Logic Patterns in Prime Numbers

In mathematics, an explicit description of a set is a definition of a set. An explicit description is a description of a set that does not rely on the axiom of choice. For example, the set of all finite initial segments of the natural numbers is an explicit description of the natural numbers. The natural numbers can also be described by means of the von Neumann ordinal definition or the inductive definition, but these methods rely on the axiom of choice. ”Explicit” is a mathematical term used to refer to an object that is specified without requiring further definition. A set can be defined by an ”explicit definition” or an ”inductive definition”. Explicitly described sets are those that are completely determined by the axioms of Zermelo–Fraenkel set theory and the axioms of boolean logic using first order logic or second order logic or perhaps third order logic. Explicitly specified sets may be defined by quantified Boolean formulae. Explicit definitions of sets do not involve the axiom of choice. Explicit definitions are to be distinguished from axiomatic descriptions of sets such as the axiom of replacement, the axiom of pairing, the axiom of union and the axiom of infinity