On a hierarchy of univalent systems of notation for countable ordinals

Abstract

In Chapter II of this thesis we develop an inductive method of construction of the hierarchy of univalent systems of notation for countable ordinals. It makes use of the normal functionλα[γα] defined on CII as follows: Each system of notation corresponds to a class of exponential polynomials in γα. We obtain the notations for ordinals by encoding their Cantor normal forms. Next we show that a countable number of fixed points of the function λα[γα] are constructive. It is done by means of a hierarchy of normal functions. Each function in the hierarchy takes fixed points of preceeding functions as its values. Further we formulate a condition under which a set of notations of a given system from the hierarchy is r.e. and can be generated simultaneously with a total linear ordering on it, isomorphic to the corresponding class of exponential polynomials. Finally, we show that this condition is satisfied for an initial segment of the hierarchy. Prior to all that, in Chapter I, we give the detailed proofs of all necessary results about countable ordinals and their arithmetic, classes of exponential polynomials, normal form and normal functions on CII.