Publisher Summary This chapter describes the regular ordinals and normal forms, demonstrates that the normal forms developed include those of Bachmann, yields a recursive ordering for a segment of the countable ordinals, and indicates how they can be used to replace ordinal diagrams in some of Takeuti's work. Although the normal forms are defined in terms of normal functions based on Bachmann collections, they are expressible as finite ordered sets of ordinals. To develop the normal forms, Bachmann collections is considered with each member θ of which will be associated finite sets of ordinals called the parts of θ. These finite sets include the ordinals occurring in the normal form of θ. Various consistency proofs and cut elimination theorems are carried through by means of an induction on the ordinal of the proof tree. Frequently, an ordinal is coordinated with a tree by assigning particular ordinals to axioms and then specifying for each rule of inference a function that determines the ordinal of the conclusion given the premiss and the ordinal of the premiss.
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