3. On the Performance of Axiom Systems

The chapter delves into the performance of axiom systems, tracing back to Hilbert’s advocacy of the axiomatic method and the quest for consistency in mathematics. It discusses the significance of Gödel’s incompleteness theorems, which wrecked Hilbert’s dream of a finitistic consistency proof for axiom systems comprising the theory of natural numbers. The chapter also highlights the work of Gerhard Gentzen, who succeeded in giving a consistency proof for an axiom system of Peano arithmetic using infinitary means. The focus then shifts to the connection between consistency and ordinal analysis, introducing the concept of proof-theoretic ordinals and their role in measuring the performance of axiom systems. The chapter further explores the genesis of proof-theoretic ordinals, their connection to abstract structures, and the development of semi-formal systems as a bridge between proofs and semantics. The use of semi-formal systems allows for a detailed analysis of the performance of axiom systems, with a particular focus on the distance between the proof-theoretic ordinal of a structure and its axiom system. The chapter concludes by discussing the potential for enhancing the performance of axiom systems and the limitations thereof, making it a comprehensive resource for understanding the intricacies of axiomatic systems and their analysis.