12.1 Interpretations between theories

Interpretations between theories are powerful tools in model theory, allowing us to compare and connect different formal systems. They map one theory's language to another's formulas, preserving truth and logical structure while revealing unexpected relationships.

By establishing interpretations, we can analyze relative consistency, expressive power, and strength of theories. This helps us understand connections between mathematical domains and provides insights into the nature of mathematical truth and proof.

Interpretation of Theories

Definition and Purpose

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• Mapping preserves truth of formulas from one theory to another
• Compares expressive power and consistency of different formal systems
• Source theory interpreted into target theory
• Maps language of source theory to formulas in target theory language
• Preserves logical connectives and quantifiers
• Demonstrates target theory's expressive power relative to source theory
• Establishes relative consistency proofs between theories

Components and Structure

• Translation function defines non-logical symbols of source theory
• Maps atomic formulas of source theory to target theory formulas
• Applies translation recursively to composite formula components
• Preserves sentence satisfaction between corresponding models
• Maintains faithfulness by not collapsing distinct elements
• Ensures definability of source theory relations in target theory
• Provides effective algorithm to compute formula translations

Properties of Interpretations

Key Characteristics

• Preserves truth and logical structure
• Demonstrates relative strength and expressiveness of theories
• Reveals connections between seemingly unrelated mathematical theories
• Allows composition for comparing indirectly related theories
• Establishes partial ordering of theories based on strength
• Indicates equivalent expressive power through mutual interpretability
• Implies consistency of source theory if target theory is consistent

Specific Examples

• Interpretation of arithmetic in set theory (von Neumann ordinals)
• Conservative extension analysis through interpretations
  • T2 conservatively extends T1 if interpretation of T1 in T2 is identity on T1's language
• Relative consistency proofs using interpretability
• Non-interpretability results establish essential differences between theories

Relationships Between Theories

Comparative Analysis

• Partial ordering based on relative strength and expressiveness
• T2 at least as strong as T1 if T1 interprets into T2
• Equivalent expressive power indicated by mutual interpretability
• Consistency of T1 implied by consistency of T2 if T1 interprets into T2
• Unexpected connections revealed between mathematical theories
• Conservative extensions analyzed through interpretations
• Composition of interpretations allows comparison of indirectly related theories

Foundational Implications

• Interpretability impacts foundations of mathematics
• Consistent T1 interpreting inconsistent T2 implies T1's inconsistency
• Interpretability in weaker theory yields surprising strength results
• Non-interpretability establishes need for stronger axioms
• Relative consistency proofs closely related to interpretability
• Comparison of consistency strength between theories

Theory Interpretation: Consequences

Establishing Interpretations

• Provide translation function preserving truth and logical structure
• Classic example of arithmetic interpretation in set theory (von Neumann ordinals)
• Significant consequences for mathematical foundations
• Inconsistency of T1 if it interprets inconsistent T2
• Surprising strength results from interpretability in weaker theory
• Essential differences and necessity of stronger axioms shown by non-interpretability
• Relative consistency proofs utilize interpretability concept

Applications and Implications

• Compares consistency strength of theories
• Reveals unexpected relationships between mathematical domains
• Establishes hierarchies of theories based on interpretability
• Provides insights into the nature of mathematical truth and proof
• Helps identify minimal axiom systems for specific mathematical results
• Facilitates transfer of results between different areas of mathematics
• Supports development of new mathematical theories through interpretation