Real number
In mathematics, the real numbers, denoted by the symbol ℝ, form the unique (up to isomorphism) complete ordered field, encompassing all rational numbers and irrational numbers to represent continuous quantities such as distances, durations, and temperatures along a number line.[1][2] This set satisfies the field axioms for addition and multiplication, including commutativity, associativity, distributive laws, and the existence of additive and multiplicative identities and inverses (except zero for multiplication), while the order axioms ensure a total ordering with compatibility under arithmetic operations.[3] The defining completeness property guarantees that every nonempty subset of ℝ bounded above has a least upper bound, enabling the solution of equations like polynomials and the foundation for calculus.[4] Real numbers can be constructed in various ways, such as through Dedekind cuts of rational numbers or Cauchy sequences, which rigorously extend the rationals to include limits of convergent sequences and fill gaps like √2 or π.[5] Every real number has a decimal expansion, which for rational numbers is either terminating (like 0.5 for 1/2) or eventually repeating, and for irrational numbers is non-terminating and non-repeating (like 3.14159... for π), allowing precise approximation and computation.[6][7] It contains subsets such as the natural numbers ℕ, the integers ℤ, and the rational numbers ℚ, with ℚ forming a dense subfield of ℝ, while the irrational numbers form the complement of ℚ in ℝ to ensure the cardinality of the continuum, which is uncountable as proven by Cantor's diagonal argument.[3][8][9] The real number system underpins much of modern mathematics and science, providing the algebraic structure for analysis, geometry, and physics, where continuity and limits are essential.[10] Its properties, such as the Archimedean axiom (no infinitesimals) and the intermediate value theorem derived from completeness, distinguish it from the rationals, which lack completeness and thus cannot model all continuous phenomena accurately.[4]Characterizing Properties
Ordered Field Structure
The real numbers, denoted \mathbb{R}, form a field under the operations of addition (+) and multiplication (\times), satisfying the standard field axioms. These include closure under both operations, associativity (x + (y + z) = (x + y) + z and x \times (y \times z) = (x \times y) \times z for all x, y, z \in \mathbb{R}), commutativity (x + y = y + x and x \times y = y \times x), the existence of additive and multiplicative identities (0 and 1, respectively), the existence of additive inverses (-x such that x + (-x) = [0](/page/0)), and multiplicative inverses for nonzero elements (x^{-1} such that x \times x^{-1} = [1](/page/1) if x \neq [0](/page/0)), along with distributivity (x \times (y + z) = x \times y + x \times z).[11] These axioms ensure that \mathbb{R} supports the algebraic structure necessary for arithmetic operations, mirroring the properties of the rational numbers but extending to a complete system.[11] In addition to the field structure, \mathbb{R} is equipped with a total order relation < that is compatible with the field operations. The order axioms include trichotomy: for any a, b \in \mathbb{R}, exactly one of a < b, a = b, or a > b holds; transitivity: if a < b and b < c, then a < c; and the definition of positives as the set \{x \in \mathbb{R} \mid 0 < x\}. Compatibility requires that addition preserves order (if a < b, then a + c < b + c for any c \in \mathbb{R}) and that multiplication by positives preserves order (if a < b and $0 < c, then a \times c < b \times c for all a, b, c \in \mathbb{R}).[12][13] These properties ensure that the order is linear and integrates seamlessly with the algebraic operations, allowing inequalities to behave predictably under addition and positive multiplication.[12] The ordered field structure of \mathbb{R} also incorporates the Archimedean property, which states that for any positive reals a, b \in \mathbb{R}, there exists a natural number n such that n a > b. This property implies that there are no "infinitesimal" elements and that the naturals are unbounded in \mathbb{R}, distinguishing \mathbb{R} from non-Archimedean ordered fields.[14] Together, the field and order axioms define \mathbb{R} as an ordered field, with completeness providing the unique feature that embeds the rationals densely and ensures the existence of limits (detailed in subsequent sections).[11]Completeness Axiom
