Set recursion and \(\Pi ^ 1_ 2\)-logic
This paper essentially has two parts. In an expository section central elements of \(\Pi^ 1_ 2\)-logic are introduced via denotation systems. The notion of a denotation system is a reformulation of the notion of a dilator. The well-founded decomposition of denotation systems is proved. The general theory is then applied on questions concerning set-recursion relativized to definable ordinal functions. If F:On\(\to On\) is set- recursive relative to a \(\Delta_ 1\)-function f:On\(\to On\), then there is a denotation system D such that F is bounded by a function G definable from f, D by recursion over a decomposition of D. A possible notion of general recursion over the decomposition of a denotation system is also discussed. The results of the paper have later been improved by J. Vauzeilles.
scientific article; zbMATH DE number 3918358
scientific article