# What is another name for the set of counting numbers

Zermelo 's construction goes as follows:. A countable non-standard model of arithmetic satisfying the Peano Arithmetic i. Walk through homework problems step-by-step from beginning to end.

Addition and multiplication are compatible, which is expressed in the distribution law: 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.

This order is compatible with the arithmetical operations in the following sense: An important property of the natural numbers is that they are well-ordered: While it is in general not possible to divide one natural number by another and get a natural number as result, the procedure of division with remainder is available as a substitute: This Euclidean division is key to several other properties divisibilityalgorithms such as the Euclidean algorithmand ideas in number theory.

Two important generalizations of natural numbers arise from the two uses of counting and ordering: For finite well-ordered sets, there is 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 i. 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 Peano axioms.

## natural numbers

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 Five by an axiom schema one obtains a weaker first-order theory called Peano Arithmetic. In the area of mathematics called set theorya special case of the von Neumann ordinal construction [34] 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. Zermelo 's construction goes as follows:. From Wikipedia, the free encyclopedia. This section needs additional citations for verification.

##### Categories you should follow

Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. October Learn how and when to remove this template message. Set-theoretic definition of natural numbers.

#### Common Number Sets

International Organization for Standardization. They follow that with their version of the Peano Postulates.

MacTutor History of Mathematics. In a footnote, Gray attributes the German quote to: Jahresbericht der Deutschen Mathematiker-Vereinigung 2: The quote is on p. See, for example, Carothersp.

**counting number**

Retrieved 19 January A review of discrete and combinatorial mathematics 5th ed. Mathematical Proof and Structures Second ed.

### Natural number

A Contemporary Approach 10th ed. Hide Ads About Ads. Common Number Sets There are sets of numbers that are used so often that they have special names and symbols: Symbol Description Natural Numbers The whole numbers from 1 upwards. Irrational Numbers Any real number that is not a Rational Number. They can also be positive, negative or zero. Includes the Algebraic Numbers and Transcendental Numbers. Numbers Index Sets Index. Natural Numbers The whole numbers from 1 upwards.

Rational Numbers The numbers you can make by dividing one integer by another but not dividing by zero. Algebraic Numbers Any number that is a solution to a polynomial equation with rational coefficients.

Real Numbers All Rational and Irrational numbers.

Imaginary Numbers Numbers that when squared give a negative result.