14.4 Proof theory in mathematical practice and foundations

Proof theory plays a crucial role in mathematical practice and foundations. It investigates the structure of mathematical proofs, exploring concepts like reverse mathematics, predicativity, and foundational programs. These approaches help us understand the minimal axioms needed for theorems and the nature of mathematical reasoning.

Consistency proofs, proof-theoretic reductions, and ordinal analysis are key tools in this field. They allow us to compare the strength of formal systems, measure their expressive power, and establish relationships between different mathematical theories. This deeper understanding of proof structures enhances our grasp of mathematical foundations.

Foundational Approaches

Reverse Mathematics and Predicativity

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• Reverse mathematics investigates the minimal axioms required to prove theorems in mathematics
• Involves proving equivalences between mathematical statements and subsystems of second-order arithmetic
• Predicative mathematics avoids impredicative definitions, which define an object using a quantifier whose domain includes the object being defined
• Impredicativity, in contrast, allows such definitions (e.g., defining a set in terms of a collection that includes the set itself)
• Predicativism is a foundational stance that rejects impredicative definitions to avoid potential circularity or paradoxes

Foundational Programs and Constructive Reverse Mathematics

• Foundational programs are systematic approaches to providing rigorous foundations for mathematics
• Examples include Hilbert's program, logicism (Russell and Whitehead), and intuitionism (Brouwer)
• These programs aim to secure the foundations of mathematics using different philosophical and logical principles
• Constructive reverse mathematics is a variant of reverse mathematics using intuitionistic logic
• Investigates the constructive content of mathematical theorems and the axioms needed to prove them constructively

Consistency and Reducibility

Consistency Proofs and Relative Consistency

• Consistency proofs demonstrate that a formal system (e.g., a set theory like ZFC) is free from contradictions
• Relative consistency proofs show that if one system (S1) is consistent, then another system (S2) is also consistent
• For example, Gödel's relative consistency proof showed that if ZF set theory is consistent, then ZFC (ZF with the Axiom of Choice) is also consistent
• Relative consistency is a powerful tool for comparing the strength and reliability of different formal systems

Proof-Theoretic Reductions and Strength

• Proof-theoretic reductions are techniques for comparing the strength of formal systems
• System S1 is proof-theoretically reducible to S2 if the consistency of S1 can be proved in S2
• Proof-theoretic strength measures the power and expressiveness of a formal system
• Stronger systems can prove the consistency of weaker ones, but not vice versa (by Gödel's Incompleteness Theorems)
• Examples of proof-theoretic reductions include reducing Peano Arithmetic to primitive recursive arithmetic and reducing second-order arithmetic to first-order arithmetic

Ordinal Analysis

Proof-Theoretic Ordinals and Measuring Strength

• Proof-theoretic ordinals are ordinal numbers used to measure the strength of formal systems
• Each system is assigned an ordinal that represents the transfinite induction principles it can prove
• For example, the proof-theoretic ordinal of Peano Arithmetic is $\varepsilon_0$, the least ordinal $\alpha$ such that $\omega^\alpha = \alpha$
• Ordinal analysis is the process of determining the proof-theoretic ordinal of a formal system
• Allows for fine-grained comparisons of proof-theoretic strength beyond what can be shown by consistency proofs alone
• Gentzen's consistency proof for Peano Arithmetic used transfinite induction up to $\varepsilon_0$, establishing its proof-theoretic ordinal