{"history":[],"article":{"id":"7a95ecd3-afc7-458e-adf0-67896f6ede33","title":"Construction of the real numbers","slug":"construction-of-the-real-numbers","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), there are several equivalent ways of defining the [real numbers](https://bliptext.com/articles/real-number).\nOne of them is that they form a [complete ordered field](https://bliptext.com/articles/complete-ordered-field) that does not contain any smaller [complete ordered field](https://bliptext.com/articles/complete-ordered-field).\nSuch a definition does not prove that such a complete ordered field exists, and the existence proof consists of constructing a [mathematical structure](https://bliptext.com/articles/mathematical-structure) that satisfies the definition.\nThe article presents several such constructions.\nThey are equivalent in the sense that, given the result of any two such constructions, there is a unique [isomorphism](https://bliptext.com/articles/isomorphism) of [ordered field](https://bliptext.com/articles/ordered-field) between them.\nThis results from the above definition and is independent of particular constructions.\nThese isomorphisms allow identifying the results of the constructions, and, in practice, to forget which construction has been chosen.\n\n# Axiomatic definitions\nAn [axiomatic definition](https://bliptext.com/articles/axiomatic-method) of the real numbers consists of defining them as the elements of a complete ordered field.\nThis means the following: The real numbers form a [set](https://bliptext.com/articles/set-mathematics), commonly denoted \\mathbb{R}, containing two distinguished elements denoted 0 and 1, and on which are defined two [binary operations](https://bliptext.com/articles/binary-operation) and one [binary relation](https://bliptext.com/articles/binary-relation); the operations are called addition and multiplication of real numbers and denoted respectively with\n+\nand\n×\nThe existence of such a [structure](https://bliptext.com/articles/mathematical-structure) is a [theorem](https://bliptext.com/articles/theorem), which is proved by constructing such a [structure](https://bliptext.com/articles/mathematical-structure).\nA consequence of the axioms is that this structure is unique [up to](https://bliptext.com/articles/up-to) an isomorphism, and thus, the real numbers can be used and manipulated, without referring to the method of construction.\n\n## Axioms\n\n### On the least upper bound property\nAxiom 4, which requires the order to be [Dedekind-complete](https://bliptext.com/articles/dedekind-complete), implies the [Archimedean property](https://bliptext.com/articles/archimedean-property).\nThe axiom is crucial in the characterization of the reals.\nFor example, the totally [ordered field](https://bliptext.com/articles/ordered-field) of the rational numbers **Q** satisfies the first three axioms, but not the fourth.\nIn other words, models of the rational numbers are also models of the first three axioms.\nNote that the axiom is [nonfirstorderizable](https://bliptext.com/articles/nonfirstorderizability), as it expresses a statement about collections of reals and not just individual such numbers.\nAs such, the reals are not given by a [first-order logic theory](https://bliptext.com/articles/list-of-first-order-theories).\n\n### On models\nA model of real numbers is a [mathematical structure](https://bliptext.com/articles/mathematical-structure) that satisfies the above axioms.\nSeveral models are given [below](https://bliptext.com/articles/).\nAny two models are isomorphic; so, the real numbers are unique [up to](https://bliptext.com/articles/up-to) isomorphisms.\nSaying that any two models are isomorphic means that for any two models and there is a [bijection](https://bliptext.com/articles/bijection) that preserves both the field operations and the order.\nExplicitly,\nf\nis both [injective](https://bliptext.com/articles/injective) and [surjective](https://bliptext.com/articles/surjective).\nf(0ℝ) = 0S\nand\nf(1ℝ) = 1S\n.\nf(x +ℝ y) = f(x) +S f(y)\nand\nf(x ×ℝ y) = f(x) ×S f(y)\n, for all\nx\nand\ny\nin \\mathbb{R}.\nx ≤ℝ y\n[if and only if](https://bliptext.com/articles/if-and-only-if)\nf(x) ≤S f(y)\n, for all\nx\nand\ny\nin \\mathbb{R}.