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All sets that can be put into a bijective relation to the natural numbers are said to have this kind of infinity. Here, S should be read as " successor ".
This monoid satisfies the cancellation property , and can be embedded in a group in the group theory sense of the word. The smallest group containing the natural numbers is the integers.
These properties of addition and multiplication make the natural numbers an instance of a commutative semiring. Semirings are an algebraic generalization of the natural numbers where multiplication is not necessarily commutative.
An important property of the natural numbers is that they are well-ordered : every non-empty set of natural numbers has a least element.
This Euclidean division is key to several other properties divisibility , algorithms such as the Euclidean algorithm , and ideas in number theory.
Two important generalizations of natural numbers arise from the two uses of counting and ordering: cardinal numbers and ordinal numbers.
For finite well-ordered sets, there is a one-to-one correspondence between ordinal and cardinal numbers; therefore they can both be expressed by the same natural number, the number of elements of the set.
This number can also be used to describe the position of an element in a larger finite, or an infinite, sequence. A countable non-standard model of arithmetic satisfying the Peano Arithmetic that is, the first-order Peano axioms was developed by Skolem in The hypernatural numbers are an uncountable model that can be constructed from the ordinary natural numbers via the ultrapower construction.
Other generalizations are discussed in the article on numbers. Many properties of the natural numbers can be derived from the five Peano axioms :  [i].
These are not the original axioms published by Peano, but are named in his honor. Some forms of the Peano axioms have 1 in place of 0. Replacing axiom 5 by an axiom schema, one obtains a weaker first-order theory called Peano arithmetic.
In the area of mathematics called set theory , a specific construction due to John von Neumann   defines the natural numbers as follows:.
Even if one does not accept the axiom of infinity and therefore cannot accept that the set of all natural numbers exists, it is still possible to define any one of these sets.
Although the standard construction is useful, it is not the only possible construction. Ernst Zermelo 's construction goes as follows: .
From Wikipedia, the free encyclopedia. This article is about "positive integers" and "non-negative integers".
For all the numbers A kind of number, used for counting. Main article: Peano axioms. Main article: Set-theoretic definition of natural numbers.
Mathematics portal. They follow that with their version of the Peano Postulates. Other tablets dated from around the same time use a single hook for an empty place.
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