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Cantor's theorem

Cantor's theorem is a foundational result in set theory, established by Georg Cantor in 1891, which asserts that for any set S, the cardinality of S is strictly less than the cardinality of its power set \mathcal{P}(S), the collection of all subsets of S.^[1] This theorem demonstrates the existence of distinct infinite cardinalities and implies that no set can be placed in one-to-one correspondence with its own power set, highlighting a fundamental limitation in comparing sizes of infinite collections.^[2] The proof of Cantor's theorem relies on a diagonal argument, a technique pioneered by Cantor that assumes a bijection f: S \to \mathcal{P}(S) exists and constructs a subset T = \{ x \in S \mid x \notin f(x) \}, which cannot be in the image of f, leading to a contradiction.^[1] This result not only establishes an infinite hierarchy of cardinal numbers—where the power set operation generates ever-larger infinities—but also precludes the existence of a largest cardinal or a universal set containing all sets, influencing the development of axiomatic set theory to avoid paradoxes like Russell's paradox.^[2] Cantor's work, building on his earlier investigations into the uncountability of the real numbers, revolutionized mathematics by revealing the counterintuitive structure of infinity and remains central to modern understandings of cardinality and transfinite numbers.^[1]

Background Concepts

Power Set

In set theory, the power set of a set A, denoted \wp A or \mathcal{P}(A), is formally defined as the set of all subsets of A, that is, \wp A = \{ B \mid B \subseteq A \}.^[3] This includes the empty set \emptyset and A itself as subsets.^[4] To illustrate the construction, consider the finite set A = \{1, 2\}. Its power set is \wp A = \{ \emptyset, \{1\}, \{2\}, \{1, 2\} \}, which contains four elements, or $2^{|A|} subsets in general for a finite set of size n. Similarly, for A = \emptyset, \wp A = \{ \emptyset \}.^[4] An alternative notation for the cardinality of the power set is | \wp A | = 2^{|A|}, reflecting the exponential growth in the number of subsets as the size of A increases for finite sets.^[3] This property arises because each element of A can either be included or excluded from a subset, yielding $2^n possibilities. Power sets provide a complete enumeration of all possible combinations of elements from A, serving as a foundational tool for measuring set sizes through cardinality comparisons.^[4]

Cardinality

In set theory, the cardinality of a set provides a measure of its size, independent of the specific elements it contains. Two sets A and B are said to have the same cardinality, denoted |A| = |B|, if there exists a bijection f: A \to B, that is, a function that is both injective (one-to-one) and surjective (onto), establishing a perfect pairing between their elements.^[5]^[6] This definition extends the intuitive notion of counting to arbitrary sets, allowing for comparisons even when direct enumeration is impossible. For finite sets, cardinality corresponds directly to the natural numbers: a set with n elements has cardinality n, where n = 0, 1, 2, \dots. Infinite sets, however, require transfinite cardinals to describe their sizes. The smallest infinite cardinal is \aleph_0 (aleph-null), which is the cardinality of the set of natural numbers \mathbb{N}; any set in bijection with \mathbb{N} is called countably infinite.^[7]^[5] Larger infinite cardinals exist, ordered by size, but \aleph_0 serves as the baseline for countable infinity. Cardinalities can be compared using the relation |A| \leq |B|, which holds if there exists an injection from A to B (a one-to-one function, not necessarily onto). A strict inequality |A| < |B| follows if such an injection exists but no bijection does, ensuring A is properly smaller than B.^[6]^[7] Examples illustrate these concepts clearly. The set \{1, 2, 3\} has cardinality 3, as it bijects with \{0, 1, 2\}. For infinite sets, \mathbb{N} has cardinality \aleph_0, and so does the set of even natural numbers E = \{2, 4, 6, \dots\}, via the bijection f(n) = 2n, demonstrating that an infinite set can have the same cardinality as one of its proper subsets.^[5]^[6] This property of infinite cardinals highlights their counterintuitive nature compared to finite ones.

