434:'s public opposition and personal attacks included describing Cantor as a "scientific charlatan", a "renegade" and a "corrupter of youth". Kronecker objected to Cantor's proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included in a standard mathematics curriculum. Writing decades after Cantor's death, Wittgenstein lamented that mathematics is "ridden through and through with the pernicious idioms of set theory", which he dismissed as "utter nonsense" that is "laughable" and "wrong". Cantor's recurring bouts of depression from 1884 to the end of his life have been blamed on the hostile attitude of many of his contemporaries, though some have explained these episodes as probable manifestations of a
2005:
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implies that these big classes are not sets, which eliminates the paradoxes since they cannot be members of any class. Von
Neumann also used his axiom to prove the well-ordering theorem: Like Cantor, he assumed that the ordinals form a set. The resulting contradiction implies that the class of all ordinals is not a set. Then his axiom provides a one-to-one correspondence between this class and the class of all sets. This correspondence well-orders the class of all sets, which implies the well-ordering theorem. In 1930, Zermelo defined
668:, and he chaired its first meeting in Halle in 1891, where he first introduced his diagonal argument; his reputation was strong enough, despite Kronecker's opposition to his work, to ensure he was elected as the first president of this society. Setting aside the animosity Kronecker had displayed towards him, Cantor invited him to address the meeting, but Kronecker was unable to do so because his wife was dying from injuries sustained in a skiing accident at the time. Georg Cantor was also instrumental in the establishment of the first
2127:, Cantor's differences with Kronecker as a quarrel between two Jews, and Cantor's madness as Romantic despair over his failure to win acceptance for his mathematics. Grattan-Guinness (1971) found that none of these claims were true, but they may be found in many books of the intervening period, owing to the absence of any other narrative. There are other legends, independent of Bell â including one that labels Cantor's father a foundling, shipped to Saint Petersburg by unknown parents. A critique of Bell's book is contained in
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having Cantor as a colleague, perceiving him as a "corrupter of youth" for teaching his ideas to a younger generation of mathematicians. Worse yet, Kronecker, a well-established figure within the mathematical community and Cantor's former professor, disagreed fundamentally with the thrust of Cantor's work ever since he had intentionally delayed the publication of Cantor's first major publication in 1874. Kronecker, now seen as one of the founders of the
1715:. First, he defined two types of multiplicities: consistent multiplicities (sets) and inconsistent multiplicities (absolutely infinite multiplicities). Next he assumed that the ordinals form a set, proved that this leads to a contradiction, and concluded that the ordinals form an inconsistent multiplicity. He used this inconsistent multiplicity to prove the aleph theorem. In 1932, Zermelo criticized the construction in Cantor's proof.
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582:, disliked much of Cantor's set theory because it asserted the existence of sets satisfying certain properties, without giving specific examples of sets whose members did indeed satisfy those properties. Whenever Cantor applied for a post in Berlin, he was declined, and the process usually involved Kronecker, so Cantor came to believe that Kronecker's stance would make it impossible for him ever to leave Halle.
5759:
775:. No one had realized that set theory had any nontrivial content. Before Cantor, there were only finite sets (which are easy to understand) and "the infinite" (which was considered a topic for philosophical, rather than mathematical, discussion). By proving that there are (infinitely) many possible sizes for infinite sets, Cantor established that set theory was not trivial, and it needed to be studied.
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numbers, Cantor produces a transcendental number. Cantor points out that his constructions prove more â namely, they provide a new proof of
Liouville's theorem: Every interval contains infinitely many transcendental numbers. Cantor's next article contains a construction that proves the set of transcendental numbers has the same "power" (see below) as the set of real numbers.
5856:. There is an error in this analysis. It states Cantor's Theorem 1 correctly: Algebraic numbers can be counted. However, it states his Theorem 2 incorrectly: Real numbers cannot be counted. It then says: "Cantor notes that, taken together, Theorems 1 and 2 allow for the redemonstration of the existence of non-algebraic real numbers âŠ" This existence demonstration is
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entity, whereas intuitionists hold that mathematical entities cannot be reduced to logical propositions, originating instead in the intuitions of the mind. Secondly, the notion of infinity as an expression of reality is itself disallowed in intuitionism, since the human mind cannot intuitively construct an infinite set. Mathematicians such as
5425:
759:. The public celebration of his 70th birthday was canceled because of the war. In June 1917, he entered a sanatorium for the last time and continually wrote to his wife asking to be allowed to go home. Georg Cantor had a fatal heart attack on January 6, 1918, in the sanatorium where he had spent the last year of his life.
