Fact-checked by Grok 2 weeks ago

Cardinality of the continuum

In set theory, the cardinality of the continuum is the cardinality of the set of real numbers \mathbb{R}, often denoted by \mathfrak{c} or $2^{\aleph_0}, which measures the "size" of the continuum as the uncountably infinite collection of points on the real line.^[1]^[2] This cardinality equals that of the power set of the natural numbers, \mathcal{P}(\mathbb{N}), and is strictly larger than the countable infinity \aleph_0 of the integers, as first proven by Georg Cantor in 1874, which demonstrates that no bijection exists between \mathbb{N} and \mathbb{R}.^[1]^[2] Sets with cardinality \mathfrak{c} include not only \mathbb{R} but also the open unit interval (0,1), the Cartesian plane \mathbb{R} \times \mathbb{R}, and the unit square, all of which are in bijection with the continuum via explicit constructions like the Cantor-Schröder-Bernstein theorem.^[3]^[1] In cardinal arithmetic, \mathfrak{c} satisfies properties such as \mathfrak{c} + \aleph_0 = \mathfrak{c}, \mathfrak{c} \times \mathfrak{c} = \mathfrak{c}, and \mathfrak{c}^{\aleph_0} = \mathfrak{c}, under the axioms of Zermelo-Fraenkel set theory with the axiom of choice (ZFC).^[1] A central question in set theory is the exact position of \mathfrak{c} in the hierarchy of infinite cardinals, formalized by the Continuum Hypothesis (CH), which asserts that \mathfrak{c} = \aleph_1, meaning there is no cardinal strictly between \aleph_0 and \mathfrak{c}.^[2]^[3] Proposed by Cantor in the late 19th century and highlighted as the first of Hilbert's 23 problems in 1900, CH was shown by Kurt Gödel in 1938 to be consistent with ZFC and by Paul Cohen in 1963 to be independent of ZFC using forcing methods, implying that \mathfrak{c} could be \aleph_1, \aleph_2, or even a larger cardinal like \aleph_{\omega_1} in different models of set theory.^[2] The Generalized Continuum Hypothesis (GCH) extends this by positing $2^{\aleph_\alpha} = \aleph_{\alpha+1} for all ordinals \alpha, further exploring the structure of power sets beyond the continuum.^[2]^[3]

Basic Properties

Uncountability

The uncountability of the real numbers was established by Georg Cantor through his diagonal argument, published in 1891 in the paper "Über eine elementare Frage der Mannigfaltigkeitslehre."^[4] This proof demonstrates that no bijection exists between the natural numbers \mathbb{N} and the real numbers \mathbb{R}, implying that the cardinality of \mathbb{R}, denoted \mathfrak{c}, strictly exceeds the cardinality of \mathbb{N}, which is \aleph_0. The argument applies directly to the power set of \mathbb{N}, showing it to be uncountable and foreshadowing its equality with \mathfrak{c}. To see the diagonal argument in action, consider the open interval (0,1), which has the same cardinality as \mathbb{R} via the bijection f(x) = \tan(\pi(x - 1/2)) for x \in (0,1).^[5] Assume for contradiction that the reals in (0,1) are countable, so they can be listed as a sequence \{r_n\}_{n=1}^\infty, where each r_n has a decimal expansion r_n = 0.d_{n1}d_{n2}d_{n3}\dots with d_{ni} \in \{0,1,\dots,9\} and expansions chosen to avoid infinite trailing 9s for uniqueness. Construct a new real s = 0.s_1 s_2 s_3 \dots \in (0,1) by setting s_k = 5 if d_{kk} \neq 5 and s_k = 6 otherwise. Then s differs from r_k in the kth decimal place for every k, so s is not in the list, yielding a contradiction. Thus, no such enumeration exists, and (0,1) is uncountable.^[4] The same diagonal construction shows that the set of infinite binary sequences, which can be mapped onto (0,1) via binary expansions $0.b_1 b_2 b_3 \dots where b_i \in \{0,1\}, is uncountable, reinforcing the result without reliance on base-10 specifics.^[4] This uncountability extends to the irrationals: the rational numbers \mathbb{Q} are countable, as proved by Cantor in 1874 by enumerating them via their reduced fractional forms and ordering by sum of numerator and denominator absolute values.^[6] Since \mathbb{R} = \mathbb{Q} \sqcup (\mathbb{R} \setminus \mathbb{Q}) is a disjoint union and |\mathbb{Q}| = \aleph_0 < \mathfrak{c}, the irrationals \mathbb{R} \setminus \mathbb{Q} must have cardinality \mathfrak{c}.^[2] In this sense, \mathfrak{c} > \aleph_0, though the exact position of \mathfrak{c} among infinite cardinals remains independent of standard set-theoretic axioms.