Statement and Proof

Statement

Cantor's theorem states that for any set A, there does not exist a bijection between A and its power set \mathcal{P}(A), the collection of all subsets of A. Equivalently, the cardinality of A, denoted |A|, is strictly less than the cardinality of \mathcal{P}(A), written |A| < |\mathcal{P}(A)|. This result, first proved by Georg Cantor in 1891, establishes a fundamental inequality in set theory regarding the sizes of sets and their collections of subsets.^[8]^[1] In cardinal arithmetic, the theorem is symbolized as \kappa < 2^\kappa for every cardinal number \kappa, where $2^\kappa denotes the cardinality of the power set of any set with cardinality \kappa. The intuitive basis lies in the observation that no function from A to \mathcal{P}(A) can cover all possible subsets; inevitably, at least one subset remains outside the image of any such mapping, ensuring the power set is strictly larger.^[8] The theorem applies universally to all sets, finite or infinite, requiring no additional axioms or properties beyond the basic notion of a set in classical set theory. It relies solely on the concepts of bijections for comparing cardinalities and the definition of the power set as the set of all subsets.^[1]

Proof

To prove Cantor's theorem, proceed by contradiction using the diagonal argument. Suppose there exists a surjective function f: A \to \mathcal{P}(A), where \mathcal{P}(A) denotes the power set of A. For each a \in A, f(a) is a subset of A, so the membership relation a \in f(a) or a \notin f(a) is well-defined for every a. Define the diagonal set D = \{ a \in A \mid a \notin f(a) \}. Clearly, D \subseteq A, so D \in \mathcal{P}(A). Since f is surjective, there must exist some b \in A such that f(b) = D. Consider whether b \in D:

If b \in D, then by the definition of D, b \notin f(b). But f(b) = D, so b \notin D, a contradiction.
If b \notin D, then by the definition of D, b \in f(b). But f(b) = D, so b \in D, again a contradiction.

In either case, a contradiction arises. Therefore, no such b exists, and f cannot be surjective. This establishes that no surjection exists from A to \mathcal{P}(A).^[9] The argument above relies on direct construction of subsets via membership conditions and does not invoke the axiom of choice; it can equivalently be framed using characteristic functions \chi_{f(a)}: A \to \{0,1\}, where the "diagonal" subset corresponds to flipping the value at each a, yielding a function not in the image.^[10] To conclude the strict inequality of cardinalities, observe that the map a \mapsto \{a\} defines an injection from A to \mathcal{P}(A), so |A| \leq |\mathcal{P}(A)|. Since no surjection exists from A to \mathcal{P}(A), the Schröder–Bernstein theorem implies there is no bijection between A and \mathcal{P}(A), hence |A| < |\mathcal{P}(A)|.

Implications and Applications

Finite Sets

Cantor's theorem applies straightforwardly to finite sets, demonstrating that the power set of any finite set has strictly greater cardinality than the set itself. If A is a finite set with |A| = n, where n is a non-negative integer, then the cardinality of the power set \mathcal{P}(A) is $2^n, and $2^n > n holds for all such n.^[2]^[11] A key edge case is the empty set \emptyset, which has cardinality 0. Its power set is \mathcal{P}(\emptyset) = \{ \emptyset \}, with cardinality 1, satisfying $1 > 0.^[12] For a non-empty example, consider A = \{1, 2, 3\} with n = 3. The power set \mathcal{P}(A) consists of 8 elements: \emptyset, \{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\}, and \{1,2,3\}, so |\mathcal{P}(A)| = 8 > 3.^[2]^[13] The inequality $2^n > n follows from the binomial theorem, which expands (1 + 1)^n = 2^n = \sum_{k=0}^n \binom{n}{k}. Each of the n+1 binomial coefficients \binom{n}{k} is a positive integer (at least 1), so the sum is at least n+1 > n.^[14] This combinatorial interpretation underscores that the subsets outnumber the elements: there are more ways to choose subsets than there are elements to pair them with bijectively. To intuit the absence of a bijection, explicit enumeration for small finite sets reveals the mismatch in sizes, as no one-to-one correspondence can pair all n elements of A with the $2^n subsets of \mathcal{P}(A). The general diagonalization argument from Cantor's proof also confirms this for finite cases, though enumeration suffices here.^[2]

