Inmathematics, there are several equivalent ways of defining thereal numbers. One of them is that they form acomplete 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 amathematical structure that satisfies the definition.

The article presents several such constructions.^[1] They are equivalent in the sense that, given the result of any two such constructions, there is a uniqueisomorphism ofordered 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.

Anaxiomatic definition of the real numbers consists of defining them as the elements of a complete ordered field.^[2][3][4] This means the following: The real numbers form aset, commonly denoted${\displaystyle \mathbb{R}}$, containing two distinguished elements denoted 0 and 1, and on which are defined twobinary operations and onebinary relation; the operations are calledaddition andmultiplication of real numbers and denoted respectively with+ and×; the binary relation isinequality, denoted${\displaystyle \le .}$ Moreover, the following properties calledaxioms must be satisfied.

The existence of such astructure is atheorem, which is proved by constructing such a structure. A consequence of the axioms is that this structure is uniqueup to an isomorphism, and thus, the real numbers can be used and manipulated, without referring to the method of construction.

1. ${\displaystyle \mathbb{R}}$ is afield under addition and multiplication. In other words,
   • For allx,y, andz in${\displaystyle \mathbb{R}}$,x + (y +z) = (x +y) +z andx × (y ×z) = (x ×y) ×z. (associativity of addition and multiplication)
   • For allx andy in${\displaystyle \mathbb{R}}$,x +y =y +x andx ×y =y ×x. (commutativity of addition and multiplication)
   • For allx,y, andz in${\displaystyle \mathbb{R}}$,x × (y +z) = (x ×y) + (x ×z). (distributivity of multiplication over addition)
   • For allx in${\displaystyle \mathbb{R}}$,x + 0 =x. (existence of additiveidentity)
   • 0 is not equal to 1, and for allx in${\displaystyle \mathbb{R}}$,x × 1 =x. (existence of multiplicative identity)
   • For everyx in${\displaystyle \mathbb{R}}$, there exists an element −x in${\displaystyle \mathbb{R}}$, such thatx + (−x) = 0. (existence of additiveinverses)
   • For everyx ≠ 0 in${\displaystyle \mathbb{R}}$, there exists an elementx⁻¹ in${\displaystyle \mathbb{R}}$, such thatx ×x⁻¹ = 1. (existence of multiplicative inverses)
2. ${\displaystyle \mathbb{R}}$ istotally ordered for${\displaystyle \le }$. In other words,
   • For allx in${\displaystyle \mathbb{R}}$,x ≤x. (reflexivity)
   • For allx andy in${\displaystyle \mathbb{R}}$, ifx ≤y andy ≤x, thenx =y. (antisymmetry)
   • For allx,y, andz in${\displaystyle \mathbb{R}}$, ifx ≤y andy ≤z, thenx ≤z. (transitivity)
   • For allx andy in${\displaystyle \mathbb{R}}$,x ≤y ory ≤x. (totality)
3. Addition and multiplication are compatible with the order. In other words,
   • For allx,y andz in${\displaystyle \mathbb{R}}$, ifx ≤y, thenx +z ≤y +z. (preservation of order under addition)
   • For allx andy in${\displaystyle \mathbb{R}}$, if 0 ≤x and 0 ≤y, then 0 ≤x ×y (preservation of order under multiplication)
4. The order ≤ iscomplete in the following sense: every non-empty subset of${\displaystyle \mathbb{R}}$ that isbounded above has aleast upper bound. In other words,
   • IfA is a non-empty subset of${\displaystyle \mathbb{R}}$, and ifA has anupper bound in${\displaystyle \mathbb{R},}$ thenA has a least upper boundu, such that for every upper boundv ofA,u ≤v.

Axiom 4, which requires the order to beDedekind-complete, implies theArchimedean property (though the converse does not hold).

The axiom is crucial in the characterization of the reals. For example, the totallyordered field of the rational numbersQ 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 isnonfirstorderizable, as it expresses a statement about collections of reals and not just individual such numbers. As such, the reals are not given by afirst-order logic theory.

