Book ChapterDOI
Recursively Mahlo Ordinals and Inductive Definitions
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A characterization of recursively Mahlo ordinals and inductive definitions is given in this article, and a characterization of the first recursive hyper-Mahlo ordinal is provided.Abstract:
Publisher Summary This chapter discusses the recursively Mahlo ordinals and inductive definitions. A characterization of the first recursively Mahlo ordinal and the first recursively hyper-Mahlo ordinalis provided. The large countable ordinals are obtained as the closure ordinals of inductive definitions. Inductive definitions play a central role in hierarchy theory. A classic example is the theory of recursive ordinals. The usual systems of notations for the recursive ordinals are inductively defined by very simple (arithmetic) operations. A version of the Candy theorem on the existence of selection operators is the basic tool. A typical system is defined by an inductive definition consisting of several cases, depending on whether the ordinal reached at a given stage was zero, a successor, notationally singular, and notationally regular.read more
Citations
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Book ChapterDOI
An Introduction to Inductive Definitions
TL;DR: This chapter discusses monotone induction and its role in extensions of recursion theory and briefly considers inductive definitions in a more general context.
Journal ArticleDOI
Fixed-point extensions of first-order logic
TL;DR: It is proved that the three extensions of first-order logic by means of positive induction, monotone inductions, and so-called non-monotone (in the terminology, inflationary) inductions respectively, all have the same expressive power in the case of finite structures.
Journal ArticleDOI
Proof-theoretic analysis of KPM
TL;DR: It is shown that a certain ordinal notation system is sufficient to measure the proof-theoretic strength of KPM, which involves a detour through an infinitary calculus RS(M), for which several cutelimination theorems are proved.
References
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Journal ArticleDOI
Recursive functionals and quantifiers of finite types. I
TL;DR: A theory of recursive functions and predicates of variables of these types has not previously been developed as mentioned in this paper, but it is possible to extend the notion of primitive recursiveness to the higher types of variables.
Journal ArticleDOI
The next admissible set
TL;DR: In this article, generalizations of several approaches to the hyperarithmetic hierarchy are described, and the Kripke-Platek theory of admissible ordinals and sets is studied.
Journal ArticleDOI
Applications of Strict Pi 1 1 Predicates to Infinitary Logic.
TL;DR: The reduction just mentioned requires proof, and the proof uses some form of the Brower-Konig Infinity Lemma.