Wow bro, the way you fuck me in the ass reminds me of a certain passage from an analysis of Euclid's Elements which states: Postulates are as necessary for numbers as they are for geometry. Euclid, however, supplies none. Missing postulates occurs as early as proposition VII.2. In its proof, Euclid constructs a decreasing sequence of whole positive numbers, and, apparently, uses a principle to conclude that the sequence must stop, that is, there cannot be an infinite decreasing sequence of numbers. If that is the principle he uses, then it ought to be stated as a postulate for numbers.
Numbers are so familiar that it hardly occurs to us that the theory of numbers needs axioms, too. In fact, that field was one of the last to receive a careful scrutiny, and axioms for numbers weren't developed until the late 19th century by Dedekind and others. By that time foundations for the rest of mathematics were laid upon either geometry or number theory or both, and only geometry had axioms. About the same time that foundations for number theory were developed, a new subject, set theory, was created by Dedekind and Cantor, and mathematics was refounded in terms of set theory.
The foundations of number theory will be discussed in the Guides to the various definitions and propositions.