Prime and atomic models

1978 ◽  
Vol 43 (3) ◽  
pp. 385-393 ◽  
Author(s):  
Julia F. Knight
Keyword(s):  
Model Theory ◽  
Existence Results ◽  
Atomic Model ◽  
Elementary Embedding ◽  
Prime Model ◽  
Atomic Models ◽  
Large Model

This paper gives some simple existence results on prime and atomic models over sets. It also contains an example in which there is no prime model over a certain set even though there is an atomic model over the set. The existence results are “local” in that they deal with just one set rather than all sets contained in models of some theory. For contrast, see the “global” results in [6] or [7, p. 200].Throughout the paper, L is a countable language, and T is a complete L-theory with infinite models. There is a “large” model of T that contains the set X and any other sets and models to be used in a particular construction of a prime or atomic model over X.A model is said to be prime over X if and every elementary monomorphism on X can be extended to an elementary embedding on all of . This notion is used in a variety of ways in model theory. It aids in distinguishing between models that are not isomorphic, as in Vaught [10]. It also aids in showing that certain models are isomorphic, as in Baldwin and Lachlan [1].

Classifying model-theoretic properties

2008 ◽  
Vol 73 (3) ◽  
pp. 885-905 ◽  
Author(s):  
Chris J. Conidis
Keyword(s):  
Model Theory ◽  
Computability Theory ◽  
Prime Model ◽  
Decidable Theory ◽  
Dense Sets ◽  
Equivalent Properties ◽  
The Relationship

AbstractIn 2004 Csima, Hirschfeldt, Knight, and Soare [1] showed that a set A ≤T 0′ is nonlow2 if and only if A is prime bounding, i.e., for every complete atomic decidable theory T, there is a prime model computable in A. The authors presented nine seemingly unrelated predicates of a set A, and showed that they are equivalent for sets. Some of these predicates, such as prime bounding, and others involving equivalence structures and abelian p-groups come from model theory, while others involving meeting dense sets in trees and escaping a given function come from pure computability theory.As predicates of A, the original nine properties are equivalent for sets; however, they are not equivalent in general. This article examines the (degree-theoretic) relationship between the nine properties. We show that the nine properties fall into three classes, each of which consists of several equivalent properties. We also investigate the relationship between the three classes, by determining whether or not any of the predicates in one class implies a predicate in another class.

scholarly journals THE NUMBER OF ATOMIC MODELS OF UNCOUNTABLE THEORIES

2018 ◽  
Vol 83 (1) ◽  
pp. 84-102
Author(s):  
DOUGLAS ULRICH
Keyword(s):  
Atomic Model ◽  
Complete Theory ◽  
Atomic Models ◽  
Image Position

AbstractWe show there exists a complete theory in a language of size continuum possessing a unique atomic model which is not constructible. We also show it is consistent with $ZFC + {\aleph _1} < {2^{{\aleph _0}}}$ that there is a complete theory in a language of size ${\aleph _1}$ possessing a unique atomic model which is not constructible. Finally we show it is consistent with $ZFC + {\aleph _1} < {2^{{\aleph _0}}}$ that for every complete theory T in a language of size ${\aleph _1}$, if T has uncountable atomic models but no constructible models, then T has ${2^{{\aleph _1}}}$ atomic models of size ${\aleph _1}$.

On nonstandard models in higher order logic

1984 ◽  
Vol 49 (1) ◽  
pp. 204-219
Author(s):  
Christian Hort ◽  
Horst Osswald
Keyword(s):  
Model Theory ◽  
Type Theory ◽  
Order Logic ◽  
Elementary Embedding ◽  
Simple Type ◽  
Order Model ◽  
Higher Order Logic ◽  
First Order ◽  
Simple Type Theory ◽  
Nonstandard Models

There are two concepts of standard/nonstandard models in simple type theory.The first concept—we might call it the pragmatical one—interprets type theory as a first order logic with countably many sorts of variables: the variables for the urelements of type 0,…, the n-ary relational variables of type (τ1, …, τn) with arguments of type (τ1,…,τn), respectively. If A ≠ ∅ then 〈Aτ〉 is called a model of type logic, if A0 = A and . 〈Aτ〉 is called full if, for every τ = (τ1,…,τn), . The variables for the urelements range over the elements of A and the variables of type (τ1,…, τn) range over those subsets of which are elements of . The theory Th(〈Aτ〉) is the set of all closed formulas in the language which hold in 〈Aτ〉 under natural interpretation of the constants. If 〈Bτ〉 is a model of Th(〈Aτ〉), then there exists a sequence 〈fτ〉 of functions fτ: Aτ → Bτ such that 〈fτ〉 is an elementary embedding from 〈Aτ〉 into 〈Bτ〉. 〈Bτ〉 is called a nonstandard model of 〈Aτ〉, if f0 is not surjective. Otherwise 〈Bτ〉 is called a standard model of 〈Aτ〉.This first concept of model theory in type logic seems to be preferable for applications in model theory, for example in nonstandard analysis, since all nice properties of first order model theory (completeness, compactness, and so on) are preserved.