\n\n## Tarski's axiomatization of the reals\nAn alternative synthetic [axiomatization](https://bliptext.com/articles/axiomatization) of the real numbers and their arithmetic was given by [Alfred Tarski](https://bliptext.com/articles/alfred-tarski), consisting of only the 8 [axioms](https://bliptext.com/articles/axiom) shown below and a mere four [primitive notions](https://bliptext.com/articles/primitive-notion): a [set](https://bliptext.com/articles/set-mathematics) called the real numbers, denoted \\mathbb{R}, a [binary relation](https://bliptext.com/articles/binary-relation) over \\mathbb{R} called order, denoted by the [infix operator](https://bliptext.com/articles/infix-operator) <, a [binary operation](https://bliptext.com/articles/binary-operation) over \\mathbb{R} called addition, denoted by the [infix operator](https://bliptext.com/articles/infix-operator) +, and the constant 1.\nAxioms of order (primitives: \\mathbb{R}, <):\n**Axiom 1**.\nIf x < y, then not y < x.\nThat is, \"<\" is an [asymmetric relation](https://bliptext.com/articles/asymmetric-relation).\n**Axiom 2**.\nIf x < z, there exists a y such that x < y and y < z.\nIn other words, \"<\" is [dense](https://bliptext.com/articles/dense-order) in \\mathbb{R}.\n**Axiom 3**.\n\"<\" is [Dedekind-complete](https://bliptext.com/articles/dedekind-complete).\nMore formally, for all X, Y ⊆ \\mathbb{R}, if for all x ∈ X and y ∈ Y, x < y, then there exists a z such that for all x ∈ X and y ∈ Y, if z ≠ x and z ≠ y, then x < z and z < y.\nTo clarify the above statement somewhat, let X ⊆ \\mathbb{R} and Y ⊆ \\mathbb{R}.\nWe now define two common English verbs in a particular way that suits our purpose:\nAxiom 3 can then be stated as:\nAxioms of addition (primitives: \\mathbb{R}, <, +):\n**Axiom 4**.\nx + (y + z) = (x + z) + y.\n**Axiom 5**.\nFor all x, y, there exists a z such that x + z = y.\n**Axiom 6**.\nIf x + y < z + w, then x < z or y < w.\nAxioms for one (primitives: \\mathbb{R}, <, +, 1):\n**Axiom 7**.\n1 ∈ \\mathbb{R}.\n**Axiom 8**.\n1 < 1 + 1.\nThese axioms imply that \\mathbb{R} is a [linearly ordered](https://bliptext.com/articles/linearly-ordered-group) [abelian group](https://bliptext.com/articles/abelian-group) under addition with distinguished element 1.\n\\mathbb{R} is also [Dedekind-complete](https://bliptext.com/articles/dedekind-complete) and [divisible](https://bliptext.com/articles/divisible-group).\n\n# Explicit constructions of models\nWe shall not prove that any models of the axioms are isomorphic.\nSuch a proof can be found in any number of modern analysis or set theory textbooks.\nWe 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.\nThe first three, due to [Georg Cantor](https://bliptext.com/articles/georg-cantor)/[Charles Méray](https://bliptext.com/articles/charles-m-ray), [Richard Dedekind](https://bliptext.com/articles/richard-dedekind)/[Joseph Bertrand](https://bliptext.com/articles/joseph-bertrand) and [Karl Weierstrass](https://bliptext.com/articles/karl-weierstrass) all occurred within a few years of each other.\nEach has advantages and disadvantages.\nA major motivation in all three cases was the instruction of mathematics students.\n\n## Construction from Cauchy sequences\nA standard procedure to force all [Cauchy sequences](https://bliptext.com/articles/cauchy-sequence) in a [metric space](https://bliptext.com/articles/metric-space) to converge is adding new points to the [metric space](https://bliptext.com/articles/metric-space) in a process called [completion](https://bliptext.com/articles/completeness-topology).\n\\mathbb{R} is defined as the completion of the set \\Q of the rational numbers with respect to the metric\nNormally, metrics are defined with real numbers as values, but this does not make the construction/definition circular, since all numbers that are implied (even implicitly) are rational numbers.\nLet R be the [set](https://bliptext.com/articles/set-mathematics) of Cauchy sequences of rational numbers.\nThat is, sequences\n(x1, x2, x3,...)\nof rational numbers such that for every rational\nε > 0\n, there exists an integer\nN\nsuch that for all natural numbers\nm, n > N\n, one has\n. Here the vertical bars denote the absolute value.\nCauchy sequences\n(xn)\nand\n(yn)\ncan be added and multiplied as follows:\n(xn) + (yn) = (xn + yn)\n(xn) × (yn) = (xn × yn)\n.\nTwo Cauchy sequences\n(xn)\nand\n(yn)\nare called equivalent if and only if the difference between them tends to zero; that is, for every rational number\nε > 0\n, there exists an integer\nN\nsuch that for all natural numbers\nn > N\n, one has\n.