange
Keyword(s):  
Analogous Result ◽  
Homogeneous Model ◽  
Countable Model ◽  
Turing Degree ◽  
Decidable Theory ◽  
Degree Spectra ◽  
Homogeneous Models ◽  
Nonzero Degree

AbstractMuch previous study has been done on the degree spectra of prime models of a complete atomic decidable theory. Here we study the analogous questions for homogeneous models. We say a countable model has a d-basis if the types realized in are all computable and the Turing degree d can list -indices for all types realized in . We say has a d-decidable copy if there exists a model ≅ such that the elementary diagram of is d-computable. Goncharov, Millar, and Peretyat'kin independently showed there exists a homogeneous with a 0-basis but no decidable copy.We prove that any homogeneous with a 0′-basis has a low decidable copy. This implies Csima's analogous result for prime models. In the case where all types of the theory T are computable and is a homogeneous model with a 0-basis, we show has copies decidable in every nonzero degree. A degree d is 0-homogeneous bounding if any automorphically nontrivial homogeneous with a 0-basis has a d-decidable copy. We show that the nonlow2 degrees are 0-homogeneous bounding.

The classification of small weakly minimal sets. II

1988 ◽  
Vol 53 (2) ◽  
pp. 625-635 ◽  
Author(s):  
Steven Buechler
Keyword(s):  
Stable Theory ◽  
Minimal Set ◽  
Minimal Sets ◽  
Finite Set ◽  
Unidimensional Theory ◽  
The Universe ◽  
Countable Models

AbstractThe main result is Vaught's conjecture for weakly minimal, locally modular and non-ω-stable theories. The more general results yielding this are the following.Theorem A. Suppose that T is a small unidimensional theory and D is a weakly minimal set, definable over the finite set B. Then for all finite A ⊂ D there are only finitely many nonalgebraic strong types over B realized in acl(A) ∩ D.Theorem B. Suppose that T is a small, unidimensional, non-ω-stable theory such that the universe is weakly minimal and locally modular. Then for all finite A there is a finite B ⊂ cl(A) such that a ∈ cl(A) iff a ∈ cl(b) for some b ∈ B.Recall the property (S) defined in the abstract of [B1].Theorem C. Let T be as in Theorem B. Then, if T does not satisfy (S), T hasmany countable models.Combining Theorem C and the results in [B1] we obtain Vaught's conjecture for such theories.

A complete, decidable theory with two decidable models

1979 ◽  
Vol 44 (3) ◽  
pp. 307-312 ◽  
Author(s):  
Terrence S. Millar
Keyword(s):  
Completeness Theorem ◽  
Countable Model ◽  
Complete Theory ◽  
Consistent Theory ◽  
Decidable Theory ◽  
Decidable Model ◽  
Recursive Set ◽  
Countable Models

A well-known result of Vaught's is that no complete theory has exactly two nonisomorphic countable models. The main result of this paper is that there is a complete decidable theory with exactly two nonisomorphic decidable models.A model is decidable if it has a decidable satisfaction predicate. To be more precise, let T be a decidable theory, let {θn∣n < ω} be an effective enumeration of all formulas in L(T), and let be a countable model of T. For any indexing E = {ai∣ i < ω} of ∣∣, and any formula ϕ ∈ L(T), let ‘ϕE’ denote the result of substituting ‘ai’ for every free occurrence of ‘xi’ in ϕ, i < ω. Then is decidable just in case, for some indexing E of ∣∣, {n ∣ ⊨ θnE} is a recursive set of integers. It is easy to show that the decidability of a model does not depend on the choice of the effective enumeration of the formulas in L(T); we omit details. By a simple ‘effectivization’ of Henkin's proof of the completeness theorem (see Chang [1]) we haveFact 1. Every decidable consistent theory has a decidable model.Assume next that T is a complete decidable theory and {θn ∣ n < ω} is an effective enumeration of all formulas of L(T).

A characterization of the 0-basis homogeneous bounding degrees

2010 ◽  
Vol 75 (3) ◽  
pp. 971-995
Author(s):  
Karen Lange
Keyword(s):  
Homogeneous Model ◽  
Countable Model

AbstractWe say a countable model has a 0-basis if the types realized in are uniformly computable. We say has a (d-)decidable copy if there exists a model ≅ such that the elementary diagram of is (d-)computable. Goncharov, Millar, and Peretyat'kin independently showed there exists a homogeneous model with a 0-basis but no decidable copy. We extend this result here. Let d ≤ 0′ be any low2 degree. We show that there exists a homogeneous model with a 0-basis but no d-decidable copy. A degree d is 0-basis homogeneous bounding if any homogenous with a 0-basis has a d-decidable copy. In previous work, we showed that the non low2 Δ20 degrees are 0-basis homogeneous bounding. The result of this paper shows that this is an exact characterization of the 0-basis homogeneous bounding Δ20 degrees.

