The transcendental rank of the formulas of an S1-categorical theory

1974 ◽  
Vol 15 (2) ◽  
pp. 182-186
Author(s):  
B. I. Zil'ber
Keyword(s):  
Categorical Theory

Indiscernibles and decidable models

1983 ◽  
Vol 48 (1) ◽  
pp. 21-32 ◽  
Author(s):  
H. A. Kierstead ◽  
J. B. Remmel
Keyword(s):  
Order Theory ◽  
Stable Theory ◽  
Categorical Theory ◽  
Large Classes ◽  
Decidable Theory ◽  
First Order Theory ◽  
Decidable Model ◽  
Decidable Theories ◽  
Basic Facts ◽  
Infinite Set

Ehrenfeucht and Mostowski [3] introduced the notion of indiscernibles and proved that every first order theory has a model with an infinite set of order indiscernibles. Since their work, techniques involving indiscernibles have proved to be extremely useful for constructing models with various specialized properties. In this paper and in a sequel [5], we investigate the effective content of Ehrenfeucht's and Mostowski's result. In this paper we consider the question of which decidable theories have decidable models with infinite recursive sets of indiscernibles. In §1, using some basic facts from stability theory, we show that certain large classes of decidable theories have decidable models with infinite recursive sets of indiscernibles. For example, we show that every ω-stable decidable theory and every stable theory which possesses a certain strong decidability property called ∃Q-decidability have such models. In §2 we construct several examples of decidable theories which have no decidable models with infinite recursive sets of indiscernibles. These examples show that our hypothesis for our positive results in §1 are necessary. Finally in §3 we give two applications of our results. First as an easy application of our results in §1, we show that every ω-stable decidable theory has uncountable models which realize only recursive types. Also our counterexamples in §2 allow us to answer negatively two questions of Baldwin and Kueker [1] concerning the effectiveness of their elimination of Ramsey quantifiers for certain theories.In [5], we show that in general the problem of finding an infinite set of indiscernibles in a decidable model is recursively equivalent to finding a path through a recursive infinite branching tree. Similarly, we show that the problem of finding an co-type of a set of indiscernibles in a decidable ω-categorical theory is recursively equivalent to finding a path through a highly recursive finitely branching tree.

scholarly journals An Uncountably Categorical Theory Whose Only Computably Presentable Model Is Saturated

2006 ◽  
Vol 47 (1) ◽  
pp. 63-71 ◽  
Author(s):  
Denis R. Hirschfeldt ◽  
Bakhadyr Khoussainov ◽  
Pavel Semukhin
Keyword(s):  
Categorical Theory

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.

Theories with recursive models

1979 ◽  
Vol 44 (1) ◽  
pp. 59-76 ◽  
Author(s):  
Manuel Lerman ◽  
James H. Schmerl
Keyword(s):  
Existential Quantifier ◽  
Categorical Theory ◽  
Recursive Model ◽  
Recursive Models ◽  
Decidable Theory ◽  
Decidable Model

A structure is recursive if the set of quantifier-free sentences in the complete diagram ⊿() of is recursive. It has been known for some time that every decidable theory has a recursive model. In fact, every decidable theory has a decidable model (that is a model such that ⊿() is recursive). In this paper we find other conditions which imply that a theory have a recursive model.In §1 we study the relation between an ℵ0-categorical theory T having a recursive model and the complexity of the quantificational hierarchy of that theory. We let ∃0 denote the set of quantifier-free sentences, and let ∃n÷1 denote the set of sentences beginning with an existential quantifier and having n alternations of quantifiers. (∀n is defined analogously.) Then we show that if T is an arithmetical ℵ0-categorical theory such that T ⋂ ∃n÷2 is Σn÷10 for each n < ω, then T has a recursive model. We show that this is a best possible result by giving an example of a ⊿n÷20 ℵ0-categorical theory T such that T ⋂ ∃n÷1 is recursive yet T has no recursive model.In §2 we consider the theory of trees. Ershov [1] had proved that every Σ10 theory of trees has a recursive model. We show this to be best possible by giving an example of a ⊿20 theory of trees which has no recursive model.

