# Chapter 15. Ordinal Analysis

Ordinal analysis studies the strength of formal theories by measuring how far their methods of proof can justify transfinite induction along well ordered structures.

The chapter begins with proof theoretic ordinals, where an ordinal is assigned to a formal theory as a measure of the induction principles and consistency arguments that the theory can support.

Transfinite induction is then introduced as a generalization of ordinary induction, allowing properties to be proved along well orders that extend beyond the natural numbers.

The strength of theories is studied by comparing which ordinals, induction principles, and consistency statements can be justified within different formal systems.

Applications to arithmetic show how ordinal methods clarify the power of systems such as Peano arithmetic and fragments of arithmetic with restricted induction.

The final part of the chapter discusses the limits of formal strength, where ordinal analysis reveals both how much can be proved by a theory and where stronger principles are needed.