1269:, "Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" ("On a Property of the Collection of All Real Algebraic Numbers"). This paper was the first to provide a rigorous proof that there was more than one kind of infinity. Previously, all infinite collections had been implicitly assumed to be
3724:, p. 51. Proof of equivalence: If a set is well-ordered, then its cardinality is an aleph since the alephs are the cardinals of well-ordered sets. If a set's cardinality is an aleph, then it can be well-ordered since there is a one-to-one correspondence between it and the well-ordered set defining the aleph.
1869:
thinkers saw the existence of an actual infinity that consisted of something other than God as jeopardizing "God's exclusive claim to supreme infinity". Cantor strongly believed that this view was a misinterpretation of infinity, and was convinced that set theory could help correct this mistake: "...
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is too big to be a set if it can be put into one-to-one correspondence with the class of all sets. He defined a set as a class that is a member of some class and stated the axiom: A class is not a set if and only if there is a one-to-one correspondence between it and the class of all sets. This axiom
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while it was in proof, writing that it was "... about one hundred years too soon." Cantor complied, but then curtailed his relationship and correspondence with Mittag-Leffler, writing to a third party, "Had Mittag-Leffler had his way, I should have to wait until the year 1984, which to me seemed
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at a more prestigious university, in particular at Berlin, at that time the leading German university. However, his work encountered too much opposition for that to be possible. Kronecker, who headed mathematics at Berlin until his death in 1891, became increasingly uncomfortable with the prospect of
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Some mathematicians consider these results to have settled the issue, and, at most, allow that it is possible to examine the formal consequences of CH or of its negation, or of axioms that imply one of those. Others continue to look for "natural" or "plausible" axioms that, when added to ZFC, will
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More often the question has been discussed of whether Georg Cantor was of Jewish origin. About this it is reported in a notice of the Danish genealogical
Institute in Copenhagen from the year 1937 concerning his father: "It is hereby testified that Georg Woldemar Cantor, born 1809 or 1814, is not
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The transfinite is increasable in magnitude, while the absolute is unincreasable. For example, an ordinal α is transfinite because it can be increased to α + 1. On the other hand, the ordinals form an absolutely infinite sequence that cannot be increased in magnitude because there are no
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contains a real number not in the sequence. Since every sequence of real numbers can be used to construct a real not in the sequence, the real numbers cannot be written as a sequence â that is, the real numbers are not countable. By applying his construction to the sequence of real algebraic
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demonstrated less than a day later that König's proof had failed, Cantor remained shaken, and momentarily questioning God. Cantor suffered from chronic depression for the rest of his life, for which he was excused from teaching on several occasions and repeatedly confined to various sanatoria. The
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in 1896 (Paul
Tannery, Memoires Scientifique 13 Correspondence, Gauthier-Villars, Paris, 1934, p. 306), Cantor states that his paternal grandparents were members of the Sephardic Jewish community of Copenhagen. Specifically, Cantor states in describing his father: "Er ist aber in Kopenhagen
1837:
are required. Intuitionism also rejects the idea that actual infinity is an expression of any sort of reality, but arrive at the decision via a different route than constructivism. Firstly, Cantor's argument rests on logic to prove the existence of transfinite numbers as an actual mathematical
661:. However, he never again attained the high level of his remarkable papers of 1874â84, even after Kronecker's death on December 29, 1891. He eventually sought, and achieved, a reconciliation with Kronecker. Nevertheless, the philosophical disagreements and difficulties dividing them persisted.
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the transfinite species are just as much at the disposal of the intentions of the
Creator and His absolute boundless will as are the finite numbers.". Prominent neo-scholastic German philosopher Constantin Gutberlet was in favor of such theory, holding that it didn't oppose the nature of God.
630:... I don't know when I shall return to the continuation of my scientific work. At the moment I can do absolutely nothing with it, and limit myself to the most necessary duty of my lectures; how much happier I would be to be scientifically active, if only I had the necessary mental freshness.
