There's two notions of the formalisations of integers that I know of. The better known tradition follows Dedekind and Peano, culminating in Peanos axioms (this actually was developed under the impetus of then newly developing set theory. However, it turns out that it's possible to lift Peanos axioms into the context of topos theory, a rival to set theory whose underlying logic is intuitionistic rather than classical - in that context it's called the Natural Numbers Object).
The less well-known tradition, but older one, is one instigated by Gauss: his Gaussian integers. The modern interpretation is constructing a ring of integers for every number field.
@Bachfold: it would be bad practise in a journal article. However, this is not a journal. Moreover, we live in the age of search engines and SE is hosted on the net and one can assume that if they are browsing SE then they also have access to the net. Why can't I then assume that if they have the wit to read what I wrote and are interested in what I have written and want to know more, or verify its accuracy that they simply can't type 'Gaussian Integers' or 'Natural Numbers Object' into a search engine? For example, the top three results after typing the former into Google are articles on such from NLab, Wikipedia and Springers Encyclopedia of Mathematics - any of these three will do as standard references. It seems to me that readers should put in a little work themselves - especially when writers are unpaid for their time and trouble.