Peano axioms: Difference between revisions
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The '''Peano axioms''' are a set of | The '''Peano axioms''' are a set of [[axiom]]s that formally describes the [[natural number]]s (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]]. | |||
They were proposed by the Italian mathematician [[Giuseppe Peano]] in 1889. | |||
== The axioms == | == The axioms == | ||
Today the Peano axioms are usually formulated as follows: | |||
# Zero is a natural number. | # Zero is a natural number. | ||
# Every natural number has a successor | # Every natural number has a unique successor that also is a natural number. | ||
# Zero is not the successor of any natural number. | # Zero is not the successor of any natural number. | ||
# Different natural numbers have different successors. | # Different natural numbers have different successors. | ||
# If Zero has property ''P'', and | # If it is true that | ||
:( | ::(a) Zero has property ''P'', and | ||
: | ::(b) if for any given natural number ''n'' that has property ''P'' its successor | ||
: then all natural numbers have property ''P''. | |||
The last axiom is called the axiom (or rule) of [[induction (mathematics)|induction]]. | |||
The last axiom is called the [[ | |||
Revision as of 06:30, 1 November 2010
The Peano axioms are a set of axioms that formally describes 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. They were proposed by the Italian mathematician Giuseppe Peano in 1889.
The axioms
Today the Peano axioms are usually formulated as follows:
- Zero is a natural number.
- Every natural number has a unique successor that also is a natural number.
- Zero is not the successor of any natural number.
- Different natural numbers have different successors.
- If it is true that
- (a) Zero has property P, and
- (b) if for any given natural number n that has property P its successor
- then all natural numbers have property P.
The last axiom is called the axiom (or rule) of induction.