Representation as 2^ℵ₀

The cardinality of the continuum, denoted by \mathfrak{c}, is defined as the cardinal number $2^{\aleph_0}, where cardinal exponentiation \kappa^\lambda is the cardinality of the set of all functions from a set of cardinality \lambda to a set of cardinality \kappa.^[7] Thus, $2^{\aleph_0} equals the cardinality of the power set \mathcal{P}(\mathbb{N}), since the set of functions from \mathbb{N} to \{0,1\} is in bijection with the subsets of \mathbb{N}. Cantor's theorem states that for any set A, the cardinality of its power set satisfies |\mathcal{P}(A)| > |A|. The proof proceeds by contradiction: assume there exists a surjection f: A \to \mathcal{P}(A); then the set D = \{ x \in A \mid x \notin f(x) \} has no preimage under f, yielding a contradiction. Applying this to A = \mathbb{N}, it follows that $2^{\aleph_0} > \aleph_0. To establish |\mathbb{R}| = 2^{\aleph_0}, consider the construction of real numbers via Dedekind cuts. Each real number corresponds to a Dedekind cut, defined as a nonempty proper subset S \subseteq \mathbb{Q} that is downward closed (if q \in S and r < q then r \in S), has no maximum element, and is bounded above. The map sending each real to its defining cut injects \mathbb{R} into \mathcal{P}(\mathbb{Q}). Since there is a bijection between \mathbb{Q} and \mathbb{N}, |\mathcal{P}(\mathbb{Q})| = 2^{\aleph_0}, so |\mathbb{R}| \leq 2^{\aleph_0}. For the reverse inequality, define an injection from \{0,1\}^\mathbb{N} (of cardinality $2^{\aleph_0}) to (0,1) \subseteq \mathbb{R} by g((a_n)_{n=1}^\infty) = \sum_{n=1}^\infty a_n / 3^n. This ternary expansion map is injective because base-3 representations with digits 0 and 1 are unique. Thus, $2^{\aleph_0} \leq |\mathbb{R}|. The Schröder-Bernstein theorem asserts that if there are injections f: X \to Y and g: Y \to X, then there exists a bijection between X and Y. Applying this with X = \{0,1\}^\mathbb{N} and Y = \mathbb{R} (composing with the bijection from \mathbb{R} to (0,1)) yields |\mathbb{R}| = 2^{\aleph_0}. An alternative application uses the earlier uncountability |\mathbb{R}| > \aleph_0 together with \aleph_0 < 2^{\aleph_0} \leq |\mathbb{R}| \leq 2^{\aleph_0} to conclude equality via Schröder-Bernstein.^[7]