Countably Infinite Sets

A countably infinite set A has cardinality |A| = \aleph_0. Cantor's theorem applied to such a set yields |P(A)| = 2^{\aleph_0} > \aleph_0.^[15] This larger cardinality $2^{\aleph_0} is denoted by \mathfrak{c}, the cardinality of the continuum.^[16] The power set P(\mathbb{N}) is in bijective correspondence with the set of real numbers \mathbb{R}. One construction establishes this by mapping each subset of \mathbb{N} to a real number in the interval [0,1] via binary expansions: the characteristic function of the subset, which assigns 1 if an element is included and 0 otherwise, serves as the binary digits after the decimal point.^[10] A variant of the diagonal argument from Cantor's proof further confirms the uncountability of \mathbb{R}, showing no enumeration of reals can be complete.^[17] These findings initiate a hierarchy of infinite cardinalities, with \aleph_0 < \mathfrak{c}, underscoring that the rational numbers, being countable, cannot be placed in bijection with the reals.^[18] For example, subsets of \mathbb{N} correspond to distinct reals in [0,1] through their characteristic sequences interpreted as binary representations, illustrating the injection from P(\mathbb{N}) into \mathbb{R}.^[19] Cantor's diagonal argument, central to the proof of his theorem, exhibits a self-referential structure that parallels the liar paradox, where a statement asserts its own falsity, leading to an inescapable contradiction. In the argument, assuming a bijection between a set and its power set allows construction of a diagonal subset that differs from every element in the enumeration, rendering the bijection impossible; this self-reference mirrors the liar's circularity, as the diagonal set is defined in opposition to the assumed listing, including or excluding itself in a way that defies resolution.^[20] This diagonal method contributes to foundational issues in set theory, particularly by illuminating inconsistencies in naive comprehension principles, as seen in Russell's paradox. Russell derived his paradox by applying Cantor's power set construction to the hypothetical universal set of all sets, defining the set of all sets not containing themselves, which leads to a contradiction regarding its own membership; Cantor's theorem exacerbates this by implying that the power set of any set, including such a universal collection, would exceed it in cardinality, generating paradoxes without restrictive axioms like separation or foundation to prevent ill-formed sets.^[20]^[21] Historically, Cantor's theorem intersects with his earlier distinction between the potential infinite—an unending process without completion, akin to Aristotelian views—and the absolute infinite, a transcendent, completed totality reserved for divine ontology. The theorem's demonstration of actual infinities of varying sizes rejected naive set theory's unrestricted formation of sets, as applying it to the "set of all sets" yields Cantor's paradox: the power set would both equal and exceed the universal set in cardinality, underscoring the need for axiomatic constraints.^[22]^[23] Philosophically, Cantor's theorem challenges intuitive notions of infinity by establishing a hierarchy of transfinite cardinals, fueling debates between actual infinity as a coherent, static totality and potential infinity as a mere potentiality. This shift from finite intuition to completed infinities provoked resistance from finitists and intuitionists, who viewed the theorem's implications—such as uncountable sets larger than the countable infinite—as paradoxical assaults on human comprehension, ultimately reshaping metaphysics by legitimizing infinities as mathematical objects.^[24]^[25]

