Classes and truths in set theory

This article studies three most basic systems of truth as well as their subsystems (and one extra system) over set theory possibly with or the axiom of global choice , and then correlates them with subsystems of Morse–Kelley class theory . The article aims at making an initial step towards the axiomatic study of truth in set theory in connection with class theory. Some new results on the side of class theory, such as conservativity, forcing and some forms of the reflection principle, are also presented. The equivalence results among systems of truth and classes obtained in this article are summarized in Theorem 104 (Section 7.1), Theorem 107 (Section 7.2) and Theorem 108 (Section 8).