Amodel of real numbers is amathematical structure that satisfies the above axioms.Several models are givenbelow. Any two models are isomorphic; so, the real numbers are uniqueup to isomorphisms.

Saying that any two models are isomorphic means that for any two models${\displaystyle (\mathbb{R},0_{\mathbb{R}},1_{\mathbb{R}},{+}_{\mathbb{R}},{\times }_{\mathbb{R}},{\le }_{\mathbb{R}})}$ and${\displaystyle (S,0_S,1_S,{+}_S,{\times }_S,{\le }_S),}$ there is abijection${\displaystyle f:\mathbb{R}\to S}$ that preserves both the field operations and the order. Explicitly,

• f is bothinjective andsurjective.
• f(0_ℝ) = 0_S andf(1_ℝ) = 1_S.
• f(x +_ℝy) =f(x) +_Sf(y) andf(x ×_ℝy) =f(x) ×_Sf(y), for allx andy in${\displaystyle \mathbb{R}.}$
• x ≤_ℝyif and only iff(x) ≤_Sf(y), for allx andy in${\displaystyle \mathbb{R}.}$

An alternative syntheticaxiomatization of the real numbers and their arithmetic was given byAlfred Tarski, consisting of only the 8axioms shown below and a mere fourprimitive notions: aset calledthe real numbers, denoted${\displaystyle \mathbb{R}}$, abinary relation over${\displaystyle \mathbb{R}}$ calledorder, denoted by theinfix operator <, abinary operation over${\displaystyle \mathbb{R}}$ calledaddition, denoted by the infix operator +, and the constant 1.

Axioms of order (primitives:${\displaystyle \mathbb{R}}$, <):

Axiom 1. Ifx <y, then noty <x. That is, "<" is anasymmetric relation.

Axiom 2. Ifx < z, there exists ay such thatx < y andy < z. In other words, "<" isdense in${\displaystyle \mathbb{R}}$.

Axiom 3. "<" isDedekind-complete. More formally, for allX, Y ⊆ ${\displaystyle \mathbb{R}}$, if for allx ∈ X andy ∈ Y,x < y, then there exists az such that for allx ∈ X andy ∈ Y, ifz ≠ x andz ≠ y, thenx < z andz < y.

To clarify the above statement somewhat, letX ⊆ ${\displaystyle \mathbb{R}}$ andY ⊆ ${\displaystyle \mathbb{R}}$. We now define two common English verbs in a particular way that suits our purpose:

X precedes Y if and only if for everyx ∈ X and everyy ∈ Y,x < y.

The real numberz separatesX andY if and only if for everyx ∈ X withx ≠ z and everyy ∈ Y withy ≠ z,x < z andz < y.

Axiom 3 can then be stated as:

"If a set of reals precedes another set of reals, then there exists at least one real number separating the two sets."

Axioms of addition (primitives:${\displaystyle \mathbb{R}}$, <, +):

Axiom 4.x + (y + z) = (x + z) + y.

Axiom 5. For allx,y, there exists az such thatx + z = y.

Axiom 6. Ifx + y < z + w, thenx < z ory < w.

Axioms for one (primitives:${\displaystyle \mathbb{R}}$, <, +, 1):

Axiom 7. 1 ∈ ${\displaystyle \mathbb{R}}$.

Axiom 8. 1 < 1 + 1.

These axioms imply that${\displaystyle \mathbb{R}}$ is alinearly orderedabelian group under addition with distinguished element 1.${\displaystyle \mathbb{R}}$ is alsoDedekind-complete anddivisible.

We shall not prove that any models of the axioms are isomorphic. Such a proof can be found in any number of modern analysis or set theory textbooks. We will sketch the basic definitions and properties of a number of constructions, however, because each of these is important for both mathematical and historical reasons. The first three, due toGeorg Cantor/Charles Méray,Richard Dedekind/Joseph Bertrand andKarl Weierstrass all occurred within a few years of each other. Each has advantages and disadvantages.

LetR be theset ofCauchy sequences of rational numbers. That is, sequences

of rational numbers such that for every rationalε > 0, there exists an integerN such that for all natural numbersm,n >N, one has |x_m −x_n| <ε. Here the vertical bars denote the absolute value.

