👉 Penologies are a type of mathematical theorem that deals with properties of real numbers and is used in several branches of mathematics, including number theory, analysis, and algebra. They were first introduced by Richard Dedekind to describe properties of the set of all integers. In essence, a penology concerns itself with properties of the real numbers that can be deduced from the axioms of arithmetic. These properties are not necessarily provable in Zermelo-Fraenkel set theory (ZF), but