Cardinal Equalities

The cardinality of the real numbers, denoted \mathfrak{c} = |\mathbb{R}|, equals the cardinality of \mathbb{R}^n for any finite positive integer n. This follows from the existence of a bijection between \mathbb{R} and \mathbb{R}^n. One explicit construction interleaves the decimal expansions of the coordinates: for a point (x_1, x_2, \dots, x_n) \in \mathbb{R}^n, where each x_i has decimal expansion $0.d_{i1}d_{i2}d_{i3}\dots, form the real number whose decimal is $0.d_{11}d_{21}\dots d_{n1}d_{12}d_{22}\dots d_{n2}\dots. This mapping is bijective, handling non-uniqueness of expansions (like terminating decimals) by consistent conventions, such as avoiding infinite 9's.^[8] Similarly, |\mathbb{R}| = |\mathbb{C}|, where \mathbb{C} is the set of complex numbers. The complex numbers are isomorphic to \mathbb{R}^2 via the identification a + bi \mapsto (a, b) for a, b \in \mathbb{R}, and since |\mathbb{R}^2| = |\mathbb{R}| as established above, the result follows immediately.^[9] The space of all functions from the natural numbers to the reals, denoted \mathbb{R}^\mathbb{N}, also has cardinality \mathfrak{c}. To see this, note that \mathbb{R}^\mathbb{N} injects into \mathfrak{c}^\aleph_0 since |\mathbb{R}| = \mathfrak{c}, and under the axiom of choice (AC), cardinal exponentiation satisfies \mathfrak{c}^\aleph_0 = (2^{\aleph_0})^{\aleph_0} = 2^{\aleph_0 \cdot \aleph_0} = 2^{\aleph_0} = \mathfrak{c}, where \aleph_0 \cdot \aleph_0 = \aleph_0 for the infinite cardinals involved. Conversely, \mathbb{R} injects into \mathbb{R}^\mathbb{N} by constant sequences, so by the Schröder–Bernstein theorem, the cardinalities are equal.^[10] The power set of the rationals, \mathcal{P}(\mathbb{Q}), has cardinality \mathfrak{c}. Since |\mathbb{Q}| = \aleph_0, Cantor's theorem gives |\mathcal{P}(\mathbb{Q})| = 2^{\aleph_0} = \mathfrak{c}. An explicit bijection can be constructed by enumerating \mathbb{Q} = \{q_1, q_2, \dots\} and identifying subsets of \mathbb{Q} with binary sequences in \{0,1\}^\mathbb{N}, which has cardinality $2^{\aleph_0}, and then mapping to reals via binary expansions.^[11] More generally, under AC, for any infinite cardinal \kappa with \aleph_0 \leq \kappa \leq \mathfrak{c}, the product \mathfrak{c} \cdot \kappa = \max(\mathfrak{c}, \kappa) = \mathfrak{c}. This holds because \kappa \leq \mathfrak{c} implies \mathfrak{c} \cdot \kappa \leq \mathfrak{c} \cdot \mathfrak{c} = 2^{\aleph_0} \cdot 2^{\aleph_0} = 2^{\aleph_0} = \mathfrak{c}, and \mathfrak{c} \leq \mathfrak{c} \cdot \kappa obviously, so equality follows by Schröder–Bernstein; the key step \mathfrak{c} \cdot \mathfrak{c} = \mathfrak{c} uses the fact that infinite cardinals satisfy \lambda \cdot \mu = \max(\lambda, \mu) when at least one is infinite and nonzero.^[12]

Infinite Cardinals and the Continuum

Beth Numbers

The beth numbers, denoted \beth_\alpha for ordinals \alpha, form a hierarchy of infinite cardinals constructed by iterating the power set operation starting from the countable infinite cardinal. They are defined via transfinite recursion as follows: \beth_0 = \aleph_0, the cardinality of the natural numbers; \beth_{\alpha+1} = 2^{\beth_\alpha}, the cardinality of the power set of a set of cardinality \beth_\alpha; and for a limit ordinal \lambda, \beth_\lambda = \sup\{\beth_\alpha \mid \alpha < \lambda\}, the least upper bound of the preceding beth numbers.^[7] The first beth number beyond the countable is the continuum: \beth_1 = 2^{\aleph_0} = \mathfrak{c}, the cardinality of the real numbers.^[7] The notation for the beth numbers was introduced by Charles Sanders Peirce in 1900.^[13] Beth numbers exhibit notable properties, including the existence of fixed points, which are cardinals \kappa such that \beth_\kappa = \kappa; these can be constructed by transfinite iteration and form a closed unbounded class of ordinals. Without the generalized continuum hypothesis (GCH), the beth numbers can grow faster than the aleph numbers, potentially exceeding them due to the power set operation. Under GCH, however, \beth_\alpha = \aleph_\alpha for all \alpha.^[7]

