Natural number

Natural numbers are used in the figures include 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, etc. Often, the zero is calculated to the natural numbers. The natural numbers form, with the addition and multiplication together, a mathematical structure is referred to as semi- commutative ring.

Naming conventions

The set of natural numbers is abbreviated with the symbols.

It includes either the positive integers

Or the non-negative integers

Both conventions are used inconsistently. The older tradition does not count the zero to the natural numbers ( zero was used in Europe until the 13th century). This definition is common in mathematical fields such as number theory, in which the multiplication of natural numbers is in the foreground. In logic, set theory, or computer science, however, is the definition with zero common and simplifies the presentation. In case of doubt, the definition used is to explicitly call.

For the set of natural numbers without zero led Dedekind in 1888 the symbol N a. Its symbol is stylized as today or as from 1894 used for the Peano natural numbers with zero, the symbol N0, which is now also stylized and defined by Peano through.

However, if the symbol of the natural numbers with zero is used, then the set of natural numbers without zero is denoted by. The DIN Standard 5473 is used for example for the non-negative integers and the positive integers. German textbooks are based in some states at this DIN standard, in others, such as Bavaria, not.

Ultimately, it is a matter of definition, which wants the two quantities seen as natural and thus let this designation come as a linguistic distinction.

Axiomatization

Richard Dedekind in 1888 for the first time defined the natural numbers implied by axioms. Regardless of him turned Giuseppe Peano in 1889 a simpler and at the same time formally precise axiom system. These so-called Peano axioms have prevailed. While the original axiom system can be formalized in predicate logic the second stage, now a weaker variant is often used in first-order logic, which is called Peano arithmetic. Other axiomatization of natural numbers which are related to the Peano arithmetic, for example, the arithmetic and Robinson primitive recursive arithmetic.

Von Neumann's model of the natural numbers

Peano described with his axiom system, although the properties of natural numbers, but saw no need to prove their existence. John von Neumann was a way to represent the natural numbers by sets, that is, it describes a set-theoretical model of the natural numbers.

Explanation: For the start element, the "0", the empty set is selected. "1", however, is that amount which contains the empty set as an element. These are different amounts, because the empty set "0" = {} does not contain any element, whereas the quantity "1" = {0} contains exactly one element. Each successor is different from its predecessor, as the successor amount an element contains more than the previous amount, namely the predecessor itself

The existence of each natural number is set-theoretically already secured by rather weak claims. For the existence of the set of all natural numbers, and however you need in the Zermelo -Fraenkel set theory own axiom, the so-called axiom of infinity.

A generalization of this construction (omission of the fifth Peano Axioms or approval of other numbers with no predecessor ) gives the ordinals.

The natural numbers as a subset of the real numbers

The introduction of natural numbers using the Peano axioms is a way to justify the theory of natural numbers. As an alternative, you can enter the axiomatic field of real numbers and define the natural numbers as a subset of. For this first you need the concept of an inductive set.

A subset of is called inductive if the following conditions are met:

Then the intersection of all inductive subsets of

The natural numbers as objects in a category

In each category there is a final object, an object of natural numbers can be defined. This is an object together with morphisms

So that for any objects and morphisms

Exactly one exists with and. is to be understood as Sukzessorfunktion.