Fixed points for the jump operator

Abstract

We construct an example of an admissible set which is a fixed point for the Σ-jump operator. Also a number of basic properties of fixed points are presented. Dedicated to the memory of my motherIn the present account, we give a positive solution to the existence problem [1, 2] for a fixed point of the Σ-jump introduced in [1]. Applying the main result of this paper makes it possible to improve one of the basic results in [3]. A fixed-point existence theorem is proved in Sec. 2 and properties of fixed points are listed in Sec. 3. PRELIMINARIESThe present paper deals with research in admissible set theory. Use is made of methods developed in computability theory, model theory, and admissible set theory proper. For basic information on these areas of research, the reader is referred to [4][5][6][7]. Here we only focus on the notation and methods of admissibility theory needed in what follows.The equality relation is denoted by ≈. For a set X, by card(X) and 2 X we denote, respectively, the cardinality and the powerset (i.e., the set of all subsets) of X. For a function f , Γ f , δf , and ρf stand for, respectively, the graph, the domain, and the range of f .Let a language consist of symbols in {U 1 , ∈ 2 , ∅}, where U is interpreted as a set of urelements (in which case the formula ¬ U defines the set of 'sets'), ∈ as the membership relation (in which *