Generalizations

Ordinal Cardinals

Cantor's theorem extends naturally to ordinal numbers, which are well-ordered sets in set theory. For any ordinal \alpha, the power set P(\alpha) cannot be placed in bijection with \alpha itself, as the diagonal argument applies directly to the set-theoretic structure of ordinals. This implies that the cardinality of P(\alpha) strictly exceeds the cardinality of \alpha, denoted |\alpha| = \kappa, yielding $2^\kappa > \kappa where $2^\kappa represents the cardinality of the power set. In the context of infinite cardinals, if \kappa = \aleph_\alpha for some ordinal \alpha, then \aleph_\alpha < 2^{\aleph_\alpha}, establishing a strict hierarchy of cardinalities without a maximum.^[26]^[27] In ordinal arithmetic, the notation $2^\alpha refers to ordinal exponentiation, defined recursively as $2^0 = 1, $2^{\beta+1} = 2^\beta \cdot 2, and for limit ordinals \gamma, $2^\gamma = \sup\{2^\beta \mid \beta < \gamma\}. While this ordinal operation does not directly measure the power set's cardinality, Cantor's theorem ensures that the cardinal exponentiation $2^{|\alpha|} surpasses |\alpha|. This distinction highlights how the theorem ensures no bijection between any ordinal \alpha and its power set P(\alpha), even though ordinal exponentiation may satisfy $2^\alpha = \alpha for some infinite ordinals like \omega (where $2^\omega = \omega); the power set cardinality remains strictly larger, reinforcing the absence of a largest ordinal: supposing a maximal ordinal \alpha would contradict the theorem, as P(\alpha) would admit a well-ordering of strictly larger order type. The aleph fixed-point issues arise in the enumeration of cardinals, where fixed points \aleph_\beta = \beta exist for limit ordinals \beta, but Cantor's theorem precludes fixed points for the power set function, ensuring the cardinal hierarchy remains unbounded.^[27]^[26] The theorem's application resolves foundational paradoxes involving ordinals, notably the Burali-Forti paradox, which posits a set \Omega of all ordinals. If \Omega were a set, it would be an ordinal greater than every ordinal, including itself, leading to a contradiction in well-ordering. Cantor's theorem ties into this by implying that no such \Omega can exist as a set, since its power set P(\Omega) would have even larger cardinality, but more critically, the assumption violates the strict increase in cardinality and order type, confirming that the class of all ordinals is a proper class rather than a set. This prevents self-inclusion issues where an ordinal might biject with a proper class of its subsets.^[27]^[26] For the first infinite ordinal \omega, the power set P(\omega) has cardinality $2^{\aleph_0}, which exceeds \aleph_0 and corresponds to the continuum. In ordinal hierarchies, such as those used in proof theory or collapsing functions, P(\omega) enables representations of ordinals far beyond \varepsilon_0 (the least fixed point of \alpha \mapsto \omega^\alpha), reaching into uncountable realms up to at least \omega_1, the first uncountable ordinal, depending on the continuum hypothesis. This illustrates how the theorem generates increasingly complex ordinal structures through iterated power sets.^[26]

Other Extensions

Cantor's theorem extends to advanced set-theoretic constructions such as forcing, where the theorem remains valid in generic extensions of the universe. In Cohen forcing, which adds new subsets to existing sets via generic filters, the forcing preserves all theorems of ZFC, including Cantor's theorem, ensuring that the cardinality of the power set of any set strictly exceeds that of the original set in the extension. Specifically, Cohen forcing adds generic reals or subsets without collapsing cardinals, so the power set cardinalities maintain their strict inequality relative to the underlying sets, preventing any bijection between a set and its power set even after adjoining generics. In the context of large cardinals, Cantor's theorem holds universally, including at inaccessible cardinals. An inaccessible cardinal κ is defined as an uncountable regular strong limit cardinal, satisfying 2^λ < κ for all λ < κ, which aligns with iterative applications of Cantor's theorem below κ. However, the theorem itself applies directly to κ, yielding 2^κ > κ, though the exact value of 2^κ may exceed κ^+ under certain assumptions; this does not contradict inaccessibility, as the definition concerns only smaller exponents. Category-theoretic analogues of Cantor's theorem appear in toposes and higher categories, where power objects play the role of power sets. In an elementary topos, for any object A, there exists a power object P(A) such that the evaluation map P(A) × A → A is universal for subobjects of A, and Lawvere's fixed-point theorem implies that no global elements can exhaust the subobjects, ensuring an analogue of the strict inequality in cardinality: there is no monomorphism from P(A) to A in well-pointed toposes satisfying certain conditions. This generalization captures Cantor's diagonal argument via the internal logic of the topos, where power objects strictly "exceed" their base objects in the order of subobject lattices.^[28] Modern variants of Cantor's theorem arise in descriptive set theory, particularly with effective power sets for Borel sets. The collection of Borel subsets of the real numbers forms an effective analogue of the power set, but its cardinality equals the continuum 2^ℵ₀, while the full power set has strictly larger cardinality by Cantor's theorem; this distinction highlights the "meagerness" of definable sets relative to all subsets. Under the generalized continuum hypothesis (GCH), which assumes 2^κ = κ^+ for every infinite cardinal κ, Cantor's theorem is strengthened by specifying the exact successor cardinal for each power set, maintaining consistency with ZFC while resolving the size of power sets in models where GCH holds.