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The religious dimension which Cantor attributed to his transfinite numbers should not be discounted as an aberration. Nor should it be forgotten or separated from his existence as a mathematician. The theological side of Cantor's set theory, though perhaps irrelevant for understanding its
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Cantor suffered his first known bout of depression in May 1884. Criticism of his work weighed on his mind: every one of the fifty-two letters he wrote to Mittag-Leffler in 1884 mentioned
Kronecker. A passage from one of these letters is revealing of the damage to Cantor's self-confidence:
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5860:. Theorem 2 stated correctly is: Given a sequence of real numbers, one can determine a real number that is not in the sequence. Taken together, Theorem 1 and this Theorem 2 produce a non-algebraic number. Cantor also used Theorem 2 to prove that the real numbers cannot be counted. See
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Cantor extended his work on the absolute infinite by using it in a proof. Around 1895, he began to regard his well-ordering principle as a theorem and attempted to prove it. In 1899, he sent
Dedekind a proof of the equivalent aleph theorem: the cardinality of every infinite set is an
1909:
In 1888, Cantor published his correspondence with several philosophers on the philosophical implications of his set theory. In an extensive attempt to persuade other
Christian thinkers and authorities to adopt his views, Cantor had corresponded with Christian philosophers such as
1938:
system are that all mathematical concepts must be devoid of internal contradiction, and that they follow from existing definitions, axioms, and theorems. This belief is summarized in his assertion that "the essence of mathematics is its freedom." These ideas parallel those of
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With a few rare exceptions the entities which are studied and analyzed in mathematics may be regarded as certain particular sets or classes of objects.... As a consequence, many fundamental questions about the nature of mathematics may be reduced to questions about set
1762:, but his proof was criticized for a variety of reasons. His response to the criticism included his axiom system and a new proof of the well-ordering theorem. His axioms support this new proof, and they eliminate the paradoxes by restricting the formation of sets.
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is an inconsistent multiplicity, which is a contradiction. Zermelo criticized Cantor's construction: "the intuition of time is applied here to a process that goes beyond all intuition, and a fictitious entity is posited of which it is assumed that it could make
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with God, and he considered his work on transfinite numbers to have been directly communicated to him by God, who had chosen Cantor to reveal them to the world. He was a devout
Lutheran whose explicit Christian beliefs shaped his philosophy of science.
1933:
Cantor's philosophy on the nature of numbers led him to affirm a belief in the freedom of mathematics to posit and prove concepts apart from the realm of physical phenomena, as expressions within an internal reality. The only restrictions on this
2049:("Even if we were descended from Jews ten times over, and even though I may be, in principle, completely in favour of equal rights for Hebrews, in social life I prefer Christians...") which could be read to imply that she was of Jewish ancestry.
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geboren, von israelitischen Eltern, die der dortigen portugisischen
Judengemeinde...." ("He was born in Copenhagen of Jewish (lit: 'Israelite') parents from the local Portuguese-Jewish community.") In addition, Cantor's maternal great uncle,
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In 1874, Cantor married Vally Guttmann. They had six children, the last (Rudolph) born in 1886. Cantor was able to support a family despite his modest academic pay, thanks to his inheritance from his father. During his honeymoon in the
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by recognizing that there are two types of multiplicities. In his set theory, when it is assumed that the ordinals form a set, the resulting contradiction implies only that the ordinals form an inconsistent multiplicity. In contrast,
1800:, where he stressed the connection between his view of the infinite and the philosophical one. To Cantor, his mathematical views were intrinsically linked to their philosophical and theological implications â he identified the
1452:, etc., in ways that would be largely acceptable now. The cardinal and ordinal arithmetic are reviewed. Cantor wanted the second paper to include a proof of the continuum hypothesis, but had to settle for expositing his theory of
1964:
his ideas were encountering: "... I realize that in this undertaking I place myself in a certain opposition to views widely held concerning the mathematical infinite and to opinions frequently defended on the nature of numbers."
