Peano axioms: Difference between revisions
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== The axioms == | == The axioms == | ||
The axioms can be formulated as follows: | |||
# Zero is a natural number. | |||
# Every natural number has a successor, which is also a natural number. | |||
# Zero is not the successor of any natural number. | |||
# Different natural numbers have different successors. | |||
# If Zero has property ''P'', and if it can be shown that: | |||
:(a) If a given natural number ''n'' has property ''P'', | |||
:(b) Then its successor S''n'' also has ''P'', | |||
:Then it follows that all natural numbers have the property ''P''. | |||
The last axiom is called the [[rule of induction]]. | |||
Revision as of 17:34, 31 October 2010
The Peano axioms are a set of formal axioms describing the natural numbers (0, 1, 2, 3 ...). Together, they describe some of the most important properties of the natural numbers: their infinitude, zero as the smallest natural number and the rule of induction.
The axioms
The axioms can be formulated as follows:
- Zero is a natural number.
- Every natural number has a successor, which is also a natural number.
- Zero is not the successor of any natural number.
- Different natural numbers have different successors.
- If Zero has property P, and if it can be shown that:
- (a) If a given natural number n has property P,
- (b) Then its successor Sn also has P,
- Then it follows that all natural numbers have the property P.
The last axiom is called the rule of induction.