# Natural number

The letter N with a double line
stands for the set of natural numbers

The natural numbers are the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, etc. used in counting . Depending on the definition, 0 ( zero ) can also be counted as a natural number. The set of natural numbers, together with addition and multiplication, form a mathematical structure called a commutative half-ring .

## Naming conventions

The set of natural numbers is abbreviated with the formula symbol . In the widely used Unicode character encoding , it is the character with the code point (with the “number”) U + 2115 (ℕ). ${\ displaystyle \ mathbb {N}}$

It includes either the positive whole numbers (i.e. without the 0)

${\ displaystyle \ mathbb {N} = \ {1; 2; 3; \ ldots \}}$

or the non-negative whole numbers (including the 0)

${\ displaystyle \ mathbb {N} = \ {0; 1; 2; 3; \ ldots \}}$.

Both conventions are used inconsistently. The older tradition does not include zero as a natural number (zero only became common in Europe from the 13th century). This definition is more common in mathematical areas such as number theory , where the multiplication of natural numbers is paramount. In logic, set theory and computer science , on the other hand, the definition with zero is more common and simplifies the representation. Only with the latter convention do the natural numbers form a monoid with addition . In case of doubt, the definition used must be stated explicitly.

In 1888 , Dedekind introduced the symbol N for the set of natural numbers without zero . Its symbol is now often stylized as the letter N with a double stroke ( or ). From 1894 Peano used the symbol N 0 for the natural numbers with zero , which today is also stylized and defined by Peano . ${\ displaystyle \ mathbb {N}}$${\ displaystyle \ mathrm {I \! N}}$${\ displaystyle \ mathbb {N} _ {0}: = \ mathbb {N} \ cup \ {0 \}}$

However, if the symbol for the natural numbers with zero is used, then the set of natural numbers without zero is denoted by. The DIN standard  5473, for example, uses for the non-negative whole numbers and for the positive whole numbers. In some federal states, German school books are based on this DIN standard, in others, e.g. B. in Bavaria , not. ${\ displaystyle \ mathbb {N}}$${\ displaystyle \ mathbb {N} _ {+} {\ text {,}} \ mathbb {N} ^ {+} {\ text {,}} \ mathbb {N} ^ {*} {\ text {,} } \ mathbb {N} _ {> 0} {\ text {,}} \ mathbb {N} _ {1} \, {\ text {or}} \, \ mathbb {N} \ setminus \ {0 \} \,}$${\ displaystyle \ mathbb {N}}$${\ displaystyle \ mathbb {N} ^ {*}}$

Ultimately, it is a question of definition which of the two sets is to be regarded as more natural and which one is to use this designation as a linguistic distinction.

## Axiomatization

Richard Dedekind first implicitly defined natural numbers using axioms in 1888 . Independently of him, Giuseppe Peano set up a simpler and at the same time formally precise system of axioms in 1889. These so-called Peano axioms have prevailed. While the original system of axioms can be formalized in second-level predicate logic , a weaker variant in first-level predicate logic, which is known as Peano arithmetic , is often used today . Other axiomatizations of natural numbers that are related to Peano arithmetic are, for example, Robinson arithmetic and primitive recursive arithmetic .

The Peano axioms can also be understood as a definition of natural numbers. A set of natural numbers is then a set that satisfies the Peano axioms. It is important that there are infinitely many such sets. However, each of these sets behaves in exactly the same way, the elements are simply labeled differently. In mathematics it is said that the sets are isomorphic . This result is also called Dedekind's uniqueness theorem. As a result, it has been conventionally agreed to say “the natural numbers”, although strictly speaking there are an infinite number of such quantities.

## Von Neumann's model of natural numbers

John von Neumann gave a way to represent the natural numbers by sets, i.e. That is, he described a set- theoretical model of natural numbers.

{\ displaystyle {\ begin {alignedat} {3} 0 &: = && \ quad \ \, \ {\} && = \ emptyset \\ 1 &: = 0 '&& = \ {0 \} && = \ {\ emptyset \ } \\ 2 &: = 1 '&& = \ {0,1 \} && = \ {\ emptyset, \ {\ emptyset \} \} \\ 3 &: = 2' && = \ {0,1,2 \} && = \ {\ emptyset, \ {\ emptyset \}, \ {\ emptyset, \ {\ emptyset \} \} \} \\ & \ vdots &&&& \\ (n + 1) &: = n '&& = \ {0,1, \ ldots, n \} && = n \ cup \ {n \} \ end {alignedat}}}

Explanation: For the start element, the "0", the empty set has been chosen. The “1”, on the other hand, is the set that contains the empty set as an element. These are different sets, because the empty set “0” = {} contains no element, whereas the set “1” = {0} contains exactly one element. ${\ displaystyle \ emptyset}$

The successor set is defined as the union of the predecessor set and the set that contains the predecessor set. The set that contains the predecessor set (it is not empty) and the predecessor set are disjoint, which is why each successor set is different from the predecessor set. This results in particular in the injectivity of the successor function defined in this way. Thus this satisfies the Peano axioms.

The existence of every single natural number is already secured in set theory by very weak requirements. For the existence of the set of all natural numbers as well as in the Zermelo-Fraenkel set theory one needs a separate axiom, the so-called infinity axiom . ${\ displaystyle \ mathbb {N}}$${\ displaystyle \ mathbb {N} _ {0}}$

A generalization of this construction (omission of the fifth Peano axiom or admission of further numbers without predecessors) gives the ordinal numbers .

## The natural numbers as a subset of the real numbers

The introduction of the natural numbers with the help of the Peano axioms is one way of establishing the theory of natural numbers. As an alternative, one can start axiomatically with the field of real numbers and define the natural numbers as a subset of . To do this, one first needs the concept of an inductive set. ${\ displaystyle \ mathbb {R}}$${\ displaystyle \ mathbb {R}}$

A subset of is called inductive if the following conditions are met: ${\ displaystyle M}$${\ displaystyle \ mathbb {R}}$

1. 0 is an element of ${\ displaystyle M.}$
2. Is an element of , then is also an element of${\ displaystyle x}$${\ displaystyle M}$${\ displaystyle x + 1}$${\ displaystyle M.}$

Then is the average of all inductive subsets of${\ displaystyle \ mathbb {N} _ {0}}$${\ displaystyle \ mathbb {R}.}$

Alternatively, the natural numbers can also be embedded in the field of the real numbers using monoid monomorphism. But this only applies if you consider 0 as an element of the natural numbers. It should be noted that the natural numbers are only interpreted as a subset of the real numbers, which, strictly speaking, are not. In the same way, one embeds the natural numbers in other known number ranges, such as in the rational numbers.

Such a canonical isomorphism is given, for example, as follows:

${\ displaystyle \ mathbb {N} \ rightarrow \ mathbb {R}, \, n \ mapsto n \ cdot 1}$,

whereby here is to be understood as the n-fold addition of the multiplicatively neutral element of the real numbers and the real numbers are to be understood as additive monoid. It is immediately apparent that the above figure is a homomorphism; so is injectivity. Consequently, the natural numbers can be identified with the picture above (and thus as a subset of the real numbers). ${\ displaystyle n \ cdot 1}$

In a completely analogous way, they can also be embedded, for example, in the ring of whole numbers, the body of rational numbers or the body of complex numbers.