The completeness axiom, also known as the Dedekind completeness property, distinguishes the real numbers from other ordered fields, such as the rationals, by ensuring the absence of "gaps" in the number line. Formally, it states that every non-empty subset of the real numbers that is bounded above has a least upper bound, or supremum, within the reals. This property was introduced by Richard Dedekind in his 1872 essay "Continuity and Irrational Numbers" to rigorously define the continuum of real numbers.[15] The axiom can be expressed mathematically as follows: For every non-empty subset S \subseteq \mathbb{[R](/page/R)} that is bounded above, there exists a supremum \sup S \in \mathbb{[R](/page/R)} such that: \forall x \in S, \, x \leq \sup S, and \forall y < \sup S, \, \exists x \in S \text{ with } y < x. This least upper bound is unique and serves as the "tightest" upper bound for S.[11] A key consequence of this axiom is the monotone convergence theorem, which asserts that every increasing sequence of real numbers that is bounded above converges to a real number. Specifically, if \{x_n\} is an increasing sequence with x_n \leq M for some M \in \mathbb{[R](/page/R)} and all n, then \lim_{n \to \infty} x_n = \sup \{ x_n : n \in \mathbb{N} \}. This theorem follows directly from the existence of the supremum for the set of sequence terms, highlighting how completeness enables convergence in analysis.[16] In the context of ordered fields, Dedekind completeness is equivalent to Cauchy completeness, meaning every Cauchy sequence converges. This equivalence holds for Archimedean ordered fields, where the reals reside, ensuring that the two formulations capture the same notion of "no gaps."[17] Finally, the completeness axiom, when added to the structure of an ordered field, yields a unique system up to isomorphism: any two complete ordered fields are order-isomorphic as fields. This uniqueness theorem implies that all constructions of the reals—whether via Dedekind cuts or Cauchy sequences—produce essentially the same mathematical object.[1]Arithmetic and Order
Fundamental Operations
The real numbers form a field under the operations of addition and multiplication, ensuring closure: for any real numbers a and b, both a + b and a \times b are also real numbers.[18] This closure property guarantees that arithmetic operations remain within the set of real numbers, preserving their structure.[19] Every real number a has an additive inverse -a, such that a + (-a) = 0, and every nonzero real number a has a multiplicative inverse $1/a, such that a \times (1/a) = 1.[20] These inverses enable the definitions of subtraction and division: subtraction is defined as a - b = a + (-b), and division as a / b = a \times (1/b) for b \neq 0.[21] The fundamental operations satisfy key algebraic properties. Addition and multiplication are commutative: a + b = b + a and a \times b = b \times a.[22] They are also associative: (a + b) + c = a + (b + c) and (a \times b) \times c = a \times (b \times c).[23] Multiplication distributes over addition: a \times (b + c) = a \times b + a \times c.[18] A specific identity arising from these properties is -(a + b) = -a + (-b), which follows from the uniqueness of additive inverses.[24] For example, adding irrational numbers like \sqrt{2} + \pi yields another real number, approximately 4.555806216, whose decimal expansion is non-terminating and non-repeating, illustrating closure while producing an irrational result. Similarly, multiplying \sqrt{2} \times \pi \approx 4.442882938 remains a real number with a non-terminating decimal expansion. These operations highlight how real numbers extend beyond rationals, maintaining the field's properties even with irrationals.[22]Order Relations and Inequalities
