Mathematics

A Functorial Model Theory
Newer Applications to Algebraic Topology, Descriptive Sets, and Computing Categories Topos

Cyrus F. Nourani, PhD

A Functorial Model Theory

Published. Available now.
Pub Date: January 2014
Hardback Price: $149.95 US
Hard ISBN: 9781926895925
Pages: 302pp
Binding Type: hardbound

The book is a preliminary introduction to a functorial model theory based on infinitary language categories. The perspective is different from the preceding authors on the areas in that functorial model theory is based on defining categories on language fragment then carriyng on functors to sets and categories to develop models. Infinitary language categories are defined and their preliminary categorical properties are presented. A foundation for infinitary language categories is presented as well. Thus the present mathematics has it own theories and application areas.

The book defines a model theory for functors starting with a countable fragment of an infinitary language. The infinite language category defined on Lω1,ω generic sets on the Lω1.K, the respective Kiesler fragment and a functor from Lω1,K to Set, called the generic functor are defined. A new technique for generating generic models with categories is defined by inventing infinite language categories and functorial model theory. Specific functor models, the String Models, are defined by infinite chains on fragment models defined on an infinite language category functor. Techniques similar to Robinson’s consistency theorem are invented to define limit models. Functorial models are further defined on a generalized diagram by limit chains. The techniques are further developed over the past several years to fill the gap between forcing and toposes with fragment consistent categories.

CONTENTS:
Preface
Chapter 1: Introduction

Chapter 2: Categorical Preliminaries
- 2.1 Categories and Functors
- 2.2 Morphisms
- 2.3 Functors
- 2.4 Categorical Products
- 2.5 Natural Transformations
- 2.7 Products on Models
- 2.6 Preservation of Limits
- 2.8 Model Theory and Topoi
- 2.9 More on Universal Constructions
Chapter Exercises
Chapter 3: Infinite Language Categories
- 3.1 Basics
- 3.2 Limits and Infinitary Languages
- 3.3 Generic Functors and Language String Models
- 3.4 FunctorialMorphic Ordered Structure Models
Chapter Exercises
Chapter 4: Functorial Fragment Model Theory
- 4.1 Introduction
- 4.2 Generic Functors and Language String Models
- 4.3 Functorial Models As ω-Chains
- 4.4 Models Glimpses From Functors
- 4.5 Structure Products
- 4.6 Higher Stratified Consistency and Completeness
- 4.7 Fragment Positive Omitting Type Algebras
- 4.8 Omitting Types and Realizability
- 4.9 Positive Categories and Consistency Models
- 4.10 More on Fragment Consistency
Chapter Exercises
Chapter 5: Algebraic Theories, Categories, and Models
- 5.1 Ultraproducts on Algebras
- 5.2 Ultraproducts and Ultrafilters
- 5.3 Ultraproduct Applications to Horn Categories
- 5.4 Algebraic Theories and Topos Models
- 5.5 Free Theories and Factor Theories
- 5.6 T-Algebras and Adjunctions
- 5.7 Theory Morphisms, Products and Co-products
- 5.8 Algebras and the Category of Algebraic Theories
- 5.9 Initial Algebraic Theories and Computable Trees
Chapter Exercises
Chapter 6: Generic Functorial Models and Topos
- 6.1 Elementary Topoi
- 6.2 GenericFunctorial Models
- 6.3 Generic Functors
- 6.4 Initial D Models
- 6.5 Positive Forcing Models
- 6.6 Functors Computing Hasse Diagram Models
- 6.7 Fragment Consistent Models
- 6.8 Homotopy theory of topos
- 6.9 Filtered colimits and comma categories
- 6.10 More on Yoneda Lemma
Chapter Exercises
Chapter 7: Models, Sheaves, and Topos
- 7.1 PreSheaves
- 7.2 Duality, Fragment Models, and Topology
- 7.3 Lifts on Topos Models on Cardinalities
Chapter Exercises
Chapter 8: Functors on Fields
- 8.1 Introduction
- 8.2 Basic Models
- 8.3 Fields
- 8.4 Prime Models
- 8.5 Omitting Types on Fields
- 8.6 Filters and Fields
- 8.7 Filters and Products
Chapter Exercises
Chapter 9: Filters and Ultraproducts on Projective Sets
- 9.1 General Definitions
- 9.2 Generic Functors and Language String Models
- 9.3 Functorial Fragment Consistency
- 9.4 Filters
- 9.5 Structure Products
- 9.6 Completing Theories and Fragments
- 9.7 Prime Models and Model Completion
- 9.8 Uniform and countably incomplete ultrafilters
- 9.9 FunctorialProjetive Set Models and Saturation
- 9.10 Ultraproducts and Ultrafliters
Chapter Exercises
Chapter 10: A Glimpse on m Algebraic Set Theory
- 10.1 Preliminaries
- 10.2 Ultraproducts and Ultrafilters on Sets
- 10.3 Ultrafilters over N
- 10.4 Saturation and Preservations
- 10.5 Functorial Models and Descriptive Sets
- 10.6 Filters, Fragment Constructible Models, and Sets
Index


About the Authors / Editors:
Cyrus F. Nourani, PhD
Independent Consultant in Computing R&D, San Francisco, California;
Research Professor, Simon Fraser University, British Columbia, Canada


Dr. Cyrus F. Nourani has a national and international reputation in computer science, artificial intelligence, mathematics, virtual haptic computation, information technology, and management. He has many years of experience in the design and implementation of computing systems. Dr. Nourani's academic experience includes faculty positions at the University of Michigan-Ann Arbor, the University of Pennsylvania, the University of Southern California, UCLA, MIT, and the University of California, Santa Barbara. He was also a Research Professor at Simon Frasier University in Burnaby, British Columbia, Canada. He was a Visiting Professor at Edith Cowan University, Perth, Australia, and a Lecturer of Management Science and IT at the the University of Auckland, New Zealand.

Dr. Nourani commenced his university degrees at MIT where he became interested in algebraic semantics. That was pursued with a category theorist at the University of California. Dr. Nourani’s dissertation on computing models and categories proved to have intutionist forcing developments that were published from the postdoctoral times on at ASL. He has taught AI to the Los Angeles aerospace industry and has authored many R&D and commercial ventures. He has written and co-authored several books. He has over 350 publications in mathematics and computer science and has written on additional topics, such as pure mathematics, AI, EC, and IT management science, decision trees, predictive economics game modeling. In 1987, he founded Ventures in 1987 for computing R&D. He began independent consulting with clients such as System Development Corporation SDC), the US Air Force Space Division, and GE Aerospace. Dr. Nourani has designed and developed AI robot planning and reasoning systems at Northrop Research and Technology Center, Palos Verdes, California. He also has comparable AI, software, and computing foundations and R&D experience at GTE Research Labs.




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