An algebraic characterization of power set in countable standard models of ZF

1975 ◽  
Vol 40 (2) ◽  
pp. 167-170
Author(s):  
George Metakides ◽  
J. M. Plotkin
Keyword(s):  
Standard Model ◽  
Boolean Algebra ◽  
Classical Result ◽  
Algebraic Characterization ◽  
Categorical Theory ◽  
Power Set ◽  
Standard Models ◽  
Image Position ◽  
Atomic Boolean Algebra

The following is a classical result:Theorem 1.1. A complete atomic Boolean algebra is isomorphic to a power set algebra [2, p. 70].One of the consequences of [3] is: If M is a countable standard model of ZF and is a countable (in M) model of a complete ℵ0-categorical theory T, then there is a countable standard model N of ZF and a Λ ∈ N such that the Boolean algebra of definable (in T with parameters from ) subsets of is isomorphic to the power set algebra of Λ in N. In particular if and T the theory of equality with additional axioms asserting the existence of at least n distinct elements for each n < ω, then there is an N and Λ ∈ N with 〈PN(Λ), ⊆〉 isomorphic to the countable, atomic, incomplete Boolean algebra of the finite and cofinite subsets of ω.From the above we see that some incomplete Boolean algebras can be realized as power sets in standard models of ZF.Definition 1.1. A countable Boolean algebra 〈B, ≤〉 is a pseudo-power set if there is a countable standard model of ZF, N and a set Λ ∈ N such thatIt is clear that a pseudo-power set is atomic.

Towards a Categorical Theory of Creativity for Music, Discourse, and Cognition

2013 ◽  
pp. 19-37 ◽  
Author(s):  
Moreno Andreatta ◽  
Andrée Ehresmann ◽  
René Guitart ◽  
Guerino Mazzola
Keyword(s):  
Categorical Theory

An (∞, 2)-Categorical Theory of Spans

2019 ◽  
pp. 169-199
Author(s):  
Tobias Dyckerhoff ◽  
Mikhail Kapranov
Keyword(s):  
Categorical Theory

On strongly minimal sets

1971 ◽  
Vol 36 (1) ◽  
pp. 79-96 ◽  
Author(s):  
J. T. Baldwin ◽  
A. H. Lachlan
Keyword(s):  
Categorical Theory ◽  
Minimal Sets ◽  
Countable Models

The purpose of this paper is twofold. In §1 and §2 which are largely expository we develop the known theory of ℵ1-categoricity in terms of strongly minimal sets. In §3 we settle affirmatively Vaught's conjecture that a complete ℵ1-categorical theory has either just one or just ℵ0 countable models, and in §4 we present an example which serves to illustrate the ideas of §3.As far as we know the only work published on strongly minimal sets is that of Marsh [3]. The present exposition goes beyond [3] in showing that any ℵ-categorical theory has a principal extension in which some formula is strongly minimal.

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.

scholarly journals MASS GROWTH OF OBJECTS AND CATEGORICAL ENTROPY

2020 ◽  
pp. 1-22
Author(s):  
AKISHI IKEDA
Keyword(s):  
Classical Theory ◽  
Stability Condition ◽  
Triangulated Category ◽  
Categorical Theory ◽  
Mass Growth ◽  
Fundamental Properties ◽  
Measured Foliation ◽  
The Stability ◽  
The Relationship

In the pioneering work by Dimitrov–Haiden–Katzarkov–Kontsevich, they introduced various categorical analogies from the classical theory of dynamical systems. In particular, they defined the entropy of an endofunctor on a triangulated category with a split generator. In the connection between the categorical theory and the classical theory, a stability condition on a triangulated category plays the role of a measured foliation so that one can measure the “volume” of objects, called the mass, via the stability condition. The aim of this paper is to establish fundamental properties of the growth rate of mass of objects under the mapping by the endofunctor and to clarify the relationship between it and the entropy. We also show that they coincide under a certain condition.

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