3.5 Soundness and completeness of first-order logic

First-order logic's soundness and completeness theorems are crucial pillars. Soundness ensures that provable formulas are valid in all models, while completeness guarantees that valid formulas are provable. These theorems bridge the gap between syntax and semantics.

The model existence theorem shows that consistent formula sets have models. This links consistency and satisfiability, key concepts in model theory. These ideas help us understand how formal systems relate to their interpretations in first-order logic.

Soundness and Completeness Theorems

Validity and Provability

Top images from around the web for Validity and Provability

• 

  Deductive reasoning - Wikipedia View original

  Is this image relevant?

• 

  Gödel's completeness theorem - Wikipedia View original

  Is this image relevant?

• 

  Deductive reasoning - Wikipedia View original

  Is this image relevant?

• 

  Gödel's completeness theorem - Wikipedia View original

  Is this image relevant?

1 of 2

Top images from around the web for Validity and Provability

• 

  Deductive reasoning - Wikipedia View original

  Is this image relevant?

• 

  Gödel's completeness theorem - Wikipedia View original

  Is this image relevant?

• 

  Deductive reasoning - Wikipedia View original

  Is this image relevant?

• 

  Gödel's completeness theorem - Wikipedia View original

  Is this image relevant?

1 of 2

• Soundness theorem proves that if a formula is provable in a formal system, then it is valid in all models of the system
  • Ensures that the formal system does not allow proving false statements
  • Provides confidence in the correctness of the formal system
• Completeness theorem demonstrates that if a formula is valid in all models of a formal system, then it is provable within that system
  • Guarantees that all valid formulas can be derived using the formal system's inference rules
  • Establishes that the formal system is powerful enough to prove all valid statements
• Gödel's completeness theorem, a specific instance of the completeness theorem, states that a formula is provable in first-order logic if and only if it is valid in all models
  • Proves the equivalence between semantic validity and syntactic provability in first-order logic
  • Fundamental result in mathematical logic, bridging the gap between syntax and semantics

Model Existence and Satisfiability

• Model existence theorem, a consequence of the completeness theorem, states that if a set of formulas is consistent, then there exists a model in which all the formulas are true
  • Demonstrates the relationship between consistency and satisfiability
  • Ensures that consistent sets of formulas have a model that satisfies them
• Satisfiability, a key concept in model theory, refers to the existence of a model that makes a formula or set of formulas true
  • Closely related to the concept of consistency, as consistent sets of formulas are satisfiable
  • Plays a crucial role in understanding the relationship between syntax and semantics in first-order logic

Constructing Models and Compactness

Henkin Construction and Consistency

• Henkin construction, a method for building models of consistent sets of formulas, involves extending the set with witness constants and ensuring its consistency
  • Starts with a consistent set of formulas and expands it by adding new constants and formulas
  • Ensures that the extended set remains consistent while providing witnesses for existential statements
  • Allows for the construction of a model that satisfies the original set of formulas
• Consistency, a property of sets of formulas, means that no contradiction can be derived from the set using the inference rules of the formal system
  • Essential for the existence of models, as inconsistent sets of formulas cannot have a model
  • Maintained throughout the Henkin construction process to guarantee the existence of a model

Compactness and Löwenheim-Skolem Theorems

• Compactness theorem states that if every finite subset of a set of formulas has a model, then the entire set has a model
  • Allows for reasoning about infinite sets of formulas by considering their finite subsets
  • Has important applications in model theory and proof theory, such as proving the existence of non-standard models of arithmetic
• Löwenheim-Skolem theorem, a consequence of the compactness theorem, states that if a countable set of formulas has an infinite model, then it has models of all infinite cardinalities
  • Demonstrates the existence of models of different sizes for a given set of formulas
  • Highlights the limitations of first-order logic in characterizing the cardinality of models
  • Leads to the concept of non-standard models, which satisfy the same formulas but have different properties than the intended model