The real numbers \mathbb{R} form a totally ordered field, where the strict order relation < satisfies trichotomy—for any a, b \in \mathbb{R}, exactly one of a < b, a = b, or b < a holds—and transitivity—if a < b and b < c, then a < c.[25] The non-strict order \leq is defined by a \leq b if a < b or a = b; this relation is reflexive (a \leq a) and antisymmetric (if a \leq b and b \leq a, then a = b).[12] The order is compatible with the field operations: for all a, b, c \in \mathbb{R}, if a < b, then a + c < b + c; and if a < b with c > 0, then a c < b c.[25] These properties ensure that addition and multiplication by positives preserve the order direction.[25] Derived inequalities follow directly from these axioms. For instance, if a < b and c < d, then a + c < b + d, obtained by adding the inequalities a < b and c < d.[25] Similarly, for positive reals, if $0 < a < b, then \frac{1}{b} < \frac{1}{a}; this holds because a < b implies a b > 0, and multiplying both sides by the positive quantity \frac{1}{a b} yields \frac{1}{b} < \frac{1}{a}.[26] Such rules underpin algebraic manipulations involving inequalities in the reals. The absolute value function on \mathbb{R} is defined by |x| = \max(x, -x), or equivalently, |x| = x if x \geq 0 and |x| = -x if x < 0; it satisfies |x| \geq 0 for all x, with equality if and only if x = 0.[27] A key property is the triangle inequality: for all a, b \in \mathbb{R}, |a + b| \leq |a| + |b|. To sketch the proof, note that -|a| \leq a \leq |a| and -|b| \leq b \leq |b| by definition of absolute value; adding these yields -(|a| + |b|) \leq a + b \leq |a| + |b|, so |a + b|, being the distance from a + b to 0, cannot exceed |a| + |b|.[27] Equality holds if a and b have the same sign (both nonnegative or both nonpositive).[27] The expression |a - b| provides an algebraic measure of the separation between a and b under the order, generalizing the absolute value as distance from 0.[27]Rational Embeddings
The natural numbers \mathbb{N} form a subset of the real numbers \mathbb{R}, where they are identified with the non-negative integers under the standard embedding that preserves their arithmetic structure. The integers \mathbb{Z} extend this embedding to include negative elements, maintaining the ring operations of addition and multiplication within \mathbb{R}.[28] The rational numbers \mathbb{Q} arise as the field of fractions of \mathbb{Z}, consisting of equivalence classes of pairs (p, q) with p \in \mathbb{Z}, q \in \mathbb{Z} \setminus \{0\}, under the relation (p, q) \sim (r, s) if ps = qr, and they embed into \mathbb{R} as a subfield that is closed under addition, multiplication, and taking inverses for non-zero elements. A key property of this embedding is the density of \mathbb{Q} in \mathbb{R}, meaning that between any two distinct real numbers x < y, there exists a rational number q such that x < q < y.[9] This density follows from the Archimedean property and the structure of \mathbb{Q}, ensuring that rationals approximate reals arbitrarily closely without gaps in the ordering.[29] For instance, given x, y \in \mathbb{R} with x < y, one can find integers m, n such that m/n lies in the interval (x, y) by scaling the difference y - x > 0.[9] The real numbers \mathbb{R} can be understood as an extension of \mathbb{Q} that fills the gaps in the rational number line, where sequences of rationals converging to irrational limits are incorporated to achieve completeness.[30] This construction preserves the ordered field properties of \mathbb{Q} while adding limits of Cauchy sequences of rationals, ensuring that \mathbb{Q} remains densely interwoven throughout \mathbb{R} without altering the existing rational structure.[31] The Archimedean property reinforces this embedding by implying that the natural numbers \mathbb{N} are unbounded in \mathbb{R}: for any real number x > 0, there exists n \in \mathbb{N} such that n > x.[14] This property guarantees the existence of an integer part for every real number, stated formally as: for any r \in \mathbb{R}, there exists n \in \mathbb{Z} such that n \leq r < n+1. [32] Here, n is the greatest integer less than or equal to r, often denoted \lfloor r \rfloor, which integrates seamlessly with the rational subfield.[14]Representations