\nThis defines an [equivalence relation](https://bliptext.com/articles/equivalence-relation) that is compatible with the operations defined above, and the set **R** of all [equivalence classes](https://bliptext.com/articles/equivalence-class) can be shown to satisfy [all axioms of the real numbers](https://bliptext.com/articles/).\n\\Q can be considered as a subset of \\R by identifying a rational number\nr\nwith the equivalence class of the Cauchy sequence\n(r, r, r, ...)\n.\nComparison between real numbers is obtained by defining the following comparison between Cauchy sequences:\n(xn) ≥ (yn)\nif and only if\nx\nis equivalent to\ny\nor there exists an integer\nN\nsuch that\nxn ≥ yn\nfor all\nn > N\n.\nBy construction, every real number\nx\nis represented by a Cauchy sequence of rational numbers.\nThis representation is far from unique; every rational sequence that converges to\nx\nis a Cauchy sequence representing\nx\n. This reflects the observation that one can often use different sequences to approximate the same real number.\nThe only real number axiom that does not follow easily from the definitions is the completeness of\n≤\n, i.e. the [least upper bound property](https://bliptext.com/articles/least-upper-bound-property).\nIt can be proved as follows: Let\nS\nbe a non-empty subset of \\R' and\nU\nbe an upper bound for\nS\n. Substituting a larger value if necessary, we may assume\nU\nis rational.\nSince\nS\nis non-empty, we can choose a rational number\nL\nsuch that\nL < s\nfor some\ns\nin\nS\n. Now define sequences of rationals\n(un)\nand\n(ln)\nas follows:\nSet\nu0 = U\nand\nl0 = L\n. For each\nn\nconsider the number\nmn = (un + ln)/2\n. If\nmn\nis an upper bound for\nS\n, set\nun+1 = mn\nand\nln+1 = ln\n. Otherwise set\nln+1 = mn\nand\nun+1 = un\n.\nThis defines two Cauchy sequences of rationals, and so the real numbers\nl = (ln)\nand\nu = (un)\n. It is easy to prove, by induction on\nn\nthat\nun\nis an upper bound for\nS\nfor all\nn\nand\nln\nis never an upper bound for\nS\nfor any\nn\nThus\nu\nis an upper bound for\nS\n. To see that it is a least upper bound, notice that the limit of\n(un − ln)\nis\n0\n, and so\nl = u\n. Now suppose\nb < u = l\nis a smaller upper bound for\nS\n. Since\n(ln)\nis monotonic increasing it is easy to see that\nb < ln\nfor some\nn\n. But\nln\nis not an upper bound for S and so neither is\nb\n. Hence\nu\nis a least upper bound for\nS\nand\n≤\nis complete.\nThe usual [decimal notation](https://bliptext.com/articles/decimal-notation) can be translated to Cauchy sequences in a natural way.\nFor example, the notation\nπ = 3.1415...\nmeans that π is the equivalence class of the Cauchy sequence\n(3, 3.1, 3.14, 3.141, 3.1415, ...)\n. The equation\n0.999... = 1\nstates that the sequences\n(0, 0.9, 0.99, 0.999,...)\nand\n(1, 1, 1, 1,...)\nare equivalent, i.e., their difference converges to\n0\n.\nAn advantage of constructing \\R as the completion of \\Q is that this construction can be used for every other metric spaces.\n\n## Construction by Dedekind cuts\nA [Dedekind cut](https://bliptext.com/articles/dedekind-cut) in an ordered field is a [partition](https://bliptext.com/articles/partition-of-a-set) of it, (A, B), such that A is nonempty and closed downwards, B is nonempty and closed upwards, and A contains no [greatest element](https://bliptext.com/articles/greatest-element).\nReal numbers can be constructed as Dedekind cuts of rational numbers.\nFor convenience we may take the lower set A\\, as the representative of any given Dedekind cut (A, B)\\,, since A completely determines B. By doing this we may think intuitively of a real number as being represented by the set of all smaller rational numbers.\nIn more detail, a real number r is any subset of the set \\textbf{Q} of rational numbers that fulfills the following conditions:\nAs an example of a Dedekind cut representing an [irrational number](https://bliptext.com/articles/irrational-number), we may take the [positive square root of 2](https://bliptext.com/articles/square-root-of-2).\nThis can be defined by the set.\nIt can be seen from the definitions above that A is a real number, and that.\nHowever, neither claim is immediate.\nShowing that A\\, is real requires showing that A has no greatest element, i.e. that for any positive rational x\\, with, there is a rational y\\, with x N\n, one has\n. Here the vertical bars denote the absolute value.\nCauchy sequences\n(xn)\nand\n(yn)\ncan be added and multiplied as follows:\n(xn) + (yn) = (xn + yn)\n(xn) × (yn) = (xn × yn)\n.\nTwo Cauchy sequences\n(xn)\nand\n(yn)\nare called equivalent if and only if the difference between them tends to zero; that is, for every rational number\nε > 0\n, there exists an integer\nN\nsuch that for all natural numbers\nn > N\n, one has\n.