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of the relationship between God and mathematics, although not in the same form as held by his critics, was long a concern of Cantor's. He directly addressed this intersection between these disciplines in the introduction to his
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that together formed an introduction to his set theory. At the same time, there was growing opposition to Cantor's ideas, led by Leopold Kronecker, who admitted mathematical concepts only if they could be constructed in a
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is a set which is either finite or denumerable; the denumerable sets are therefore the infinite countable sets. However, this terminology is not universally followed, and sometimes "denumerable" is used as a synonym for
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mission in Saint Petersburg, and his correspondence with his son shows both of them as devout Lutherans. Very little is known for sure about Georg Waldemar's origin or education. Cantor's mother, Maria Anna Böhm, was an
1816:
regarding the nature of actual infinity. Some held to the view that infinity was an abstraction which was not mathematically legitimate, and denied its existence. Mathematicians from three major schools of thought
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In 1882, the mathematical correspondence between Cantor and Dedekind came to an end, apparently as a result of Dedekind's declining the chair at Halle. Cantor also began another important correspondence, with
1769:
developed an axiom system that eliminates the paradoxes by using an approach similar to Cantor'sânamely, by identifying collections that are not sets and treating them differently. Von Neumann stated that a
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Although the presentation is axiomatic rather than naive, Suppes proves and discusses many of Cantor's results, which demonstrates Cantor's continued importance for the edifice of foundational mathematics.
2056:, Cantor was of Jewish descent, although both parents were baptized. In a 1971 article entitled "Towards a Biography of Georg Cantor", the British historian of mathematics Ivor Grattan-Guinness mentions (
8124:
3563:. Cantor actually applies his construction to the irrationals rather than the transcendentals, but he knew that it applies to any set formed by removing countably many numbers from the set of reals (
748:
repeatedly cited Cantor's work, but the encounter did not come about. The following year, St. Andrews awarded Cantor an honorary doctorate, but illness precluded his receiving the degree in person.
482:, Russian Empire, was brought up in that city until the age of eleven. The oldest of six children, he was regarded as an outstanding violinist. His grandfather Franz Böhm (1788â1846) (the violinist
2155:, to infect Italian mathematics ... Any acceptance of infinitesimals necessarily meant that his own theory of number was incomplete. Thus to accept the work of Thomae, du Bois-Reymond, Stolz and
423: â a proposition that Cantor vigorously rejected. Not all theologians were against Cantor's theory; prominent neo-scholastic philosopher Constantin Gutberlet was in favor of it and Cardinal
2525:, p. 170): "Aus dem Paradies, das Cantor uns geschaffen, soll uns niemand vertreiben können." (Literally: "Out of the Paradise that Cantor created for us, no one must be able to expel us.")
2109:(1927) â largely the correspondence with Mittag-Leffler â and Fraenkel (1930). Both were at second and third hand; neither had much on his personal life. The gap was largely filled by
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also both expressed their admiration. At that Congress, Cantor renewed his friendship and correspondence with Dedekind. From 1905, Cantor corresponded with his British admirer and translator
5664:. With acknowledgement of Dauben's pioneering historical work, this article further discusses Cantor's relation to the philosophy of Spinoza and Leibniz in depth, and his engagement in the
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in modern mathematics, in the sense that it interprets propositions about mathematical objects (for example, numbers and functions) from all the traditional areas of mathematics (such as
1892:, he said that such an explanation could only come about by drawing on the resources of the philosophy of Spinoza and Leibniz. In making these claims, Cantor may have been influenced by
709:
events of 1904 preceded a series of hospitalizations at intervals of two or three years. He did not abandon mathematics completely, however, lecturing on the paradoxes of set theory (
692:), and this tragedy drained Cantor of much of his passion for mathematics. Cantor was again hospitalized in 1903. One year later, he was outraged and agitated by a paper presented by
573:
Cantor was promoted to extraordinary professor in 1872 and made full professor in 1879. To attain the latter rank at the age of 34 was a notable accomplishment, but Cantor desired a
2087:, for I am born 3 March 1845 at Saint Peterborough, Capital of Russia, but I went with my father and mother and brothers and sister, eleven years old in the year 1856, into Germany.
4619:, I.; Hawkins, T.; Pedersen, K. From the calculus to set theory, 1630â1910. An introductory history. Edited by I. Grattan-Guinness. Gerald Duckworth & Co. Ltd., London, 1980.
