Peano's Axioms
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.
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