The theory of Peano Arithmetic, or PA, is the theory in the formal language with usual logical symbols, symbols for +, ., <, 0, 1, variables x,y,z,... (thought as ranging over natural numbers). The axioms of PA state that the domain is the nonnegative part of a discretely ordered ring, satisfying the induction principle

for all statements phi(x) expressible in the above language.phi(0) and forall x ( phi(x) -> phi(x+1) ) implies forall x phi(x)

The theory of Presburger Arithmetic is essentially the same, but the language is restricted so that it only contains symbols for +, <, 0, 1, and the usual logical symbols.

Some introductory texts on Peano arithmetic include

- Richard Kaye,
*Models of Peano arithmetic*. Oxford Logic Guides 15, OUP 1991, ISBN 0 19 853213 X, 292 pages. ---Concentrates of the model theory of PA, should be readable by any student with a minimal amount of background in first-order logic and in computability. - Craig Smorynski,
*Logical Number Theory I*. Springer Verlag Universitext, 1991, ISBN 0-387-52236-0. ---Idiosyncratic, but highly entertaining. Includes some material on Presburger[-Skolem] arithmetic. - Petr Hajek and Pavel Pudlak,
*Metamathematics of First-order Arithmetic*. Springer Verlag Perspectives in Mathematical Logic, 1993, ISBN 0-387-50632-2, ISBN 3-540-50632-2. ---Almost encyclopaedic in places, and probably difficult reading for a newcomer to the area. Contains a useful account of bounded arithmetic.