Some developments in the theory of numerations
A. H. Kruse · Transactions of the American Mathematical Society · 1960
This paper has arisen from attempts to sharpen Specker's result, which is sharpened in 7.1 and 7.2 (first cf. the definition of H(m; a) in ?7 prior to 7.1). The writer's efforts along these lines led to developments in the theory of numerations (defined in the first paragraph of ?2) of independent interest, and most of this paper is concerned with these developments. The content of this paper may be developed in an axiomatic set theory of the von Neumann-Bernays-Godel kind (cf., e.g., [1]) modified as follows to allow (but not to imply the existence of) elements which are not sets. Each object is either an element or a class. A set is any element which is a class. An atom is an element which is not a class. The usual axioms may be modified in the obvious way to accommodate atoms. We shall assume all the usual axioms so modified except the restrictive axiom and the axiom of choice (cf. ?7). We indicate briefly our use of some terminology and notation. Elements