Aleph Numbers

In set theory, the aleph numbers \aleph_\alpha, where \alpha ranges over all ordinals, enumerate the infinite cardinal numbers in a canonical way based on well-orderings. The aleph numbers are defined recursively: \aleph_0 is the cardinality of the set of natural numbers \mathbb{N}, \aleph_{\alpha+1} is the successor cardinal to \aleph_\alpha, and for a limit ordinal \lambda, \aleph_\lambda = \sup\{\aleph_\beta \mid \beta < \lambda\}.^[14] This hierarchy assumes the axiom of choice, which guarantees that every set can be well-ordered, allowing infinite cardinals to be indexed by ordinals as alephs.^[14] The successor aleph \aleph_{\alpha+1} is constructed using the Hartogs number of the cardinal \aleph_\alpha. The Hartogs number h(X) of a set X is the least ordinal that cannot be injected into X, ensuring that |h(X)| > |X| and that h(X) itself is a cardinal ordinal.^[15] With the axiom of choice, this yields \aleph_{\alpha+1} = h(\aleph_\alpha), the smallest cardinal strictly larger than \aleph_\alpha. All infinite cardinals are limit ordinals in this framework.^[14] Key properties of aleph numbers include their cofinality, defined as the smallest cardinal \lambda such that the aleph in question is the sum of \lambda many smaller cardinals.^[14] An aleph \aleph_\alpha is regular if its cofinality equals itself, as holds for successor alephs like \aleph_{\beta+1}, and singular if its cofinality is strictly smaller, as in the case of \aleph_\omega = \sup\{\aleph_n \mid n < \omega\}, where the cofinality is \aleph_0 < \aleph_\omega.^[14] The aleph number \aleph_1 is the smallest uncountable cardinal.^[14] The cardinality of the continuum \mathfrak{c} = 2^{\aleph_0} equals some \aleph_\alpha with \alpha \geq 1, but determining the exact index \alpha remains independent of the standard axioms of set theory.^[14] In contrast to beth numbers, which generate cardinals via iterated power sets starting from finite sets, aleph numbers arise from the well-ordering of infinite sets.^[14]

The Continuum Hypothesis

The Continuum Hypothesis (CH) asserts that the cardinality of the continuum \mathfrak{c} = 2^{\aleph_0} equals \aleph_1, the least uncountable cardinal, implying no infinite cardinal lies strictly between \aleph_0 and \mathfrak{c}.^[2] The Generalized Continuum Hypothesis (GCH) generalizes this assertion, stating that for every infinite cardinal \kappa, $2^\kappa = \kappa^+, where \kappa^+ is the successor cardinal of \kappa.^[2] Georg Cantor first conjectured CH in 1878 as part of his investigations into transfinite numbers and the structure of the real line.^[16] In 1900, David Hilbert elevated it to prominence by designating it as the first of his 23 unsolved problems in mathematics, emphasizing its foundational importance for set theory.^[17] Kurt Gödel advanced the understanding in 1938 by constructing the inner model L, the universe of constructible sets, and proving that ZFC (Zermelo-Fraenkel set theory with the axiom of choice) is consistent with both CH and GCH relative to the consistency of ZFC itself.^[18] Paul Cohen completed the independence results in 1963, introducing the method of forcing to show that the negation of CH is also consistent with ZFC, thereby establishing that CH cannot be proved or disproved within standard set-theoretic axioms. In modern set theory, large cardinal axioms provide constraints on possible violations of CH. For instance, assuming the existence of a proper class of Woodin cardinals, W. Hugh Woodin constructed in 1999 a model where $2^{\aleph_0} = \aleph_2 and all projective sets of reals satisfy a form of determinacy, yielding a controlled failure of CH while preserving key inner model properties.^[19] Regarding the Suslin Hypothesis (SH), which posits that every ccc (countable chain condition) dense linear order without endpoints is order-isomorphic to the reals (equivalently, no Suslin trees exist), Ronald Jensen proved in the 1970s that SH is independent of ZFC + GCH; specifically, he constructed models where GCH holds alongside both SH and its negation.^[20] CH admits several equivalent formulations within ZFC. The interpolant version states there exists no set S with \aleph_0 < |S| < \mathfrak{c}.^[2] The well-ordering version asserts that no subset of \mathbb{R} admits a well-ordering of order type \omega_2, reflecting that all well-orderable subsets of \mathbb{R} have cardinality at most \aleph_0 or exactly \mathfrak{c}.^[2] Another equivalent, in the context of measure theory, is that \mathbb{R} cannot be expressed as a union of \aleph_1 many Lebesgue null sets under the axiom of determinacy (AD), where AD implies the failure of CH and enhances regularity properties of sets of reals.^[2]