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1324:, .... In other words, the real algebraic numbers are countable. Cantor starts his second construction with any sequence of real numbers. Using this sequence, he constructs
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225:
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Neither my father nor my mother were of German blood, the first being a Dane, borne in Kopenhagen, my mother of Austrian Hungar descension. You must know, Sir, that I am not a
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number of steps from the natural numbers, which he took as intuitively given. For Kronecker, Cantor's hierarchy of infinities was inadmissible, since accepting the concept of
1829:) opposed Cantor's theories in this matter. For constructivists such as Kronecker, this rejection of actual infinity stems from fundamental disagreement with the idea that
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419:) saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God â on one occasion equating the theory of transfinite numbers with
2008:
The title on the memorial plaque (in Russian): "In this building was born and lived from 1845 till 1854 the great mathematician and creator of set theory Georg Cantor",
1992:'s philosophy, in the realms of both the philosophy of mathematics and metaphysics. He shared B. Russell's motto "Kant or Cantor", and defined Kant "yonder sophistical
1727:
treated all collections as sets, which leads to paradoxes. In Russell's set theory, the ordinals form a set, so the resulting contradiction implies that the theory is
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again until 1899. Soon after that second hospitalization, Cantor's youngest son Rudolph died suddenly on December 16 (Cantor was delivering a lecture on his views on
771:. Prior to this work, the concept of a set was a rather elementary one that had been used implicitly since the beginning of mathematics, dating back to the ideas of
3658:. The paper had been submitted in July 1877. Dedekind supported it, but delayed its publication due to Kronecker's opposition. Weierstrass actively supported it.
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In 1878, Cantor submitted another paper to Crelle's Journal, in which he defined precisely the concept of a 1-to-1 correspondence and introduced the notion of "
2072:, a Hungarian violinist, has been described as Jewish, which may imply that Cantor's mother was at least partly descended from the Hungarian Jewish community.
1214:. Dedekind, whom Cantor befriended in 1872, cited this paper later that year, in the paper where he first set out his celebrated definition of real numbers by
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or CH: there exists no set whose power is greater than that of the naturals and less than that of the reals (or equivalently, the cardinality of the reals is
993:
praised Cantor's set theory and, following public lectures delivered by Cantor at the first International Congress of Mathematicians, held in ZĂŒrich in 1897,
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2045:
Mögen wir zehnmal von Juden abstammen und ich im Princip noch so sehr fĂŒr Gleichberechtigung der HebrĂ€er sein, im socialen Leben sind mir Christen lieber ...
795:) in a single theory, and provides a standard set of axioms to prove or disprove them. The basic concepts of set theory are now used throughout mathematics.
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was to deny the perfection of Cantor's own creation. Understandably, Cantor launched a thorough campaign to discredit Veronese's work in every way possible.
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were false. Since the paper had been read in front of his daughters and colleagues, Cantor perceived himself as having been publicly humiliated. Although
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1667:
The difficulty Cantor had in proving the continuum hypothesis has been underscored by later developments in the field of mathematics: a 1940 result by
1218:. While extending the notion of number by means of his revolutionary concept of infinite cardinality, Cantor was paradoxically opposed to theories of
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1968:
Hence he devotes much space to justifying his earlier work, asserting that mathematical concepts may be freely introduced as long as they are free of
1273:(that is, of "the same size" or having the same number of elements). Cantor proved that the collection of real numbers and the collection of positive
2494:, p. 1. Text includes a 1964 quote from psychiatrist Karl Pollitt, one of Cantor's examining physicians at Halle Nervenklinik, referring to Cantor's
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27, pp. 345â391, 1971) that he was unable to find evidence of Jewish ancestry. (He also states that Cantor's wife, Vally Guttmann, was Jewish).
3887:, pp. 288, 290â291. Cantor had pointed out that inconsistent multiplicities face the same restriction: they cannot be members of any multiplicity. (
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In 1891, he published a paper containing his elegant "diagonal argument" for the existence of an uncountable set. He applied the same idea to prove
597:, in that order, but each declined the chair after being offered it. Friedrich Wangerin was eventually appointed, but he was never close to Cantor.
1611:. While he made free use of countability as a concept, he did not write the word "countable" until 1883. Cantor also discussed his thinking about
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and ordinal numbers. In 1885, Cantor extended his theory of order types so that the ordinal numbers simply became a special case of order types.
