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well-ordering principle for natural numbers (Axiom)

Every nonempty set $ S$ of natural numbers contains a least element; that is, there is some number $ a$ in $ S$ such that $ a \leq b$ for all $ b$ belonging to $ S$.

Beware that there is another statement (which is equivalent to the axiom of choice) called the well-ordering principle. It asserts that every set can be well-ordered.

Note that the well-ordering principle for natural numbers is equivalent to the principle of mathematical induction (or, the principle of finite induction).



"well-ordering principle for natural numbers" is owned by smw. [ full author list (2) | owner history (1) ]
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See Also: maximality principle, well ordered set, existence and uniqueness of the gcd of two integers


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well-ordering principle for natural numbers proven from the principle of finite induction (Proof) by smw
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Cross-references: principle of finite induction, induction, well-ordered, well-ordering principle, axiom of choice, number, least element, contains, natural numbers
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This is version 13 of well-ordering principle for natural numbers, born on 2001-10-16, modified 2007-06-23.
Object id is 244, canonical name is WellOrderingPrinciple.
Accessed 5646 times total.

Classification:
AMS MSC: 06F25 (Order, lattices, ordered algebraic structures :: Ordered structures :: Ordered rings, algebras, modules)
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Arithmetic Progressions of Prime Numbers by Manoj on 2003-06-26 21:23:47
There exist many examples of (finite) Arithmetic progressions consisting of primes like 5,11,17,23,29 and 61,67,73,79. the longest such example known has 22 primes in arithmetic progression. However, it is not known
if there are arbitrarily long arithmetic progressions of primes.
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general well-ordering principle by vampyr on 2001-10-18 17:44:40
Maybe we should be specific, and say "well-ordering principle for natural numbers" and "general well-ordering principle", because when you say it, it seems to mean "the naturals can be well-ordered", and when I say it, it means "all sets can be well-ordered." I don't know which is the standard definition.
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