Sets Involving the Continuum

Sets of Cardinality 𝔠

Prominent examples of sets with cardinality \mathfrak{c} arise in topology. The closed unit interval [0,1] has cardinality \mathfrak{c}.^[21] Finite-dimensional Euclidean spaces \mathbb{R}^n for n \geq 1 also have cardinality \mathfrak{c}.^[21] The middle-thirds Cantor set, a compact nowhere dense perfect subset of [0,1] with Lebesgue measure zero, similarly has cardinality \mathfrak{c}.^[22] In measure theory and descriptive set theory, Lebesgue measurable sets provide further instances. Any Lebesgue measurable subset of \mathbb{R} with positive measure has cardinality \mathfrak{c}.^[23] The space of continuous real-valued functions on a compact Hausdorff space K with |K| \leq \mathfrak{c}, denoted C(K, \mathbb{R}), has cardinality \mathfrak{c}. Algebraic structures yield additional examples. The field \mathbb{R} of real numbers, regarded as a vector space over the field \mathbb{Q} of rational numbers, has dimension \mathfrak{c}.^[24] Consequently, any Hamel basis for this vector space has cardinality \mathfrak{c}, and the extension degree [\mathbb{R}:\mathbb{Q}] equals \mathfrak{c}.^[24] In probability theory, the unit interval [0,1] equipped with the Lebesgue measure forms a standard probability space whose sample space has cardinality \mathfrak{c}. The continuum hypothesis (CH) implies specific well-orderability properties. Under CH, \mathfrak{c} = \aleph_1, so \mathbb{R} admits a well-ordering of order type \omega_1. Without CH, \aleph_1 < \mathfrak{c} holds, and thus sets of cardinality \aleph_1 strictly smaller than \mathfrak{c} exist, such as the set \omega_1 of countable ordinals.

Sets of Cardinality Greater than 𝔠

The power set of the real numbers, denoted \mathcal{P}(\mathbb{R}), consists of all subsets of \mathbb{R} and has cardinality $2^\mathfrak{c}, which is strictly greater than \mathfrak{c}. This follows from Cantor's theorem, which asserts that for any set S, the cardinality of its power set \mathcal{P}(S) exceeds that of S itself, as no surjection from S onto \mathcal{P}(S) can exist.^[25] Cantor's theorem, first proved in his 1891 paper, establishes this strict inequality and implies an unending hierarchy of larger cardinals beyond \mathfrak{c}.^[7] In 1874, Cantor demonstrated the uncountability of the continuum and explored its properties, laying foundational groundwork for recognizing cardinalities larger than \mathfrak{[c](/page/c)} through operations like power sets. The power set \mathcal{P}(\mathbb{R}) exemplifies such a set, as its elements include not only familiar subsets like the rationals or irrationals but also uncountably many more abstract collections, underscoring the exponential growth in size. The set of all functions from \mathbb{R} to \mathbb{R}, denoted \mathbb{R}^\mathbb{R}, provides another example of a set with cardinality exceeding \mathfrak{[c](/page/c)}. This space has cardinality \mathfrak{[c](/page/c)}^\mathfrak{[c](/page/c)}, and under the axiom of choice, cardinal exponentiation yields \mathfrak{[c](/page/c)}^\mathfrak{[c](/page/c)} = (2^{\aleph_0})^{2^{\aleph_0}} = 2^{\aleph_0 \cdot 2^{\aleph_0}} = 2^{2^{\aleph_0}} = 2^\mathfrak{[c](/page/c)}, matching the cardinality of \mathcal{P}(\mathbb{R}).^[7] Functions in \mathbb{R}^\mathbb{R} range from continuous ones like polynomials to highly discontinuous mappings, but the total count surpasses \mathfrak{[c](/page/c)} due to the vast possibilities for arbitrary assignments across the uncountable domain. The beth numbers extend this hierarchy systematically: \beth_0 = \aleph_0, \beth_1 = 2^{\aleph_0} = \mathfrak{c}, \beth_2 = 2^\mathfrak{c}, and for successor ordinals, \beth_{\alpha+1} = 2^{\beth_\alpha}, with limits defined by supremum. Thus, \beth_2 captures the cardinality immediately above \mathfrak{c} via power set iteration, while higher beths like \beth_3 = 2^{2^\mathfrak{c}} and beyond generate even larger cardinals, forming an infinite ascending sequence.^[7] Ordinal examples illustrate these cardinalities in well-ordered contexts. The initial ordinal of cardinality $2^\mathfrak{c} is the smallest ordinal \omega_\gamma such that |\omega_\gamma| = 2^\mathfrak{c}, where \gamma is the least ordinal indexing that cardinal in the aleph sequence; under the generalized continuum hypothesis, \gamma = 2 so it is \omega_2, but under the continuum hypothesis alone, \gamma \geq 2 and the exact position varies.^[2] In ZFC set theory, it is provable that $2^\mathfrak{c} \geq \aleph_2, as \aleph_1 \leq \mathfrak{c} < 2^\mathfrak{c} forces the next cardinal after \mathfrak{c} to be at least \aleph_2, but the precise identification of $2^\mathfrak{c} with some \aleph_\delta (for \delta \geq 2) remains independent of ZFC, allowing models where it equals \aleph_2, \aleph_{17}, or larger.^[2] This independence highlights how the continuum hypothesis influences the "gap" immediately following \mathfrak{c} in the aleph hierarchy, though larger cardinals like $2^\mathfrak{c} always exceed it regardless. The indefinite continuation of this hierarchy via repeated exponentiation ensures no largest cardinal exists.

