This chapter focuses on how to formulate constructive analogues for large regular ordinals and how to obtain notation systems for them using non-monotone inductive definitions. The chapter presents results locating the ordinals of inductive definitions in relation to the ordinals of certain wellorderings. Alternative characterizations of some of the reflecting properties of admissible ordinals are presented. The chapter presents a comparative analysis of reflecting properties of admissible ordinals with the reflecting properties for the indescribable cardinals, and investigate their relative magnitudes. The chapter examines the first order inductive definitions. The notion of a closed class of operators is formulated. The construction of the notation systems and the associated coding lemma are the key to getting lower bounds for the ordinals of inductive definitions. The coding lemma is proved in the chapter.
|Number of pages
|Studies in Logic and the Foundations of Mathematics
|Published - Jan 1 1974