Decimal Expansions
Every real number in the interval [0, 1) admits a decimal expansion of the form $0.d_1 d_2 d_3 \dots, where each d_i is an integer digit satisfying $0 \leq d_i \leq 9, and the value of the expansion is the infinite series \sum_{k=1}^\infty d_k / 10^k.[33] This representation arises from repeatedly multiplying the fractional part by 10 and extracting the integer part as the next digit, a process that generates the sequence of digits uniquely except in specific cases.[5] For a general non-negative real number r \geq 0, the decimal expansion extends by separating the integer part: r = n + \sum_{k=1}^\infty d_k / 10^k, where n = \lfloor r \rfloor is the greatest integer less than or equal to r, and the sum represents the fractional part in [0, 1).[33] Negative real numbers are represented by prefixing a minus sign to the expansion of their absolute value.[5] The algorithm to compute these digits for any real r begins with the fractional part f_0 = r - n; the first digit is d_1 = \lfloor 10 f_0 \rfloor, the next fractional part is f_1 = 10 f_0 - d_1, and iteratively, d_{k+1} = \lfloor 10 f_k \rfloor with f_{k+1} = 10 f_k - d_{k+1}, yielding remainders that remain in [0, 1).[34] Although decimal expansions provide a standard way to represent real numbers, they are not always unique. Real numbers with terminating expansions, such as those ending in infinite zeros, also possess an equivalent representation ending in infinite nines.[34] For instance, the expansions $0.999\dots and $1.000\dots represent the same real number 1. To prove this equivalence, let s = 0.999\dots; multiplying by 10 gives $10s = 9.999\dots, and subtracting the original yields $9s = 9, so s = 1.[35] This non-uniqueness occurs precisely when a real number can be expressed with a finite decimal, leading to exactly two distinct infinite expansions differing at the terminating point.[34]Alternative Bases
Real numbers can be represented using positional notation in any integer base b > 1, generalizing the decimal system. For a real number r \geq 0, the representation consists of an integer part and a fractional part, where the fractional part is given by r = n + \sum_{k=1}^{\infty} d_k b^{-k}, with n a non-negative integer and digits d_k integers satisfying $0 \leq d_k < b.[36] This allows every real number to be approximated arbitrarily closely by finite truncations of the expansion. In base 2 (binary), this notation is particularly useful for computational purposes, as it aligns with binary hardware. For instance, the irrational number \sqrt{2} has the binary expansion beginning $1.0110101000001001111\ldots_2, which can be used to compute approximations like \sqrt{2} \approx 1.01101_2 = 1 + 1/4 + 1/8 + 1/32 = 1.40625.[37] While representations in integer bases b \geq 2 are generally unique except for terminating expansions (which admit dual forms, such as $1/2 = 0.1000\ldots_2 = 0.0111\ldots_2), non-integer bases introduce significant uniqueness issues. In bases like the golden ratio \phi = (1 + \sqrt{5})/2 \approx 1.618, some real numbers possess multiple expansions, while others may lack any representation using digits 0 and 1; the set of uniquely representable numbers forms a Cantor-like set of measure zero. The Cantor set further illustrates limitations in integer bases, as it is an uncountable subset of [0,1] consisting precisely of numbers whose base-3 (ternary) expansions use only digits 0 and 2, implying that no finite ternary expansion can represent its points exhaustively, since the set of finite expansions is countable. As a non-positional alternative, continued fractions represent any real number x as x = [a_0; a_1, a_2, \ldots] = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{\ddots}}}, where the a_i (with a_0 \in \mathbb{Z}, a_i \geq 1 for i \geq 1) are unique for irrationals and provide optimal rational approximations.[38]Completeness in Analysis
Dedekind Cuts
