Peano axioms: Difference between revisions
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imported>Johan Förberg |
imported>Johan Förberg |
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# If Zero has property ''P'', and if it can be shown that: | # If Zero has property ''P'', and if it can be shown that: | ||
:(a) If a given natural number ''n'' has property ''P'', | :(a) If a given natural number ''n'' has property ''P'', | ||
:(b) | :(b) It follows that its successor S''n'' also has ''P'', | ||
:Then | :Then all natural numbers have the property ''P''. | ||
The last axiom is called the [[rule of induction]]. | The last axiom is called the [[rule of induction]]. | ||
Revision as of 17:35, 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) It follows that its successor Sn also has P,
- Then all natural numbers have the property P.
The last axiom is called the rule of induction.