Key Concepts of Incompleteness Theorems to Know for Proof Theory

Incompleteness theorems reveal the limits of formal systems in mathematics. Gödel's work shows that true statements can exist without proof, highlighting the challenges of consistency and completeness in formal logic, which connects deeply to the study of proof theory.

1. Gödel's First Incompleteness Theorem

   • States that in any consistent formal system that is capable of expressing basic arithmetic, there exist true statements that cannot be proven within the system.
   • Introduces the concept of self-reference through Gödel sentences, which assert their own unprovability.
   • Demonstrates the limitations of formal systems, showing that completeness cannot be achieved if the system is consistent.

2. Gödel's Second Incompleteness Theorem

   • Extends the first theorem by showing that no consistent system can prove its own consistency.
   • Implies that if a system can prove its own consistency, it must be inconsistent.
   • Highlights the inherent limitations of formal proofs regarding the consistency of mathematical systems.

3. Tarski's Undefinability Theorem

   • States that the truth of statements in a sufficiently rich language cannot be defined within that same language.
   • Establishes that truth is a meta-mathematical concept, leading to the conclusion that a complete and consistent theory cannot fully capture its own truth.
   • Relates to Gödel's theorems by emphasizing the limitations of formal systems in expressing certain properties.

4. Rosser's Theorem

   • Provides a stronger version of Gödel's First Incompleteness Theorem by showing that a system can be made to avoid certain self-referential constructions.
   • Demonstrates that there are true statements that cannot be proven even in systems that are more robust than those considered by Gödel.
   • Offers a method to construct undecidable propositions without relying on the notion of self-reference.

5. Church-Turing Thesis

   • Proposes that any function that can be computed algorithmically can be computed by a Turing machine.
   • Establishes a foundation for understanding computability and the limits of what can be computed.
   • Connects to incompleteness by suggesting that there are mathematical truths that cannot be computed or proven algorithmically.

6. Löb's Theorem

   • Addresses provability in formal systems, stating that if a statement can prove itself, then it is true.
   • Provides a framework for understanding self-referential statements in formal systems.
   • Relates to Gödel's work by exploring the implications of self-reference in the context of provability.

7. Chaitin's Incompleteness Theorem

   • Introduces the concept of algorithmic randomness and shows that there are true mathematical statements that cannot be proven due to their complexity.
   • Establishes a connection between incompleteness and the notion of randomness in mathematics.
   • Suggests that the complexity of certain mathematical truths leads to inherent limitations in formal proofs.

8. Paris-Harrington Theorem

   • A combinatorial statement that is true but cannot be proven within Peano Arithmetic, illustrating the limitations of formal systems.
   • Serves as an example of a mathematical truth that is independent of standard axiomatic systems.
   • Highlights the richness of mathematical structures that exceed the capabilities of formal proofs.

9. Goodstein's Theorem

   • A statement about sequences of natural numbers that is true but not provable in Peano Arithmetic.
   • Demonstrates the existence of true mathematical statements that transcend the capabilities of formal systems.
   • Connects to incompleteness by showing that certain properties of numbers can be established without formal proof.

10. Henkin's Completeness Theorem

    • States that if a set of sentences is consistent, then there exists a model in which all those sentences are true.
    • Establishes the relationship between syntactic consistency and semantic truth in formal systems.
    • Provides a foundation for understanding the completeness of logical systems, contrasting with the incompleteness results of Gödel.