A Dedekind cut is a partition of the rational numbers \mathbb{Q} into two nonempty subsets L and U such that every element of L is less than every element of U, and L has no greatest element.[39] This construction, introduced by Richard Dedekind, provides a way to define the real numbers \mathbb{R} as equivalence classes of such cuts, where each cut corresponds to a real number represented by the supremum of L.[39] The order relation on Dedekind cuts is defined by (L_1, U_1) < (L_2, U_2) if and only if L_1 is a proper subset of L_2.[39] This induces a total order on the set of all cuts, preserving the order properties of \mathbb{Q}. Rational numbers embed naturally into this structure: for a rational q, the cut has L = \{ r \in \mathbb{Q} \mid r < q \} and U = \{ r \in \mathbb{Q} \mid r \geq q \}.[39] The density of rationals ensures that gaps in cuts correspond precisely to irrational numbers, filling the incompleteness of \mathbb{Q}.[39] Arithmetic operations on cuts are defined to make the set of cuts an ordered field. Addition of two cuts (L_1, U_1) and (L_2, U_2) yields the cut with lower set L_1 + L_2 = \{ q_1 + q_2 \mid q_1 \in L_1, q_2 \in L_2 \} and upper set the complement.[39] For positive cuts, the lower set of the product consists of all non-positive rationals together with all positive rationals q such that there exist positive p \in L_1 and r \in L_2 with q \leq p r; this is extended to general cases using sign rules.[40] These operations are well-defined and satisfy the field axioms, with the order compatible. The set of Dedekind cuts forms a complete ordered field: it is an ordered field by verification of associativity, commutativity, distributivity, and order preservation in the operations; completeness follows because any nonempty subset of cuts bounded above has a least upper bound given by the cut whose lower set is the union of the lower sets of the subset.[39] This structure is unique up to isomorphism as the complete ordered field containing \mathbb{Q}, establishing the cuts as a rigorous construction of \mathbb{R}.[39]Cauchy Sequences
A Cauchy sequence of rational numbers is a sequence (q_n)_{n=1}^\infty with each q_n \in \mathbb{Q} such that for every \varepsilon > 0, there exists N \in \mathbb{N} with the property that |q_m - q_n| < \varepsilon whenever m, n > N.[41] This condition ensures that the terms of the sequence become arbitrarily close to each other as the index increases, capturing the intuitive notion of convergence within the rationals.[42] Two Cauchy sequences (q_n) and (r_n) are said to be equivalent, written (q_n) \sim (r_n), if \lim_{n \to \infty} |q_n - r_n| = 0.[41] This equivalence relation partitions the set of all Cauchy sequences of rationals into equivalence classes, where each class consists of sequences that "converge to the same limit" in an informal sense.[42] The real numbers \mathbb{R} are then constructed as the quotient set of these equivalence classes, with each real number represented by [ (q_n) ], the class containing (q_n).[31] Arithmetic operations on \mathbb{R} are induced from those on \mathbb{Q} by defining [ (q_n) ] + [ (r_n) ] = [ (q_n + r_n) ] and [ (q_n) ] \cdot [ (r_n) ] = [ (q_n r_n) ] for representative sequences (q_n) and (r_n).[41] These definitions are independent of the choice of representatives, as the equivalence relation preserves the operations, yielding a field structure isomorphic to the rationals extended by limits.[42] The order on \mathbb{R} inherits from the absolute value on \mathbb{Q}, as detailed in the order relations section.[43] The metric on \mathbb{R} is given by d([ (q_n) ], [ (r_n) ]) = \lim_{n \to \infty} |q_n - r_n|, which is well-defined and turns \mathbb{R} into a metric space.[41] This construction completes the rationals: every Cauchy sequence in \mathbb{Q} defines a real number, and \mathbb{R} itself is complete, meaning every Cauchy sequence of reals converges to a real limit.[44] Proofs involve verifying that Cauchy sequences of equivalence classes remain Cauchy and converge within the quotient, ensuring no "gaps" persist as in \mathbb{Q}.[42] This Cauchy completion yields a complete ordered field isomorphic as an ordered field to the reals constructed via Dedekind cuts.[45] The isomorphism maps each equivalence class to the Dedekind cut it generates, preserving addition, multiplication, and order.[46]Topological Structure
