• There is a natural number 0.
• Every natural number a has a natural number successor, denoted by S(a).^ Successor rule : For every natural number n there is exactly one other natural number called its successor s(n) .
  • Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]
  
  ^ Additive Inverse: For any natural number n other than zero, n, there is exactly one number -n which not a natural number, and which called the additive inverse of n, where n + -n = 0 .
  • Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]
  
  ^ John Foster says that Natural numbers “are the ones we use to count things that are there.” But that’s his definition…his opinion.
  • Is 0 a Natural Number? :: squareCircleZ 1 February 2010 3:28 UTC www.squarecirclez.com [Source type: FILTERED WITH BAYES]
  
  .Intuitively, S(a) is a+1.
• There is no natural number whose successor is 0.
• Distinct natural numbers have distinct successors: if a ≠ b, then S(a) ≠ S(b).
• If a property is possessed by 0 and also by the successor of every natural number which possesses it, then it is possessed by all natural numbers.^ In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself.
  • The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]
  
  ^ Because all non-prime numbers can be decomposed into a product of underlying primes, then either this resultant number is prime itself, or there is a prime number or prime numbers which the resultant number could be decomposed into but are not in the original finite set of primes.
  • The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]
  
  ^ However there is no evidence to suggest that starfish have 5 arms because 5 is a prime number.
  • The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]
  
  .(This postulate ensures that the proof technique of mathematical induction is valid.^ Proof:  Its contra positive is a subset of the natural numbers, which does not possess a least element, is null.  Hence, it obviously is equivalent to the Mathematical Induction axiom.  QED. .
  • Natural to Complex Numbers 1 February 2010 3:28 UTC www.rism.com [Source type: Academic]
  
  ^ Proof by Mathematical Induction.
  • Natural to Complex Numbers 1 February 2010 3:28 UTC www.rism.com [Source type: Academic]
  
  )

Constructions based on set theory

A standard construction

We set 0 := { }, the empty set,

and define for every set a. S(a) is the successor of a, and S is called the successor function.

If the axiom of infinity holds, then the set of all natural numbers exists and is the intersection of all sets containing 0 which are closed under this successor function.

If the set of all natural numbers exists, then it satisfies the Peano axioms.

Each natural number is then equal to the set of natural numbers less than it, so that

• 0 = { }
• 1 = {0} = {{ }}
• 2 = {0,1} = {0, {0}} = {{ }, {{ }}}
• 3 = {0,1,2} = {0, {0}, {0, {0}}} = {{ }, {{ }}, {{ }, {{ }}}}
• 

and so on. When a natural number is used as a set, this is typically what is meant. Under this definition, there are exactly n elements (in the naïve sense) in the set n and n ≤ m (in the naïve sense) if and only if n is a subset of m.

Also, with this definition, different possible interpretations of notations like Rⁿ (n-tuples versus mappings of n into R) coincide.

Even if the axiom of infinity fails and the set of all natural numbers does not exist, it is possible to define what it means to be one of these sets. A set n is a natural number means that it is either 0 (empty) or a successor, and each of its elements is either 0 or the successor of another of its elements.

Other constructions

one could define 0 = { }

and S(a) = {a},

producing

0 = { }

1 = {0} = {{ }}

2 = {1} = {{{ }}}, etc.

and S(a) = a ∪ {a}

producing

0 = {{ }}

1 = {{ }, 0} = {{ }, {{ }}}

2 = {{ }, 0, 1}, etc.

See also

Integer Whole number Countable set

Mathematics portal

Notes

References

• Edmund Landau, Foundations of Analysis, Chelsea Pub Co. ISBN 0-8218-2693-X.
• Richard Dedekind, Essays on the theory of numbers, Dover, 1963, ISBN 0486210103 / Kessinger Publishing, LLC , 2007, ISBN 054808985X
• N. L. Carothers. Real analysis. .Cambridge University Press, 2000. ISBN 0521497566
• Brian S. Thomson, Judith B. Bruckner, Andrew M. Bruckner.^ In Out of the Shadows: Contributions of 20 th Century Women to Physics ; Byers, N.; Williams, G., Eds.; Cambridge University Press: Cambridge, England, 2005; pp 371384.
  • Francium (Atomic Number 87), the Last Discovered Natural Element 1 February 2010 3:28 UTC chemeducator.org [Source type: FILTERED WITH BAYES]
  
  Elementary real analysis. ClassicalRealAnalysis.com, 2000. ISBN 0130190756

External links

• Axioms and Construction of Natural Numbers
• Essays on the Theory of Numbers by Richard Dedekind at Project Gutenberg

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