In the formal sciences, the domain of discourse or universe of discourse (borrowing from the mathematical concept of universe) is the set of entities over which certain variables of interest in some formal treatment may range.
It is also defined as the collection of objects being discussed in a specific discourse.In model-theoretical semantics, a universe of discourse is the set of entities that a model is based on.
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated ZFC, where C stands for "choice", and ZF refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded.
Informally, Zermelo–Fraenkel set theory is intended to formalize a single primitive notion, that of a hereditary well-founded set, so that all entities in the universe of discourse are such sets. Thus the axioms of Zermelo–Fraenkel set theory refer only to pure sets and prevent its models from containing urelements (elements that are not themselves sets). Furthermore, proper classes (collections of mathematical objects defined by a property shared by their members where the collections are too big to be sets) can only be treated indirectly. Specifically, Zermelo–Fraenkel set theory does not allow for the existence of a universal set (a set containing all sets) nor for unrestricted comprehension, thereby avoiding Russell's paradox. Von Neumann–Bernays–Gödel set theory (NBG) is a commonly used conservative extension of Zermelo–Fraenkel set theory that does allow explicit treatment of proper classes.
Coherence theories of truth characterize truth as a property of whole systems of propositions that can be ascribed to individual propositions only derivatively according to their coherence with the whole. While modern coherence theorists hold that there are many possible systems to which the determination of truth may be based upon coherence, others, particularly those with strong religious beliefs, hold that the truth only applies to a single absolute system. In general, truth requires a proper fit of elements within the whole system. Very often, though, coherence is taken to imply something more than simple formal coherence. For example, the coherence of the underlying set of concepts is considered to be a critical factor in judging validity for the whole system. In other words, the set of base concepts in a universe of discourse must first be seen to form an intelligible paradigm before many theorists will consider that the coherence theory of truth is applicable.