Metric Space Properties
The real line \mathbb{R} forms a metric space with the standard metric d: \mathbb{R} \times \mathbb{R} \to [0, \infty)$ defined by d(x, y) = |x - y|, where the absolute value function |\cdot|is induced by the order relation on\mathbb{R}via|z| = zifz \geq 0and|z| = -zifz < 0.[49] This metric satisfies three fundamental properties: non-negativity, where d(x, y) \geq 0andd(x, y) = 0if and only ifx = y; symmetry, where d(x, y) = d(y, x)for allx, y \in \mathbb{R}; and the triangle inequality, where d(x, z) \leq d(x, y) + d(y, z)for allx, y, z \in \mathbb{R}.[49] These axioms ensure that dmeasures distances consistently, enabling the study of convergence and continuity in\mathbb{R}$.[47] The metric d induces a topology on \mathbb{R}, where open sets are arbitrary unions of open balls B(x, \epsilon) = \{ y \in \mathbb{R} \mid d(x, y) < \epsilon \} for x \in \mathbb{R} and \epsilon > 0.[48] In \mathbb{R}, each open ball B(x, \epsilon) coincides with the open interval (x - \epsilon, x + \epsilon), and the collection of all such open intervals forms a basis for the topology, meaning every open set is a union of these intervals.[47] A neighborhood of a point x \in \mathbb{R} is any set containing an open ball B(x, \epsilon) for some \epsilon > 0, and thus open sets consist precisely of sets where every point has such a neighborhood contained within it.[48] Closed intervals [a, b] = \{ x \in \mathbb{R} \mid a \leq x \leq b \} for a \leq b are the complements of open sets in certain cases, but more generally, they are the closures of their corresponding open intervals under this topology.[49] The metric space (\mathbb{R}, d) is complete, meaning that every Cauchy sequence \{x_n\} in \mathbb{R}—where for every \epsilon > 0 there exists N \in \mathbb{N} such that d(x_m, x_n) < \epsilon for all m, n > N—converges to some limit x \in \mathbb{R}.[47] This completeness distinguishes \mathbb{R} from incomplete metric spaces like the rationals \mathbb{Q} under the same metric. Additionally, bounded closed subsets of \mathbb{R} exhibit sequential compactness, as established by the Bolzano-Weierstrass theorem: every bounded sequence in \mathbb{R} has a subsequence that converges to a point in \mathbb{R}.[50] Consequently, a subset of \mathbb{R} is sequentially compact if and only if it is closed and bounded, ensuring that sequences in such sets always admit convergent subsequences within the set itself.[48]Heine-Borel Compactness
In the context of subsets of the real numbers \mathbb{R}, a set S is defined to be compact if every open cover of S has a finite subcover. An open cover of S is a collection of open sets \{U_\alpha\}_{\alpha \in A} such that S \subseteq \bigcup_{\alpha \in A} U_\alpha, and a finite subcover is a finite subfamily \{U_{\alpha_1}, \dots, U_{\alpha_n}\} that still covers S. This topological property captures the idea that S cannot be "spread out" indefinitely without being reducible to a finite portion of any covering by open sets./04%3A_Topology_of_the_Real_Line/4.04%3A_Compact_Sets) A subset S \subseteq \mathbb{R} is bounded if there exists some M > 0 such that |x| \leq M for all x \in S. Equivalently, S is contained in some finite interval [-M, M]. Boundedness ensures that S does not extend infinitely in either direction along the real line, providing a foundational constraint for compactness in \mathbb{R}.[51] The Heine-Borel theorem characterizes compactness for subsets of \mathbb{R}: a subset S \subseteq \mathbb{R} is compact if and only if it is closed and bounded. This result, first stated and proved in a restricted form (for countable covers) by Émile Borel in 1895, and later generalized, highlights the interplay between closure (containing all limit points) and boundedness in the Euclidean topology of \mathbb{R}. The theorem fails in infinite-dimensional spaces but holds in finite-dimensional Euclidean spaces, underscoring the special structure of \mathbb{R}.