The theory of natural numbers defined by the five Peano's axioms. Paris and Harrington (1977) gave the first "natural" example of a statement which is true for the integers but unprovable in Peano arithmetic (Spencer 1983).

Kirby, L. and Paris, J. "Accessible Independence Results for Peano Arithmetic." Bull. London Math. Soc. 14, 285-293, 1982.

Paris, J. and Harrington, L. "A Mathematical Incompleteness in Peano Arithmetic." In Handbook of Mathematical Logic (Ed. J. Barwise). Amsterdam, Netherlands: North-Holland, pp. 1133-1142, 1977.

Spencer, J. "Large Numbers and Unprovable Theorems." Amer. Math. Monthly 90, 669-675, 1983.

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Weisstein, Eric W. "Peano Arithmetic." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PeanoArithmetic.html