References

[1]
cardinality of the continuum - PlanetMath
Mar 22, 2013 · The cardinality of the continuum , often denoted by c 𝔠 , is the cardinality of the set R ℝ of real numbers. A set of cardinality c 𝔠 is ...
[2]
The Continuum Hypothesis - Stanford Encyclopedia of Philosophy
May 22, 2013 · The first formulation of CH is that there is no interpolant, that is, there is no infinite set A of real numbers such that the cardinality of A ...Definable Versions of the... · The Case for ¬CH · The Multiverse
[3]
https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Gentle_Introduction_to_the_Art_of_Mathematics_(Fields)/08%3A_Cardinality/8.05%3A_The_Continuum_Hypothesis_and_The_Generalized_Continuum_Hypothesis
[4]
Ueber eine elementare Frage der Mannigfaltigketislehre. - EuDML
Cantor, Georg. "Ueber eine elementare Frage der Mannigfaltigketislehre.." Jahresbericht der Deutschen Mathematiker-Vereinigung 1 (1890/91): 72-78. <http://eudml ...
[5]
[PDF] Naive set theory. - Whitman People
Naive set theory. Halmos, Paul R. (Paul Richard), 1916-2006. Princeton, N.J., Van Nostrand, [1960].
[6]
Ueber eine Eigenschaft des Inbegriffs aller reellen algebraischen ...
Cantor, G.. "Ueber eine Eigenschaft des Inbegriffs aller reellen algebraischen Zahlen.." Journal für die reine und angewandte Mathematik 77 (1874): 258-262.
[7]
Set Theory - Stanford Encyclopedia of Philosophy
Oct 8, 2014 · Set theory is the mathematical theory of well-determined collections, called sets, of objects that are called members, or elements, of the set.
[8]
Bijection from R to RN - Math Stack Exchange
Nov 24, 2012 · The nicest trick in the book is to find a bijection between R and NN, in this case we are practically done. Why? (NN)N∼NN×N∼NN.
[9]
Do the real numbers and the complex numbers have the same ...
Nov 26, 2012 · Yes, you are right. However, they all all (complex) rational hence of no interest for the sets of continuum cardinality.Cantor's infinities, and the cardinality of reals vs. complexGroups of cardinality greater than the continuumMore results from math.stackexchange.com
[10]
cardinality of all real sequences - Math Stack Exchange
Jun 7, 2013 · The one on the left you know from your own research; i.e. the cardinality of the sequences of the real numbers is the same as that of the reals.basis of vector space of real sequences over R - Math Stack ExchangeWhat is the cardinality of the set of all sequences in R converging to ...More results from math.stackexchange.com
[11]
Proof of the Cardinality of the power set of Q - Math Stack Exchange
Oct 17, 2020 · Proof of the Cardinality of the power set of Q, i realize the continuum hypotheses if taken says there no cardinality between |N| and |R i also ...What is known about the power set of the real number line?cardinal numbers and the power-set - Math Stack ExchangeMore results from math.stackexchange.com
[12]
A×B|=max(|A|,|B|) for infinite sets - Math Stack Exchange
Oct 18, 2014 · To be explicit, I'm asking for a proof of the statement in the question title, where A,B are sets with infinite cardinalities. set-theory ...