Towards a computational theory of social groups: A finite set of cognitive primitives for representing any and all social groups in the context of conflict

2021 ◽  
pp. 1-62
Author(s):  
David Pietraszewski
Keyword(s):  
Social Group ◽  
Group Representation ◽  
Social Groups ◽  
Study Groups ◽  
Human Mind ◽  
Computational Theory ◽  
Mistaken Belief ◽  
Finite Set ◽  
The One ◽  
Definition Of

Abstract We don't yet have adequate theories of what the human mind is representing when it represents a social group. Worse still, many people think we do. This mistaken belief is a consequence of the state of play: Until now, researchers have relied on their own intuitions to link up the concept social group on the one hand, and the results of particular studies or models on the other. While necessary, this reliance on intuition has been purchased at considerable cost. When looked at soberly, existing theories of social groups are either (i) literal, but not remotely adequate (such as models built atop economic games), or (ii) simply metaphorical (typically a subsumption or containment metaphor). Intuition is filling in the gaps of an explicit theory. This paper presents a computational theory of what, literally, a group representation is in the context of conflict: it is the assignment of agents to specific roles within a small number of triadic interaction types. This “mental definition” of a group paves the way for a computational theory of social groups—in that it provides a theory of what exactly the information-processing problem of representing and reasoning about a group is. For psychologists, this paper offers a different way to conceptualize and study groups, and suggests that a non-tautological definition of a social group is possible. For cognitive scientists, this paper provides a computational benchmark against which natural and artificial intelligences can be held.

Symbolic Macroconcept of the Universe in the Aspect of the First Cognitive Sign in Russian Linguoculture

2021 ◽  
Vol 16 (1) ◽  
pp. 92-101
Author(s):  
Marina V. Pimenova ◽  
◽  
Aigul A. Bakirova ◽  
Keyword(s):  
Middle Part ◽  
Russian Language ◽  
The World ◽  
Symbolic Meanings ◽  
New Type ◽  
Human World ◽  
Internal Volume ◽  
Definition Of ◽  
Universe Structure ◽  
The Universe

The article analyzes the cognitive signs of the macroconcept universe in Russian linguoculture. The relevance of the research is determined by the prospect of studying a new type of mental structures - symbolic macroconcepts. The purpose of the article is to describe the specifics of the macroconcept universe structure formation from the standpoint of the definition of syncretic primordial signs. The main methods in the work are the historical and etymological analysis of the studied macroconcept representative, descriptive and interpretive methods. During the study, seven motivating signs of the macroconcept universe were noted: 'earth', 'live', ‘world’，‘inhabit’，‘inhabited’，‘settlement’，‘light’. All identified motivating signs are syncretic symbolic primordial signs 'house' (conceptum, according to V. V. Kolesov). Motivating signs express two main symbolic meanings of Russian linguoculture: home is a place where people live, settle; home is the world of people and all living beings, this world-light (unlike that world-light where the souls of the dead go: that world-light is located in the sky), it is built on earth. The macroconcept universe is objectified by erased metaphors of a closed space (in particular, the metaphor of a key), which has an internal volume, center-middle, limits, parts, edges, corners, people live in this house, they live and exist in it, it is inhabited and settle down in Russian linguoculture. The model of the universe in the Russian language picture of the world is three-parted: the middle part in it represents the human world, in which the principle of anthropocentrism is manifested - a person measures space and chooses himself as a reference point. The syncretic primary sign ‘house’ unites in itself all the motivating signs of the studied macroconcept, keeping their relevance to our days. Keywords: macroconcept, motivating signs, first sign, language picture of the world, linguoculture, comparative studies

scholarly journals Formation of skills to create stories of different type in senior preschool children

2018 ◽  
pp. 111-119
Author(s):  
Alexander Savchin
Keyword(s):  
Preschool Children ◽  
Theoretical Analysis ◽  
School Environment ◽  
Preschool Age ◽  
Different Types ◽  
Definition Of ◽  
Main Definition

The article gives an analysis of problems of formation of skills in children of the senior preschool age to build stories of different types. On the basis of the theoretical analysis of psycho-pedagogical and special literature, based on the main definition of «building a story», a subordinate term is formulated in relation to the topic of the study: «building stories of different types» The psychological and pedagogical conditions of formation of sustainable skills of creation and expedient use of stories of different types in children of the senior preschool age are revealed and substantiated. The components, criteria, indicators and levels of formation of abilities to build a different type of narration in children of the sixth year of life are determined. The stages of formation of persistent skills for building different types of stories by senior preschoolers are singled out. The lack of effective pedagogical technologies in relation to the formation of skills in the children of the senior preschool age to build and expedient use of narration of comprehensive typing significantly reduces the effective preparation of children to school. The proposed pedagogical and motivational technology for the formation of children of the sixth year of a sustainable ability to create stories of various types provides the small person an effective communicative tool, which will definitely promote its self-realization both in school environment and in career, society, own life. The prospects for further research are outlined, such independent types of stories are presented as a story - an instruction and a story - a presentation.

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