4.1 Axioms, theories, and models

Axioms, theories, and models form the backbone of mathematical logic. They're like building blocks that help us understand complex mathematical structures. Axioms are the starting points, theories are the rules we build, and models are the concrete examples that bring it all to life.

This trio works together to create a framework for exploring mathematical truths. By studying how axioms shape theories and how models satisfy them, we gain insights into the nature of mathematical reasoning and the foundations of logic.

Axioms, Theories, and Models

Fundamental Concepts in Model Theory

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• Axioms function as foundational statements assumed true without proof within mathematical systems or logical frameworks
• Theories in model theory encompass sets of sentences in formal languages closed under logical consequence
• Models represent mathematical structures satisfying all axioms and sentences of given theories
• Model theory language incorporates symbols for relations, functions, and constants to express axioms and construct theories
• Satisfiability determines whether sets of sentences have models making all sentences true

Key Theorems and Results

• Löwenheim-Skolem theorem states theories in countable languages with infinite models have models of all infinite cardinalities
• Completeness theorem bridges proof-theoretic and model-theoretic approaches by equating consistency with model existence
• Compactness theorem proves existence of models for infinite theories by considering finite subsets of axioms

Relationships between Axioms, Theories, and Models

Foundational Connections

• Axioms provide basic assumptions for theories from which all other statements derive
• Theories achieve consistency when they possess models, linking syntactic consistency to semantic satisfiability
• Models offer concrete realizations of abstract concepts defined by axioms and theories

Advanced Relationships

• Elementary equivalence relates different models satisfying identical sentences within the same theory
• Categoricity describes theories with exactly one model (up to isomorphism) of each infinite cardinality
• Interplay between axioms, theories, and models enables study of mathematical structures through syntactic (proof-theoretic) and semantic (model-theoretic) methods

Constructing Theories and Models

Theory Construction Process

• Select formal language appropriate for expressing desired concepts and relationships
• Identify fundamental properties and relationships serving as basis for theory through axiom selection
• Perform consistency checking to ensure chosen axioms avoid contradictions
• Form complete theory through deductive closure of axiom set, including all logically derivable sentences
• Find or create mathematical structures satisfying all axioms and sentences of theory for model construction

Model Building Techniques

• Apply method of diagrams to systematically build models for consistent theories
• Utilize Henkin construction for model creation of consistent theories
• Employ compactness theorem to prove model existence for infinite theories by examining finite axiom subsets

Properties of Theories and Models

Formal System Properties

• Soundness ensures provable statements hold true in all models
• Completeness guarantees all true statements remain provable
• Categoricity in power describes theories with unique models (up to isomorphism) of given cardinalities

Model-Theoretic Concepts

• Analyze subsets or relations definable using theory language formulas through model-theoretic definability
• Löwenheim-Skolem-Tarski theorem extends Löwenheim-Skolem theorem to uncountable languages
• Saturation describes richness of types realized in models
• Homogeneity characterizes symmetry of model structures
• Stability classifies theories based on non-isomorphic model quantities across different cardinalities
• Quantifier elimination simplifies formulas and determines completeness and decidability of theories