Cauchy sequences(x_n) and(y_n) can be added and multiplied as follows:

Two Cauchy sequences(x_n) and(y_n) are calledequivalent if and only if the difference between them tends to zero; that is, for every rational numberε > 0, there exists an integerN such that for all natural numbersn >N, one has |x_n −y_n| <ε.

This defines anequivalence relation that is compatible with the operations defined above, and the setR of allequivalence classes can be shown to satisfyall axioms of the real numbers.${\displaystyle \mathbb{Q}}$ can be considered as a subset of${\displaystyle \mathbb{R}}$ by identifying a rational numberr with the equivalence class of the Cauchy sequence (r,r,r, ...).

Comparison between real numbers is obtained by defining the following comparison between Cauchy sequences:(x_n) ≥ (y_n) if and only ifx is equivalent toy or there exists an integerN such thatx_n ≥y_n for alln >N.

By construction, every real numberx is represented by a Cauchy sequence of rational numbers. This representation is far from unique; every rational sequence that converges tox is a Cauchy sequence representingx. This reflects the observation that one can often use different sequences to approximate the same real number.^[5]

The only real number axiom that does not follow easily from the definitions is the completeness of≤, i.e. theleast upper bound property. It can be proved as follows: LetS be a non-empty subset of${\displaystyle {\mathbb{R}}^{\prime }}$ andU be an upper bound forS. Substituting a larger value if necessary, we may assumeU is rational. SinceS is non-empty, we can choose a rational numberL such thatL <s for somes inS. Now define sequences of rationals(u_n) and(l_n) as follows:

Setu₀ =U andl₀ =L. For eachn consider the numberm_n = (u_n +l_n)/2.Ifm_n is an upper bound forS, setu_n+1 =m_n andl_n+1 =l_n.Otherwise setl_n+1 =m_n andu_n+1 =u_n.

This defines two Cauchy sequences of rationals, and so the real numbersl = (l_n) andu = (u_n). It is easy to prove, by induction onn thatu_n is an upper bound forS for allnandl_n is never an upper bound forS for anyn.

Thusu is an upper bound forS. To see that it is a least upper bound, notice that the limit of(u_n −l_n) is0, and sol =u. Now supposeb <u =l is a smaller upper bound forS. Since(l_n) is monotonic increasing it is easy to see thatb <l_n for somen. Butl_n is not an upper bound forS and so neither isb. Henceu is a least upper bound forS and≤ is complete.

The usualdecimal notation can be translated to Cauchy sequences in a natural way. For example, the notationπ = 3.1415... means thatπ is the equivalence class of the Cauchy sequence(3, 3.1, 3.14, 3.141, 3.1415, ...). The equation0.999... = 1 states that the sequences(0, 0.9, 0.99, 0.999,...) and(1, 1, 1, 1,...) are equivalent, i.e., their difference converges to0.

An advantage of constructing${\displaystyle \mathbb{R}}$ as the completion of${\displaystyle \mathbb{Q}}$ is that the method can be used for thecompletion of any metric space, simply by replacing everywhere⁠${\displaystyle |x-y|}$⁠ with⁠${\displaystyle d(x,y)}$⁠, where⁠${\displaystyle d}$⁠ denotes the distance of the metric space.In particular, thefields ofp-adic numbers can be defined as the completions of the rationals with respect to other absolute values, thep-adic absolute values.