[13]
[PDF] Set Theory
Page 1. Thomas Jech. Set Theory. The Third Millennium Edition, revised and ... Ordinal Numbers. Induction and. Recursion. Ordinal Arithmetic. Well-Founded ...
[14]
Hartogs number - PlanetMath
Mar 22, 2013 · A set A A is said to be embeddable in another set B B if there is a one-to-one function f:A→B f : A → B . For example, every subset of a set ...
[15]
[PDF] Cantor's Continuum Hypothesis: consequences in mathematics and ...
Georg Cantor first conjectured this fact in 1878, and spent the rest of his life in trying to prove it, an endeavour undertaken by many other valiant ...
[16]
[PDF] Mathematical Problems
Examples of mathematical problems include Fermat's problem, the problem of three bodies, the duplication of the cube, and the squaring of the circle.
[17]
The Consistency Of The Axiom Of Choice and Of The Generalized ...
Oct 21, 2020 · ... Continuum Hypothesis With The Axioms Of Set Theory. by: Godel Kurt. Publication date: 1940 ... PDF WITH TEXT download · download 1 file · SINGLE ...
[18]
[PDF] The Continuum Hypothesis, Part I - Cornell Mathematics
[Gödel 1940] K. GÖDEL, The Consistency of the Axiom of. Choice and of the Generalized Continuum Hypothe- sis with the Axioms of Set Theory, Ann. of Math ...
[19]
Higher Souslin trees and the generalized continuum hypothesis
Mar 12, 2014 · The existence of an ω2-Souslin tree will be proved (Theorem 2.2 or §3) from the Generalized Continuum Hypothesis (GCH) plus Jensen's ...Missing: Suslin | Show results with:Suslin
[20]
[PDF] MATH 2513-001 Discrete Mathematics Fall 2005
6. Examples. The following sets of real numbers all have the same cardinality; R, (0,∞), (−π/2, π/2), (0,π), (0,1), (a, b) for any a<b, [a, b), (a, b], [a, b]. ...
[21]
[PDF] Cardinality of the Cantor Set
Jun 13, 2015 · Cardinality of the Cantor Set. Claim: The Cantor is uncountable, and in fact card (^) = C. To see. • this, let's think again about how eve ...
[22]
Do sets with positive Lebesgue measure have same cardinality as R?
Dec 15, 2009 · A proof is given that every measurable subset with cardinality less than that of R has Lebesgue measure zero.Cardinalities larger than the continuum in areas besides set theoryHow to prove that the Lebesgue $\sigma$-Algebra is not countably ...More results from mathoverflow.net
[23]
[PDF] A Brief Study of Real-Valued Continuous Functions on Various Spaces
The cardinality of a space X is the most basic cardinal function defined as IX I = number of points in X + w. Thus, in the previous example, the cardinality of ...
[24]
[PDF] Numbers and Vector spaces - Purdue Math
Sep 30, 2011 · Real numbers form a vector space over rational numbers. Think, what ... It is also a real vector space of dimension 2n (find a basis!).
[25]
Cantor's theorem | Set theory, cardinality, countability | Britannica
Oct 30, 2025 · The 1891 proof of Cantor's theorem for infinite sets rested on a version of his so-called diagonalization argument, which he had earlier used to ...