Bounding prime models

2004 ◽  
Vol 69 (4) ◽  
pp. 1117-1142 ◽  
Author(s):  
Barbara F. Csima ◽  
Denis R. Hirschfeldt ◽  
Julia F. Knight ◽  
Robert I. Soare
Keyword(s):  
Model Theory ◽  
Computable Function ◽  
Prime Model ◽  
Computable Model ◽  
Principal Type ◽  
Combinatorial Properties ◽  
Computable Model Theory ◽  
Monotonic Functions ◽  
Limitwise Monotonic Functions

Abstract.A set X is prime bounding if for every complete atomic decidable (CAD) theory T there is a prime model of T decidable in X. It is easy to see that X = 0′ is prime bounding. Denisov claimed that every X <T 0′ is not prime bounding, but we discovered this to be incorrect. Here we give the correct characterization that the prime bounding sets X ≤τ 0′ are exactly the sets which are not low2. Recall that X is low2 if X″ ≤τ 0″. To prove that a low2 set X is not prime bounding we use a 0′ -computable listing of the array of sets {Y : Y ≤τX } to build a CAD theory T which diagonalizes against all potential X-decidable prime models of T, To prove that any non-low2X is indeed prime bounding. we fix a function f ≤TX that is not dominated by a certain 0′-computable function that picks out generators of principal types. Given a CAD theory T. we use f to eventually find, for every formula φ(x̄) con sistent with T. a principal type which contains it. and hence to build an X-decidable prime model of T. We prove the prime bounding property equivalent to several other combinatorial properties, including some related to the limitwise monotonic functions which have been introduced elsewhere in computable model theory.

CONSTRUCTING MANY ATOMIC MODELS IN ℵ1

2016 ◽  
Vol 81 (3) ◽  
pp. 1142-1162 ◽  
Author(s):  
JOHN T. BALDWIN ◽  
MICHAEL C. LASKOWSKI ◽  
SAHARON SHELAH
Keyword(s):  
Order Theory ◽  
Atomic Model ◽  
First Order ◽  
Atomic Models ◽  
First Order Theory ◽  
Image Position

AbstractWe introduce the notion of pseudoalgebraicity to study atomic models of first order theories (equivalently models of a complete sentence of ${L_{{\omega _1},\omega }}$). Theorem: Let T be any complete first-order theory in a countable language with an atomic model. If the pseudominimal types are not dense, then there are 2ℵ0 pairwise nonisomorphic atomic models of T, each of size ℵ1.

scholarly journals Building a pseudo-atomic model of the anaphase-promoting complex

2013 ◽  
Vol 69 (11) ◽  
pp. 2236-2243 ◽  
Author(s):  
Kiran Kulkarni ◽  
Ziguo Zhang ◽  
Leifu Chang ◽  
Jing Yang ◽  
Paula C. A. da Fonseca ◽  
...  
Keyword(s):  
Cell Cycle ◽  
Homology Modelling ◽  
Active Form ◽  
Protein Crystallography ◽  
Hybrid Approach ◽  
Atomic Model ◽  
Anaphase Promoting Complex ◽  
Atomic Models ◽  
Entire Complex ◽  
Cell Cycle Regulatory Proteins

The anaphase-promoting complex (APC/C) is a large E3 ubiquitin ligase that regulates progression through specific stages of the cell cycle by coordinating the ubiquitin-dependent degradation of cell-cycle regulatory proteins. Depending on the species, the active form of the APC/C consists of 14–15 different proteins that assemble into a 20-subunit complex with a mass of approximately 1.3 MDa. A hybrid approach of single-particle electron microscopy and protein crystallography of individual APC/C subunits has been applied to generate pseudo-atomic models of various functional states of the complex. Three approaches for assigning regions of the EM-derived APC/C density map to specific APC/C subunits are described. This information was used to dock atomic models of APC/C subunits, determined either by protein crystallography or homology modelling, to specific regions of the APC/C EM map, allowing the generation of a pseudo-atomic model corresponding to 80% of the entire complex.

