Abstract
In this chapter we develop a constructive system of ordinals which we shall use in §16 and Chapter VIII for the proof-theoretic treatment of pure number theory and predicative analysis. In §13 we start from a non-constructive presentation of the. classical theory of ordinals in which we take as basis a corresponding axiomatic characterization of the set O, of all finite and denumerably infinite ordinals where, however, we do not go very far into set theory but use sets in a naive way. In this non-constructive framework we develop a hierarchy of normal functions and critical ordinals which determine a certain initital segment of the ordinals (in fact up to Γ0). This initial segment of the ordinals will be constructively characterized in §14 (by ordinal terms) and all the properties of this system of ordinal terms which we need later will be developed constructively. The full system of these ordinal terms (up to Γ0) is first used in Chapter VIII (for predicative analysis) while a subsystem (up to ε0) occurs in §16 (for pure number theory).
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