[52][53] A proof of the "if" direction—that every closed and bounded set is compact—relies on the completeness of \mathbb{R} via the nested interval theorem. Consider a closed bounded set S \subseteq [a, b] and an arbitrary open cover \{U_\alpha\}_{\alpha \in A} of S. Define the auxiliary set T = \{x \in [a, b] \mid [a, x] \cap S can be covered by finitely many sets from \{U_\alpha\}\}. The set T is nonempty (as a \in T) and bounded above by b, so let \gamma = \sup T. Since S is closed, \gamma \in S. There exists an open set U_{\alpha_0} \in \{U_\alpha\} containing \gamma with radius \delta > 0 such that (\gamma - \delta, \gamma + \delta) \subseteq U_{\alpha_0}. Choose x \in T with \gamma - \delta < x \leq \gamma; then [a, x] \cap S has a finite subcover \{U_{\alpha_1}, \dots, U_{\alpha_n}\}, and adjoining U_{\alpha_0} yields a finite cover of [a, \gamma + \delta/2] \cap S, implying \gamma + \delta/2 \in T. Iterating this process constructs nested closed intervals whose intersection is nonempty by completeness, yielding a contradiction unless the entire [a, b] \cap S has a finite subcover. The converse—that compact sets are closed and bounded—follows from the fact that unbounded compact sets would admit covers without finite subcovers (e.g., balls of increasing radius) and non-closed compact sets would miss limit points coverable only infinitely.[54] One key application of the Heine-Borel theorem is the extreme value theorem: if f: K \to \mathbb{R} is continuous on a compact set K \subseteq \mathbb{R}, then f attains both its maximum and minimum values on K. To see this, consider the image f(K), which is compact (as the continuous image of a compact set) and thus bounded and closed in \mathbb{R}. Hence, \sup f(K) and \inf f(K) are achieved at some points in K. This result, proved using Heine-Borel compactness, is fundamental for optimization and guarantees the existence of extrema without explicit computation.[55]Set-Theoretic Aspects
Cardinality
The cardinality of the set of real numbers \mathbb{R} is $2^{\aleph_0}, commonly denoted by \mathfrak{c} and known as the cardinality of the continuum.[56] This cardinality is strictly greater than \aleph_0, the cardinality of the natural numbers \mathbb{N}, establishing that \mathbb{R} is uncountable.[56] In contrast, the set of rational numbers \mathbb{Q} is countable, with |\mathbb{Q}| = \aleph_0 < \mathfrak{c}.[57] The power set \mathcal{P}(\mathbb{N}), consisting of all subsets of \mathbb{N}, also has cardinality |\mathcal{P}(\mathbb{N})| = 2^{\aleph_0} = \mathfrak{c}.[58] This equality follows from the fact that binary expansions of real numbers in the interval (0,1) provide a surjection from \{0,1\}^{\mathbb{N}} (in bijection with \mathcal{P}(\mathbb{N})) onto (0,1), while the uncountability of (0,1) matches that of \mathcal{P}(\mathbb{N}).[58] The set \mathbb{R} is in bijection with the open interval (0,1), confirming they share the same cardinality \mathfrak{c}.[57] One such bijection is the composition h(x) = \tan(\pi(x - 1/2)) for x \in (0,1), which maps (0,1) continuously and bijectively onto \mathbb{R}.[57] There is no bijection between \mathbb{R} and \mathbb{N}, as \mathbb{R} cannot be enumerated in a countable list.[56] Cantor's diagonal argument demonstrates the uncountability of [0,1] (and thus \mathbb{R}) by contradiction, relying on decimal expansions of real numbers.[56] Assume there exists a bijection f: \mathbb{N} \to [0,1], listing the numbers as \begin{align*} f(1) &= 0.d_{11} d_{12} d_{13} \dots , \\ f(2) &= 0.d_{21} d_{22} d_{23} \dots , \\ &\vdots \\ f(n) &= 0.d_{n1} d_{n2} d_{n3} \dots , \end{align*} where each d_{ij} \in \{0,1,\dots,9\}. Construct a real number r = 0.d_1 d_2 d_3 \dots \in [0,1] such that d_i \neq d_{ii} for all i (e.g., d_i = d_{ii} + 1 if d_{ii} \leq 8, else d_i = 0). Then r differs from f(n) in the nth decimal place for every n, so r is not in the list, contradicting the assumption of a bijection.[56] This proof, originally presented by Georg Cantor, establishes that no such enumeration exists.[59]Continuum Hypothesis
