The N0 Natural Number Sequence

The N0 Natural Number Sequence is an Infinite Number Sequence defined in terms the Successor Function, Peano's Axioms, and the starting element of Zero.

• AKA: N0, The Natural Number Sequence.
• Context:
  • It can be represented as: {0 < 1 < 2 < 3... < ∞}.
  • It can be used to define a Countable Set.
  • …
• Counter-Example(s):
  • N1: {1 < 2 < 3... < ∞}.
  • The Integers {-∞ ...<-2<-1<0<1<2,...<∞}
• See: The N1 Natural Number Sequence, Countable Set, Integer Number Set, Real Number Set.

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References

2009

• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Natural_number
  • In mathematics, a natural number means either an element of the set {1, 2, 3, ...} (the positive integers) or an element of the set {0, 1, 2, 3, ...} (the non-negative integers).
  • Natural numbers have two main purposes: counting ("there are 3 apples on the table") and ordering ("this is the 3rd largest city in the country").
  • Properties of the natural numbers related to divisibility, such as the distribution of prime numbers, are studied in number theory. Problems concerning counting, such as Ramsey theory, are studied in combinatorics.
  • Mathematicians use N or \mathbb{N} (an N in blackboard bold, displayed as ℕ in Unicode) to refer to the set of all natural numbers. This set is countably infinite: it is infinite but countable by definition. This is also expressed by saying that the cardinal number of the set is aleph-null (\aleph_0).
  • To be unambiguous about whether zero is included or not, sometimes an index "0" is added in the former case, and a superscript "*" or subscript "1" is added in the latter case:
    • \mathbb{N}_0 = \{ 0, 1, 2, \ldots \}; \quad \mathbb{N}^* = \mathbb{N}_1 = \{ 1, 2, \ldots \}.
• WordNet http://wordnet.princeton.edu/perl/webwn?s=natural%20number
  • the number 1 and any other number obtained by adding 1 to it repeatedly

• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Natural_number#Peano_axioms
  • The Peano axioms give a formal theory of the natural numbers starting with 0. The axioms are:
    • There is a natural number 0.
    • Every natural number a has a natural number successor, denoted by S(a). 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. (This postulate ensures that the proof technique of mathematical induction is valid.)
  • It should be noted that the "0" in the above definition need not correspond to what we normally consider to be the number zero. “0" simply means some object that when combined with an appropriate successor function, satisfies the Peano axioms.

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