The theory of Natural Numbers defined by the five Peano's Axioms. Any universal statement which is undecidable in Peano arithmetic is necessarily True. Undecidable statements may be either True or False. 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).

**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.

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1999-05-26