On the existence of atomic models

1993 ◽  
Vol 58 (4) ◽  
pp. 1189-1194 ◽  
Author(s):  
M. C. Laskowski ◽  
S. Shelah
Keyword(s):  
Atomic Model ◽  
Atomic Models ◽  
Countable Theory

AbstractWe give an example of a countable theory T such that for every cardinal λ ≥ ℵ2 there is a fully indiscernible set A of power λ such that the principal types are dense over A, yet there is no atomic model of T over A. In particular, T(A) is a theory of size λ where the principal types are dense, yet T(A) has no atomic model.

Generic expansions of structures

1973 ◽  
Vol 38 (4) ◽  
pp. 561-570 ◽  
Author(s):  
Julia F. Knight
Keyword(s):  
Positive Result ◽  
Set Theory ◽  
Model Theory ◽  
The Other ◽  
Elementary Embedding ◽  
Elementary Substructure ◽  
Skolem Function ◽  
Countable Type ◽  
Theoretic Technique ◽  
The Axiom Of Choice

In this paper, Cohen's forcing technique is applied to some problems in model theory. Forcing has been used as a model-theoretic technique by several people, in particular, by A. Robinson in a series of papers [1], [10], [11]. Here forcing will be used to expand a family of structures in such a way that weak second-order embeddings are preserved. The forcing situation resembles that in Solovay's proof that for any theorem φ of GB (Godel-Bernays set theory with a strong form of the axiom of choice), if φ does not mention classes, then it is already a theorem of ZFC. (See [3, p. 105] and [2, p. 77].)The first application of forcing here is to the problem (posed by Keisler) of when is it possible to add a Skolem function to a pair of structures, one of which is an elementary substructure of the other, in such a way that the elementary embedding is preserved.It is not always possible to find such a Skolem function. Payne [9] found an example involving countable structures with uncountably many relations. The author [4], [6] found an example involving uncountable structures with only two relations. The problem remains open in case the structures are required both to be countable and to have countable type. Forcing is used to obtain a positive result under some special conditions.

scholarly journals Model theory of R-trees

2020 ◽  
Vol 12 ◽  
Author(s):  
Sylvia Carlisle ◽  
C Ward Henson
Keyword(s):  
Group Theory ◽  
Model Theory ◽  
Quantifier Elimination ◽  
Geometric Group Theory ◽  
Atomic Model ◽  
Infinite Cardinal ◽  
Model Companion ◽  
Canonical Bases ◽  
Type Spaces

We show the theory of pointed $\R$-trees with radius at most $r$ is axiomatizable in a suitable continuous signature. We identify the model companion $\rbRT_r$ of this theory and study its properties. In particular, the model companion is complete and has quantifier elimination; it is stable but not superstable. We identify its independence relation and find built-in canonical bases for non-algebraic types. Among the models of $\rbRT_r$ are $\R$-trees that arise naturally in geometric group theory. In every infinite cardinal, we construct the maximum possible number of pairwise non-isomorphic models of $\rbRT_r$; indeed, the models we construct are pairwise non-homeomorphic. We give detailed information about the type spaces of $\rbRT_r$. Among other things, we show that the space of $2$-types over the empty set is nonseparable. Also, we characterize the principal types of finite tuples (over the empty set) and use this information to conclude that $\rbRT_r$ has no atomic model.

scholarly journals Near-atomic model of microtubule-tau interactions

Science ◽  
2018 ◽  
Vol 360 (6394) ◽  
pp. 1242-1246 ◽  
Author(s):  
Elizabeth H. Kellogg ◽  
Nisreen M. A. Hejab ◽  
Simon Poepsel ◽  
Kenneth H. Downing ◽  
Frank DiMaio ◽  
...  
Keyword(s):  
Electron Microscopy ◽  
Alzheimer’S Disease ◽  
Alzheimer's Disease ◽  
Atomic Model ◽  
Developmentally Regulated ◽  
Computational Approaches ◽  
Atomic Models ◽  
Cryo Electron Microscopy ◽  
Tubulin Binding ◽  
Axonal Protein

Tau is a developmentally regulated axonal protein that stabilizes and bundles microtubules (MTs). Its hyperphosphorylation is thought to cause detachment from MTs and subsequent aggregation into fibrils implicated in Alzheimer’s disease. It is unclear which tau residues are crucial for tau-MT interactions, where tau binds on MTs, and how it stabilizes them. We used cryo–electron microscopy to visualize different tau constructs on MTs and computational approaches to generate atomic models of tau-tubulin interactions. The conserved tubulin-binding repeats within tau adopt similar extended structures along the crest of the protofilament, stabilizing the interface between tubulin dimers. Our structures explain the effect of phosphorylation on MT affinity and lead to a model of tau repeats binding in tandem along protofilaments, tethering together tubulin dimers and stabilizing polymerization interfaces.

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