ADedekind cut in an ordered field is apartition of it, (A,B), such thatA is nonempty and closed downwards,B is nonempty and closed upwards, andA contains nogreatest element. Real numbers can be constructed as Dedekind cuts of rational numbers.^[6][7]

For convenience we may take the lower set${\displaystyle A}$ as the representative of any given Dedekind cut${\displaystyle (A,B)}$, since${\displaystyle A}$ completely determines${\displaystyle B}$. By doing this we may think intuitively of a real number as being represented by the set of all smaller rational numbers. In more detail, a real number${\displaystyle r}$ is any subset of the set${\displaystyle \mathbf{\text{Q}}}$ of rational numbers that fulfills the following conditions:^[8]

1. ${\displaystyle r}$ is not empty
2. ${\displaystyle r\ne \mathbf{\text{Q}}}$
3. ${\displaystyle r}$ is closed downwards. In other words, for all${\displaystyle x,y\in \mathbf{\text{Q}}}$ such that, if${\displaystyle y\in r}$ then${\displaystyle x\in r}$
4. ${\displaystyle r}$ contains no greatest element. In other words, there is no${\displaystyle x\in r}$ such that for all${\displaystyle y\in r}$,${\displaystyle y\le x}$

• We form the set${\displaystyle \mathbf{\text{R}}}$ of real numbers as the set of all Dedekind cuts${\displaystyle A}$ of${\displaystyle \mathbf{\text{Q}}}$, and define atotal ordering on the real numbers as follows:${\displaystyle x\le y\iff x\subseteq y}$
• Weembed the rational numbers into the reals by identifying the rational number${\displaystyle q}$ with the set of all smaller rational numbers.^[8] Since the rational numbers aredense, such a set can have no greatest element and thus fulfills the conditions for being a real number laid out above.
• Addition.${\displaystyle A+B:=\{a+b:a\in A\wedge b\in B\}}$^[8]
• Subtraction.${\displaystyle A-B:=\{a-b:a\in A\wedge b\in (\mathbf{\text{Q}}\setminus B)\}}$ where${\displaystyle \mathbf{\text{Q}}\setminus B}$ denotes therelative complement of${\displaystyle B}$ in${\displaystyle \mathbf{\text{Q}}}$,${\displaystyle \{x:x\in \mathbf{\text{Q}}\wedge x\notin B\}}$
• Negation is a special case of subtraction:
• Definingmultiplication is less straightforward.^[8]
  • if${\displaystyle A,B\ge 0}$ then
  • if either${\displaystyle A}$ or${\displaystyle B}$ is negative, we use the identities${\displaystyle A\times B=-(A\times -B)=-(-A\times B)=(-A\times -B)}$ to convert${\displaystyle A}$ and/or${\displaystyle B}$ to positive numbers and then apply the definition above.
• We definedivision in a similar manner:
  • if${\displaystyle A\ge 0\text{ and }B>0}$ then${\displaystyle A/B:=\{a/b:a\in A\wedge b\in (\mathbf{\text{Q}}\setminus B)\}}$
  • if either${\displaystyle A}$ or${\displaystyle B}$ is negative, we use the identities${\displaystyle A/B=-(A/-B)=-(-A/B)=-A/-B}$ to convert${\displaystyle A}$ to a non-negative number and/or${\displaystyle B}$ to a positive number and then apply the definition above.
• Supremum. If a nonempty set${\displaystyle S}$ of real numbers has any upper bound in${\displaystyle \mathbf{\text{R}}}$, then it has a least upper bound in${\displaystyle \mathbf{\text{R}}}$ that is equal to${\displaystyle \bigcup S}$.^[8]

As an example of a Dedekind cut representing anirrational number, we may take thepositive square root of 2. This can be defined by the set.^[9] It can be seen from the definitions above that${\displaystyle A}$ is a real number, and that${\displaystyle A\times A=2}$. However, neither claim is immediate. Showing that${\displaystyle A}$ is real requires showing that${\displaystyle A}$ has no greatest element, i.e. that for any positive rational${\displaystyle x}$ with, there is a rational${\displaystyle y}$ with and The choice${\displaystyle y=\frac{2x+2}{x+2}}$ works. Then${\displaystyle A\times A\le 2}$ but to show equality requires showing that if${\displaystyle r}$ is any rational number with, then there is positive${\displaystyle x}$ in${\displaystyle A}$ with.

An advantage of this construction is that each real number corresponds to a unique cut. Furthermore, by relaxing the first two requirements of the definition of a cut, theextended real number system may be obtained by associating${\displaystyle -\mathrm{\infty }}$ with the empty set and${\displaystyle \mathrm{\infty }}$ with all of${\displaystyle \mathbf{\text{Q}}}$.

