Gupta-Belnap-style circular definitions use all real numbers as possible starting points of revision sequences. In that sense they are boldface definitions. We discuss lightface versions of circular definitions and boldface versions of inductive definitions.

Gupta-Belnap-style circular definitions use all real numbers as possible starting points of revision sequences. In that sense they are boldface definitions. We discuss lightface versions of circular definitions and boldface versions of inductive definitions.

Let T be a suitable system of classical set theory. We will show, that the Σ1 spectrum of T, i.e. the set of ordinals having good Σ1 definition in T is an initial segment of the ordinals.

Determinacy axioms state the existence of winning strategies for infinite games played by two players on natural numbers. We show that a base theory enriched by a certain scheme of determinacy axioms is proof-theoretically equivalent to -comprehension.