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.