As in thehyperreal numbers, one constructs the hyperrationals${\displaystyle {}^{\ast }\mathbb{Q}}$ from the rational numbers by means of anultrafilter.^[10] Here a hyperrational is by definition a ratio of twohyperintegers. Consider thering${\displaystyle B}$ of all limited (i.e. finite) elements in${\displaystyle {}^{\ast }\mathbb{Q}}$. Then${\displaystyle B}$ has a uniquemaximal ideal${\displaystyle I}$, theinfinitesimal hyperrational numbers. The quotient ring${\displaystyle B/I}$ gives thefield${\displaystyle \mathbb{R}}$ of real numbers.^[11] This construction uses a non-principal ultrafilter over the set of natural numbers, the existence of which is guaranteed by theaxiom of choice.

It turns out that the maximal ideal respects the order on${\displaystyle {}^{\ast }\mathbb{Q}}$. Hence the resulting field is an ordered field. Completeness can be proved in a similar way to the construction from the Cauchy sequences.

Every ordered field can be embedded in thesurreal numbers. The real numbers form a maximal subfield that isArchimedean (meaning that no real number is infinitely large or infinitely small). This embedding is not unique, though it can be chosen in a canonical way.

A relatively less known construction allows to define real numbers using only the additive group of integers${\displaystyle \mathbb{Z}}$ with different versions.^[12][13][14]Arthan (2004), who attributes this construction to unpublished work byStephen Schanuel, refers to this construction as theEudoxus reals, naming them after ancient Greek astronomer and mathematicianEudoxus of Cnidus. As noted byShenitzer (1987) andArthan (2004), Eudoxus's treatment of quantity using the behavior ofproportions became the basis for this construction. This construction has beenformally verified to give a Dedekind-complete ordered field by the IsarMathLib project.^[15]

Let analmost homomorphism be a map${\displaystyle f:\mathbb{Z}\to \mathbb{Z}}$ such that the set${\displaystyle \{f(n+m)-f(m)-f(n):n,m\in \mathbb{Z}\}}$ is finite (equivalently,${\displaystyle f(n+m)-f(m)-f(n)}$ is bounded). (Note that${\displaystyle f(n)=\lfloor \alpha n\rfloor }$ is an almost homomorphism for every${\displaystyle \alpha \in \mathbb{R}}$.) Almost homomorphisms form an abelian group under pointwise addition. We say that two almost homomorphisms${\displaystyle f,g}$ arealmost equal if the set${\displaystyle \{f(n)-g(n):n\in \mathbb{Z}\}}$ is finite (equivalently,${\displaystyle f(n)-g(n)}$ is bounded). This defines an equivalence relation on the set of almost homomorphisms. Real numbers are defined as the equivalence classes of this relation. Alternatively, the almost homomorphisms taking only finitely many values form a subgroup, and the underlying additive group of the real number is the quotient group. To add real numbers defined this way we add the almost homomorphisms that represent them. Multiplication of real numbers corresponds to functional composition of almost homomorphisms. If${\displaystyle [f]}$ denotes the real number represented by an almost homomorphism${\displaystyle f}$ we say that${\displaystyle 0\le [f]}$ if${\displaystyle f}$ is bounded or${\displaystyle f}$ takes an infinite number of positive values on${\displaystyle {\mathbb{Z}}^{+}}$ (equivalently, if${\displaystyle f}$ has no upper bound). This defines thelinear order relation on the set of real numbers constructed this way.

Faltin et al. (1975) write: "Few mathematical structures have undergone as many revisions or have been presented in as many guises as the real numbers. Every generation reexamines the reals in the light of its values and mathematical objectives."^[16]

A number of other constructions have been given, by:

• de Bruijn (1976),de Bruijn (1977)
• Rieger (1982)
• Knopfmacher & Knopfmacher (1987),Knopfmacher & Knopfmacher (1988)

For an overview, seeWeiss (2015).

As a reviewer of one noted: "The details are all included, but as usual they are tedious and not too instructive."^[17]

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