6.1 Compactness theorem and its applications

The Compactness Theorem is a powerful tool in logic, stating that a set of sentences is satisfiable if every finite subset is satisfiable. It's a key property of first-order logic, with far-reaching consequences in model theory, algebra, and topology.

This theorem enables the construction of non-standard models, proves the existence of ultrafilters, and demonstrates limitations of first-order logic. It's crucial for consistency proofs and connects closely to the Löwenheim-Skolem theorems, forming a cornerstone of model theory.

The Compactness Theorem in Logic

Fundamental Principles of Compactness

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• Compactness Theorem states a set of first-order sentences proves satisfiable if and only if every finite subset proves satisfiable
• Equivalent to the statement that every consistent set of first-order sentences has a model
• Distinguishes first-order logic from higher-order logics as a fundamental property
• Proves far-reaching consequences in model theory, algebra, and topology (algebraic closures, Stone spaces)
• Demonstrates using either the completeness theorem for first-order logic or ultraproduct constructions
• Implies first-order logic cannot express certain properties (finiteness, countability)
• Relates closely to the Löwenheim-Skolem theorems, forming a cornerstone of model theory

Proof Techniques and Implications

• Utilizes proof by contradiction to establish the theorem
  • Assumes a set of sentences is unsatisfiable but every finite subset proves satisfiable
  • Leads to a contradiction, proving the original statement
• Employs the notion of consistency in first-order logic
  • Consistent set of sentences cannot derive a contradiction
  • Inconsistent set can prove any formula, including falsehood
• Connects to the notion of logical consequence
  • If $\Sigma \models \phi$, then there exists a finite subset $\Sigma_0 \subseteq \Sigma$ such that $\Sigma_0 \models \phi$
• Demonstrates limitations of first-order logic in capturing certain mathematical structures
  • Cannot axiomatize the standard model of arithmetic uniquely
  • Fails to characterize finite structures up to isomorphism

Applying the Compactness Theorem

Constructing Mathematical Structures

• Constructs models with specific properties by formulating appropriate sets of sentences
• Creates non-standard models (non-Archimedean ordered fields)
• Proves existence of ultrafilters and non-principal ultrafilters on infinite sets
• Shows existence of infinitely large cardinals in set theory (inaccessible cardinals)
• Enables construction of saturated models, important in model theory
• Proves existence of non-standard analysis models, including infinitesimals
• Involves crafting infinite sets of sentences capturing desired properties
  • Example: Constructing a non-standard model of arithmetic
    • Add new constant symbol c and axioms stating c > n for each natural number n
    • Compactness ensures consistency of this expanded theory

Applications in Various Mathematical Fields

• Demonstrates existence of non-principal ultrafilters on infinite sets
  • Crucial for constructing ultraproducts and ultrapowers
• Proves existence of algebraically closed fields of arbitrary characteristic
• Establishes properties of topological spaces (compactness in product topology)
• Constructs models in set theory with specific cardinal properties
• Applies to graph theory (infinite chromatic number, infinite cliques)
• Utilizes in algebra to prove existence of certain algebraic structures
  • Example: Constructing a field of characteristic 0 containing all finite fields
    • Create language with constant symbols for each element of each finite field
    • Add axioms stating field properties and relationships between elements
    • Compactness ensures existence of a model satisfying all these properties

The Compactness Theorem for Consistency

Consistency Proofs and Reductions

• Provides powerful tool for proving consistency of mathematical theories
• Reduces consistency proofs to checking consistency of finite subsets of axioms
• Proves instrumental in showing consistency of theories with infinitely many axioms
• Uses to prove consistency of extensions of well-known theories (Peano arithmetic)
• Enables construction of models for consistent theories, establishing their consistency
• Plays crucial role in independence proofs, showing certain statements unprovable from given axioms
• Demonstrates limitations of first-order axiomatizations of mathematical structures

Applications in Mathematical Logic

• Proves consistency of set theories (ZFC) relative to weaker theories
• Establishes independence results in arithmetic (Gödel's incompleteness theorems)
• Shows consistency of non-standard analysis relative to standard analysis
• Demonstrates consistency of infinitary combinatorial principles (large cardinal axioms)
• Applies to prove relative consistency of alternative set theories (New Foundations)
• Utilizes in reverse mathematics to analyze strength of mathematical theorems
• Establishes conservativity results between different mathematical theories
  • Example: Proving consistency of PA + Con(PA)
    • Add new constant symbol c and axioms stating c codes a proof of contradiction in PA
    • Show any finite subset proves consistent with PA
    • Compactness implies consistency of entire theory, establishing Con(PA) unprovable in PA

Non-Standard Models of Arithmetic

Construction and Properties

• Constructs non-standard models of arithmetic satisfying all first-order statements true in standard model
• Contains elements behaving like "infinite" natural numbers
• Adds new constant symbol c and axioms stating c greater than each standard natural number
• Ensures expanded theory consistent through Compactness Theorem
• Produces uncountable models containing copies of standard natural numbers as initial segment
• Demonstrates limitations of first-order logic in capturing full structure of natural numbers
• Implies important consequences for foundations and philosophy of mathematics

Implications and Applications

• Reveals inadequacy of first-order Peano Arithmetic to categorically axiomatize natural numbers
• Provides framework for non-standard analysis and infinitesimal calculus
• Offers new perspective on concepts like infinity and infinitesimals in mathematics
• Applies to study of formal theories of arithmetic and their limitations
• Enables construction of models satisfying certain number-theoretic properties
• Utilizes in proof theory to analyze strength of various arithmetical systems
• Demonstrates existence of non-standard prime numbers and their properties
  • Example: Constructing a non-standard prime number
    • Formulate theory stating existence of number greater than all standard primes
    • Add axioms ensuring this number proves prime (not divisible by any standard prime)
    • Compactness guarantees existence of model with non-standard prime