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).
Peano Arithmetic
See also
First-Order Logic, Kreisel Conjecture, Natural Independence Phenomenon, Number Theory, Peano's Axioms, Propositional CalculusExplore with Wolfram|Alpha
References
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.Referenced on Wolfram|Alpha
Peano ArithmeticCite this as:
Weisstein, Eric W. "Peano Arithmetic." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PeanoArithmetic.html