These axioms are based on Whitehead, "The Axioms of Projective Geometry". These principles should uniformly adhere to sound logical axioms or postulates. Ancient geometers maintained some distinction between axioms and postulates. Many mathematical axioms are based upon recursive rules. Algebraic structures are defined through different configurations of axioms. Some authors require theories to also include all logical consequences of the axioms. Babylonian thought was also based on an open-systems ontology which is compatible with ergodic axioms. This means that the second-order Peano axioms are categorical. This has been carried out for Arithmetic using Peano axioms. The smallest geometry satisfying all three axioms contains seven points. With the proper generalization of the group axioms this gives rise to an "n"-ary group. Fraenkel axioms (ZF) of set theory. In calculus, "foundations" refers to the rigorous development of the subject from axioms and definitions. Examples of such properties include connectedness, compactness, and various separation axioms. In logic programming, a program consists of a set of axioms and rules.