History

Cantor's Discovery

Georg Cantor developed his theorem during his investigations into the nature of infinite sets in the late 19th century, motivated by problems in real analysis. His work originated from efforts to understand the representation of functions via Fourier series, where he sought to determine the uniqueness of trigonometric expansions and the sizes of point sets on the real line. Around 1874, while grappling with the convergence and uniqueness of such series, Cantor realized that the set of irrational numbers is uncountable, meaning it cannot be placed in one-to-one correspondence with the natural numbers, a breakthrough that highlighted the existence of different infinities.^[29]^[30] In December 1873, Cantor shared a preliminary proof sketch of the uncountability of the real numbers with his colleague Richard Dedekind in a private letter, employing a contradiction argument based on nested intervals to show that no enumeration could cover all points in a perfect set. This correspondence marked an early articulation of ideas central to his theorem, though it focused on specific cases within analysis rather than the general form. Cantor's insights arose directly from comparing the cardinalities of point sets derived from Fourier representations, underscoring the limitations of countable infinity in describing continuous domains.^[31] Cantor formalized the general statement of his theorem—that for any set, its power set has strictly greater cardinality—in his 1891 paper "Über eine elementare Frage der Mannigfaltigkeitslehre," published in Acta Mathematica. In this work, he presented a proof using a diagonalization technique to construct an element not in any assumed enumeration, effectively demonstrating the absence of a surjection from the set to its power set; this approach refined earlier injection-based arguments from his analytic proofs. The publication established the theorem as a cornerstone of set theory, independent of the geometric intuitions from his prior research. The revolutionary nature of Cantor's discoveries on infinities contributed to significant personal strain, exacerbating his preexisting mental health challenges. He experienced his first depressive episode in 1884, with recurrent institutionalizations beginning in 1899, partly attributed to the intellectual isolation and opposition from contemporaries like Leopold Kronecker, who rejected transfinite methods as paradoxical. These struggles persisted until his death in 1918, intertwining his groundbreaking work with profound psychological toll.^[32]

Subsequent Developments

Following Cantor's original work, his theorem played a pivotal role in the development of axiomatic set theory. In 1908, Ernst Zermelo formulated the first axiomatic system for set theory, incorporating the power set axiom, which guarantees the existence of the power set for any set; Cantor's theorem demonstrates the non-triviality of this axiom by proving that the power set always has a strictly greater cardinality than the original set, thus ensuring the existence of ever-larger infinite cardinals.^[33] This integration into Zermelo's framework, later refined by Abraham Fraenkel and others into Zermelo-Fraenkel set theory (ZF), provided a rigorous foundation that resolved paradoxes arising from naive set comprehension while preserving the theorem's implications for infinite sizes. The theorem's influence extended into mid-20th-century investigations of foundational questions in set theory. In 1938, Kurt Gödel established the relative consistency of the axiom of choice and the generalized continuum hypothesis with ZF axioms using his constructible universe model, a construction that relies on Cantor's theorem to iterate power sets and generate a hierarchy of cardinals.^[34] Building on this, Paul Cohen's 1963 forcing technique proved the independence of the continuum hypothesis from Zermelo-Fraenkel set theory with the axiom of choice (ZFC), showing that models can be constructed where the continuum's cardinality is neither the smallest uncountable cardinal nor strictly larger in a way forbidden by the hypothesis; these results underscore how Cantor's theorem delimits the possible sizes of power sets within consistent axiomatic systems.^[34] In the pedagogical landscape, Cantor's theorem transitioned from its roots in real analysis—where it initially proved the uncountability of the reals—to a cornerstone of pure set theory education in textbooks after the 1920s, reflecting the field's shift toward axiomatic rigor amid foundational crises.^[35] David Hilbert, in his program for securing the foundations of mathematics, explicitly acknowledged the value of Cantor's transfinite methods, including the theorem, while advocating finitary consistency proofs to justify their use, as articulated in his 1925 address "On the Infinite."^[36] More recently, from the 2000s onward, Cantor's theorem has undergone formal verification in interactive proof assistants, enhancing confidence in its logical soundness within computerized mathematics. For instance, mechanized proofs have been constructed in Coq, a dependently typed proof assistant based on the calculus of inductive constructions, confirming the theorem's validity without reliance on informal reasoning; such verifications, including those integrating it with broader set-theoretic libraries, continue to appear in formal mathematics projects up to the 2020s.

