See the paper [Chris Mortensen, Inconsistent Number Systems, Notre
Dame
Journal of Formal Logic, Volume 29, Number 1, Winter 1988](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.ndjfl/1093637770).

This paper continues the study of so-called "inconsistent" number
systems, that is, structures considered in a non-classical logic that allows
at least some statements to be both true and false (bizarre, I know). He uses a
three-valued logic containing *true*, *false* and a third value
fruitfully interpreted as *true-and-false*, but please see his paper for the details. He is
particularly interested in finite structures in which all the
classical theorems of PA remain true (and some of them also
false). He writes, "it is one purpose of this paper to...[display]
inconsistent theories which contain various well-known classical
consistent complete subtheories."