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blamed Kronecker's persistent criticism and Cantor's inability to confirm his continuum hypothesis" for Cantor's recurring bouts of depression.
1926:. Although later this Cardinal accepted the theory as valid, due to some clarifications from Cantor's. Cantor even sent one letter directly to
1577:) or "equivalence" of sets: two sets are equivalent (have the same power) if there exists a 1-to-1 correspondence between them. Cantor defined
400:
3244:
1053:. Cantor solved this problem in 1869. It was while working on this problem that he discovered transfinite ordinals, which occurred as indices
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mathematical content, is nevertheless essential for the full understanding of his theory and why it developed in its early stages as it did.
1865:
Some Christian theologians saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God. In particular,
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551:
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was regarded as counter-intuitive â even shocking. This caused it to encounter resistance from mathematical contemporaries such as
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permit either a proof or refutation of CH, or even for direct evidence for or against CH itself; among the most prominent of these is
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3115:"The rise of non-Archimedean mathematics and the roots of a misconception. I. The emergence of non-Archimedean systems of magnitudes"
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4490:, pp. 93â94, from Louis' trip to Chicago in 1863. It is ambiguous in German, as in English, whether the recipient is included.
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609:. But in 1885, Mittag-Leffler was concerned about the philosophical nature and new terminology in a paper Cantor had submitted to
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1562:" ("I see it, but I don't believe it!") The result that he found so astonishing has implications for geometry and the notion of
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1187:, and so on. He had examples that went on forever, and so here was a naturally occurring infinite sequence of infinite numbers
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534:, then and later a center for mathematical research. Cantor was a good student, and he received his doctoral degree in 1867.
498:, seeking milder winters than those of Saint Petersburg. In 1860, Cantor graduated with distinction from the Realschule in
486:'s brother) was a well-known musician and soloist in a Russian imperial orchestra. Cantor's father had been a member of the
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would open the door to paradoxes which would challenge the validity of mathematics as a whole. Cantor also introduced the
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1961:
404:
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Dauben, J.: The development of the Cantorian set theory, pp.~181â219. See pp.216â217. In Bos, H.; Bunn, R.; Dauben, J.;
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are then introduced as the order types of well-ordered sets. Cantor then defines the addition and multiplication of the
507:
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4448:, Grattan-Guinness's only evidence on the grandfather's date of death is that he signed papers at his son's engagement.
3336:. Gray (pp. 821â822) describes a computer program that uses Cantor's constructions to generate a transcendental number.
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In 1911, Cantor was one of the distinguished foreign scholars invited to the 500th anniversary of the founding of the
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1896:, whose lecture courses he attended at Berlin, and in turn Cantor produced a Latin commentary on Book 1 of Spinoza's
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in Zurich. After receiving a substantial inheritance upon his father's death in June 1863, Cantor transferred to the
311:
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912:, which extended the arithmetic of the natural numbers. His notation for the cardinal numbers was the Hebrew letter
634:
This crisis led him to apply to lecture on philosophy rather than on mathematics. He also began an intense study of
550:
at the University of Berlin in 1867. After teaching briefly in a Berlin girls' school, he took up a position at the
332:
19 February] 1845 â 6 January 1918) was a mathematician who played a pivotal role in the creation of
1630:
This paper displeased Kronecker and Cantor wanted to withdraw it; however, Dedekind persuaded him not to do so and
1754:. He had two motivations for developing the axiom system: eliminating the paradoxes and securing his proof of the
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1533:, though he did not use that phrase. He then began looking for a 1-to-1 correspondence between the points of the
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were closed, they contained their limit points, and the intersection of the infinite decreasing sequence of sets
989:
in Paris. Cantor's work also attracted favorable notice beyond Hilbert's celebrated encomium. The US philosopher
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4190:
Davenport, Anne A. (1997). "The Catholics, the Cathars, and the Concept of Infinity in the Thirteenth Century".
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in particular, were noted. In August 1862, he then graduated from the "Höhere Gewerbeschule Darmstadt", now the
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5973:
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3066:(2012). "A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography".
2119:(1937), which one of Cantor's modern biographers describes as "perhaps the most widely read modern book on the
2024:. There is very little direct information on them. Cantor's father, Georg Waldemar Cantor, was educated in the
1549:, there exists a 1-to-1 correspondence between the points on the unit line segment and all of the points in an
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present in the registers of the Jewish community, and that he completely without doubt was not a Jew ..."