References

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https://math.libretexts.org/Bookshelves/Analysis/Real_Analysis_(Boman_and_Rogers)/09%3A_Back_to_the_Real_Numbers/9.03%3A_Cantor%E2%80%99s_Theorem_and_Its_Consequences
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4.10 Cantor's Theorem
Cantor's theorem implies that there are infinitely many infinite cardinal numbers, and that there is no largest cardinal number.
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None
Below is a merged summary of the power set definition and notation from Enderton’s *Elements of Set Theory* (1977), consolidating all information from the provided segments into a single, comprehensive response. To maximize detail and clarity, I will use a table in CSV format for key information (e.g., definitions, notation, cardinality, and page references), followed by additional narrative details and useful URLs. This approach ensures all data is retained and presented efficiently.
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Power Set -- from Wolfram MathWorld
Given a set S, the power set of S, sometimes also called the powerset, is the set of all subsets of S. The order of a power set of a set of order n is 2^n.Missing: formal definition textbook
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Cardinality of Finite and Infinite Sets. Recall that the cardinality of a set A is the number of distinct elements in A.
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Ueber eine elementare Frage der Mannigfaltigketislehre. Georg Cantor · Jahresbericht der Deutschen Mathematiker-Vereinigung (1890/91). Volume: 1, page 72-78 ...Missing: pdf | Show results with:pdf
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[PDF] Ueber eine elementare Frage der Mannigfaltigketislehre. (7 Seiten)
Titel: Ueber eine elementare Frage der Mannigfaltigketislehre. Autor: Cantor, Georg. PURL: https://resolver.sub.uni-goettingen.de/purl?37721857X_0001 ...Missing: 1891 | Show results with:1891
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(PDF) A Translation of G. Cantor's “Ueber eine elementare Frage ...
Aug 23, 2019 · Vereinigung, 1:75–78, 1891. Citations (1). References (1) ... Über eine elementare Frage der Mannigfaltigkeitslehre. Article. G. Cantor · View ...
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Infinity
It's easy to show this for finite sets, since the power set \wp(B) of set B of cardinality n is 2^n. And even for the empty set, of cardinality 0, the power set ...
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CSCI 2824: Lecture 15
Feb 27, 2014 · The empty set has cardinality zero whereas the power set of empty has cardinality 1 . Counting the Elements of the Power Set. If set A ...
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[PDF] Infinity and its cardinalities
... set and if a set has n elements it has 2n subsets. So the cardinality of the power set is: n(P(A))= 2n(A). The power set is itself a set. 64. Power set of a set ...
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1.3 Binomial coefficients
The entries in Pascal's Triangle are the coefficients of the polynomial produced by raising a binomial to an integer power.
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Mar 22, 2013 · A set of cardinality c 𝔠 is said to have continuum many elements. Cantor's diagonal argument shows that c 𝔠 is uncountable. Furthermore ...
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[PDF] Cantor's 1874 Proof of Non- Denumerability - English Translation
The original German text can be viewed online at: Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen. ... has a new proof of the theorem, ...Missing: uncountability | Show results with:uncountability
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[PDF] On Cantor's uncountability proofs - arXiv
... (1908). 276. [3] G. Cantor: Über eine elementare Frage der Mannigfaltigkeitslehre, Jahresbericht der Deutschen. Math. Vereinigung I (1890-91) 75 - 78. 2.
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Russell's initial reaction to Cantor's theorem was to regard it as guilty of error.² Cantor himself concluded from the theorem that there was no greatest ...
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Georg Cantor - Dartmouth Mathematics
In strictly mathematical terms, Cantor translated this as follows: "Thus every potential infinity (the wandering limit) leads to a Transfinitum (the sure path ...
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May 30, 2006 · It must be noted that our formalization of naive set theory is an anachronism. Cantor did not fully formalize his set theory, so it cannot be ...
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In this article, we recall the story of the discovery of set theory and point set topology by Georg Cantor (1845– ... exists. The convergence of a Fourier series ...Missing: timeline 1878
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In 1873 Cantor proved the rational numbers countable, i.e. they may be placed in one-one correspondence with the natural numbers. He also showed that the ...
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