The continuum hypothesis (CH), proposed by Georg Cantor, asserts that there is no infinite set S whose cardinality satisfies \aleph_0 < |S| < 2^{\aleph_0}, where \aleph_0 is the cardinality of the natural numbers and $2^{\aleph_0} is the cardinality of the real numbers.[60] In 1900, David Hilbert elevated this conjecture to the first of his 23 unsolved problems in mathematics, emphasizing its fundamental role in understanding infinite cardinalities during his address at the International Congress of Mathematicians in Paris. Kurt Gödel proved in 1938 that the axiom of choice and the generalized continuum hypothesis (GCH)—which extends CH by stating that $2^\kappa = \kappa^+ for every infinite cardinal \kappa, where \kappa^+ is the successor cardinal—are consistent with the Zermelo-Fraenkel axioms of set theory including the axiom of choice (ZFC), assuming ZFC itself is consistent.[60] This result constructed an inner model of ZFC, known as the constructible universe L, in which CH holds. In 1963, Paul Cohen established the independence of CH from ZFC by developing the forcing technique, demonstrating that CH can be false in some models of ZFC while remaining consistent. Cohen's work showed that if ZFC is consistent, then so is ZFC together with the negation of CH, implying that no contradiction arises from assuming the existence of a cardinal between \aleph_0 and $2^{\aleph_0}. If CH holds, then the cardinality of the real numbers equals \aleph_1, the smallest uncountable cardinal, making |\mathbb{R}| = 2^{\aleph_0} = \aleph_1. The generalized CH, consistent via Gödel's model, further constrains the power set cardinalities across all infinite levels. Modern investigations explore axioms beyond ZFC that influence the continuum's size; for instance, Martin's axiom (MA), introduced by Donald A. Martin and Robert M. Solovay in 1970, combined with the negation of CH, implies that $2^{\aleph_0} is large, often greater than many successor cardinals, and prohibits certain small continuum values in forcing extensions.[61]Formal Constructions
Axiomatic Approach
The axiomatic approach to the real numbers begins within Zermelo-Fraenkel set theory with the axiom of choice (ZFC), starting from the empty set ∅. The axiom of infinity postulates the existence of an infinite set, which allows the construction of the natural numbers ℕ as the smallest inductive set containing ∅ and closed under the successor operation, typically defined via von Neumann ordinals: 0 = ∅, 1 = {∅}, 2 = {∅, {∅}}, and so on.[62] The integers ℤ are then formed as equivalence classes of ordered pairs from ℕ × ℕ under the relation (a, b) ∼ (c, d) if a + d = b + c, with addition and multiplication defined componentwise to yield an ordered ring. The rational numbers ℚ arise similarly as equivalence classes of ℤ × (ℤ \ {0}) under (a, b) ∼ (c, d) if a d = b c, forming an ordered field dense in itself.[62] The real numbers ℝ are axiomatized as the unique (up to isomorphism) complete ordered field extending ℚ, where completeness ensures that every nonempty subset bounded above has a least upper bound. This structure satisfies the field axioms for addition and multiplication, the order axioms for a total order compatible with the operations, and the completeness axiom. Formally, let F be a set with operations + and ·, constants 0 and 1, and relation ≤. The axioms are: Field Axioms:- Commutativity: For all x, y ∈ F, x + y = y + x and x · y = y · x.
- Associativity: For all x, y, z ∈ F, (x + y) + z = x + (y + z) and (x · y) · z = x · (y · z).
- Identities: There exists 0 ∈ F such that x + 0 = x for all x ∈ F, and 1 ∈ F such that x · 1 = x for all x ∈ F.
- Inverses: For each x ∈ F, there exists -x ∈ F such that x + (-x) = 0; for x ≠ 0, there exists x^{-1} ∈ F such that x · x^{-1} = 1.
- Distributivity: For all x, y, z ∈ F, x · (y + z) = (x · y) + (x · z).
- Trichotomy: For all x, y ∈ F, exactly one of x < y, x = y, or y < x holds, where x < y iff x ≤ y and x ≠ y.
- Transitivity: If x ≤ y and y ≤ z, then x ≤ z.
- Addition preservation: If x ≤ y, then x + z ≤ y + z for all z ∈ F.
- Multiplication preservation: If x ≤ y and 0 ≤ z, then x · z ≤ y · z.