1. Zero is a number.

2. If
is a number, the successor of is a number.

3. zero is not the successor of a number.

4. Two numbers of which the successors are equal are themselves equal.

5. (induction axiom.) If a set of numbers contains zero and also
the successor of every number in , then every number is in .

Peano's axioms are the basis for the version of number
theory known as Peano arithmetic.

## See also

Induction Axiom,

Peano
Arithmetic
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## References

Wolfram, S. *A New Kind of Science.* Champaign, IL: Wolfram Media, p. 1152,
2002.## Referenced on Wolfram|Alpha

Peano's Axioms
## Cite this as:

Weisstein, Eric W. "Peano's Axioms." From
*MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/PeanosAxioms.html

## Subject classifications