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Construction of the real numbers
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Construction of the real numbers
In mathematics, there are several equivalent ways of defining the real numbers. One of them is that they form a complete ordered field that does not contain any smaller complete ordered field. Such a definition does not prove that such a complete ordered field exists, and the existence proof consists of constructing a mathematical structure that satisfies the definition.
The article presents several such constructions. They are equivalent in the sense that, given the result of any two such constructions, there is a unique isomorphism of ordered field between them. This results from the above definition and is independent of particular constructions. These isomorphisms allow identifying the results of the constructions, and, in practice, to forget which construction has been chosen.
An axiomatic definition of the real numbers consists of defining them as the elements of a complete ordered field. This means the following: The real numbers form a set, commonly denoted , containing two distinguished elements denoted 0 and 1, and on which are defined two binary operations and one binary relation; the operations are called addition and multiplication of real numbers and denoted respectively with + and ×; the binary relation is inequality, denoted Moreover, the following properties called axioms must be satisfied.
The existence of such a structure is a theorem, which is proved by constructing such a structure. A consequence of the axioms is that this structure is unique up to an isomorphism, and thus, the real numbers can be used and manipulated, without referring to the method of construction.
Axiom 4, which requires the order to be Dedekind-complete, implies the Archimedean property (though the converse does not hold).
The axiom is crucial in the characterization of the reals. For example, the totally ordered field of the rational numbers Q satisfies the first three axioms, but not the fourth. In other words, models of the rational numbers are also models of the first three axioms.
Note that the axiom is nonfirstorderizable, as it expresses a statement about collections of reals and not just individual such numbers. As such, the reals are not given by a first-order logic theory.
A model of real numbers is a mathematical structure that satisfies the above axioms. Several models are given below. Any two models are isomorphic; so, the real numbers are unique up to isomorphisms.
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Construction of the real numbers AI simulator
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Construction of the real numbers
In mathematics, there are several equivalent ways of defining the real numbers. One of them is that they form a complete ordered field that does not contain any smaller complete ordered field. Such a definition does not prove that such a complete ordered field exists, and the existence proof consists of constructing a mathematical structure that satisfies the definition.
The article presents several such constructions. They are equivalent in the sense that, given the result of any two such constructions, there is a unique isomorphism of ordered field between them. This results from the above definition and is independent of particular constructions. These isomorphisms allow identifying the results of the constructions, and, in practice, to forget which construction has been chosen.
An axiomatic definition of the real numbers consists of defining them as the elements of a complete ordered field. This means the following: The real numbers form a set, commonly denoted , containing two distinguished elements denoted 0 and 1, and on which are defined two binary operations and one binary relation; the operations are called addition and multiplication of real numbers and denoted respectively with + and ×; the binary relation is inequality, denoted Moreover, the following properties called axioms must be satisfied.
The existence of such a structure is a theorem, which is proved by constructing such a structure. A consequence of the axioms is that this structure is unique up to an isomorphism, and thus, the real numbers can be used and manipulated, without referring to the method of construction.
Axiom 4, which requires the order to be Dedekind-complete, implies the Archimedean property (though the converse does not hold).
The axiom is crucial in the characterization of the reals. For example, the totally ordered field of the rational numbers Q satisfies the first three axioms, but not the fourth. In other words, models of the rational numbers are also models of the first three axioms.
Note that the axiom is nonfirstorderizable, as it expresses a statement about collections of reals and not just individual such numbers. As such, the reals are not given by a first-order logic theory.
A model of real numbers is a mathematical structure that satisfies the above axioms. Several models are given below. Any two models are isomorphic; so, the real numbers are unique up to isomorphisms.