\nThis defines an [equivalence relation](https://bliptext.com/articles/equivalence-relation) that is compatible with the operations defined above, and the set **R** of all [equivalence classes](https://bliptext.com/articles/equivalence-class) can be shown to satisfy [all axioms of the real numbers](https://bliptext.com/articles/).\n\\Q can be considered as a subset of \\R by identifying a rational number\nr\nwith the equivalence class of the Cauchy sequence\n(r, r, r, ...)\n.\nComparison between real numbers is obtained by defining the following comparison between Cauchy sequences:\n(xn) ≥ (yn)\nif and only if\nx\nis equivalent to\ny\nor there exists an integer\nN\nsuch that\nxn ≥ yn\nfor all\nn > N\n.\nBy construction, every real number\nx\nis represented by a Cauchy sequence of rational numbers.\nThis representation is far from unique; every rational sequence that converges to\nx\nis a Cauchy sequence representing\nx\n. This reflects the observation that one can often use different sequences to approximate the same real number.\nThe only real number axiom that does not follow easily from the definitions is the completeness of\n≤\n, i.e. the [least upper bound property](https://bliptext.com/articles/least-upper-bound-property).\nIt can be proved as follows: Let\nS\nbe a non-empty subset of \\R' and\nU\nbe an upper bound for\nS\n. Substituting a larger value if necessary, we may assume\nU\nis rational.\nSince\nS\nis non-empty, we can choose a rational number\nL\nsuch that\nL < s\nfor some\ns\nin\nS\n. Now define sequences of rationals\n(un)\nand\n(ln)\nas follows:\nSet\nu0 = U\nand\nl0 = L\n. For each\nn\nconsider the number\nmn = (un + ln)/2\n. If\nmn\nis an upper bound for\nS\n, set\nun+1 = mn\nand\nln+1 = ln\n. Otherwise set\nln+1 = mn\nand\nun+1 = un\n.\nThis defines two Cauchy sequences of rationals, and so the real numbers\nl = (ln)\nand\nu = (un)\n. It is easy to prove, by induction on\nn\nthat\nun\nis an upper bound for\nS\nfor all\nn\nand\nln\nis never an upper bound for\nS\nfor any\nn\nThus\nu\nis an upper bound for\nS\n. To see that it is a least upper bound, notice that the limit of\n(un − ln)\nis\n0\n, and so\nl = u\n. Now suppose\nb < u = l\nis a smaller upper bound for\nS\n. Since\n(ln)\nis monotonic increasing it is easy to see that\nb < ln\nfor some\nn\n. But\nln\nis not an upper bound for S and so neither is\nb\n. Hence\nu\nis a least upper bound for\nS\nand\n≤\nis complete.\nThe usual [decimal notation](https://bliptext.com/articles/decimal-notation) can be translated to Cauchy sequences in a natural way.\nFor example, the notation\nπ = 3.1415...\nmeans that π is the equivalence class of the Cauchy sequence\n(3, 3.1, 3.14, 3.141, 3.1415, ...)\n. The equation\n0.999... = 1\nstates that the sequences\n(0, 0.9, 0.99, 0.999,...)\nand\n(1, 1, 1, 1,...)\nare equivalent, i.e., their difference converges to\n0\n.\nAn advantage of constructing \\R as the completion of \\Q is that this construction can be used for every other metric spaces.\n\n## Construction by Dedekind cuts\nA [Dedekind cut](https://bliptext.com/articles/dedekind-cut) in an ordered field is a [partition](https://bliptext.com/articles/partition-of-a-set) of it, (A, B), such that A is nonempty and closed downwards, B is nonempty and closed upwards, and A contains no [greatest element](https://bliptext.com/articles/greatest-element).\nReal numbers can be constructed as Dedekind cuts of rational numbers.\nFor convenience we may take the lower set A\\, as the representative of any given Dedekind cut (A, B)\\,, since A completely determines B. By doing this we may think intuitively of a real number as being represented by the set of all smaller rational numbers.\nIn more detail, a real number r is any subset of the set \\textbf{Q} of rational numbers that fulfills the following conditions:\nAs an example of a Dedekind cut representing an [irrational number](https://bliptext.com/articles/irrational-number), we may take the [positive square root of 2](https://bliptext.com/articles/square-root-of-2).\nThis can be defined by the set.\nIt can be seen from the definitions above that A is a real number, and that.\nHowever, neither claim is immediate.\nShowing that A\\, is real requires showing that A has no greatest element, i.e. that for any positive rational x\\, with, there is a rational y\\, with x

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