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1448:'s editorship; these were his last significant papers on set theory. The first paper begins by defining set,
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Cantor established these results using two constructions. His first construction shows how to write the real
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numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact he was well aware of.
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It was important to Cantor that his philosophy provided an "organic explanation" of nature, and in his 1883
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as its set of zeros, Cantor had discovered a procedure that produced another trigonometric series that had
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was an important shared concern within the realms of mathematics, philosophy and religion. Preserving the
1731:. From 1901 to 1903, Russell discovered three paradoxes implying that his set theory is inconsistent: the
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For this, and more information on the mathematical importance of Cantor's work on set theory, see e.g.,
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8129:
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6171:
6074:
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1809:
has traced the effect Cantor's Christian convictions had on the development of transfinite set theory.
1672:
1202:
Between 1870 and 1872, Cantor published more papers on trigonometric series, and also a paper defining
780:
653:
Cantor recovered soon thereafter, and subsequently made further important contributions, including his
337:
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138:
8134:
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8046:
7968:
7214:
7099:
7087:
7082:
6627:
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6212:
5963:
4595:, p. 1 and notes. (Bell's Jewish stereotypes appear to have been removed from some postwar editions.)
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1062:
811:
735:
341:
17:
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4165:
3672:. One of Gödel's last papers argues that the CH is false, and the continuum has cardinality Aleph-2.
2091:
There were documented statements, during the 1930s, that called this Jewish ancestry into question:
2004:
1833:
such as Cantor's diagonal argument are sufficient proof that something exists, holding instead that
1009:
and on Cantor's religious ideas. This was later published, as were several of his expository works.
7015:
6483:
6191:
6186:
6176:
5886:
5816:
5794:
4697:. Proceedings of the 9th ACMS Conference (Westmont College, Santa Barbara, Calif.). pp. 1â22.
3761:
for each ordinal. If this construction runs out of elements, then the function well-orders the set
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324:
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6232:
6227:
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6069:
4991:
4669:[Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite
3024:
Cooke, Roger (1993). "Uniqueness of trigonometric series and descriptive set theory, 1870â1985".
2263:
558:
for his thesis, also on number theory, which he presented in 1869 upon his appointment at Halle.
487:
5784:
4500:
Men of Mathematics: The Lives and Achievements of the Great Mathematicians from Zeno to Poincaré
2041:
upon marriage. However, there is a letter from Cantor's brother Louis to their mother, stating:
1675:
together imply that the continuum hypothesis can be neither proved nor disproved using standard
602:
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7675:
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635:
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4984:"Ăber Grenzzahlen und Mengenbereiche: neue Untersuchungen ĂŒber die Grundlagen der Mengenlehre"
1881: â and was shocked when he realized that he was the only faculty member at Halle who did
1664:
it, in vain. His inability to prove the continuum hypothesis caused him considerable anxiety.
8074:
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2034:
1859:
1830:
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1704:
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1368:), published in 1883, was the most important of the six and was also published as a separate
1338:
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The beginning of set theory as a branch of mathematics is often marked by the publication of
1026:
982:
943:
915:
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744:
3628:
3622:
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1660:
aleph-one). Cantor believed the continuum hypothesis to be true and tried for many years to
814:(hereinafter denoted "1-to-1 correspondence") in set theory. He used this concept to define
8154:
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3205:
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1261:. The sequence at the bottom cannot occur anywhere in the infinite list of sequences above.
1227:
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973:
776:
718:
710:
589:
died. Halle accepted Cantor's suggestion that Heine's vacant chair be offered to Dedekind,
454:
5563:
Contains a detailed treatment of both Cantor's and Dedekind's contributions to set theory.
4909:
1634:
supported its publication. Nevertheless, Cantor never again submitted anything to Crelle.
8:
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7482:
7390:
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6650:
6533:
6518:
6451:
6327:
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6095:
5958:
5802:
5780:
5690:
5643:
Georg Cantor, Leben, Werk und Wirkung (Georg Cantor, Life, Work and Influence, in German)
5520:
Dauben, Joseph W. (June 1983). "Georg Cantor and the Origins of Transfinite Set Theory".
4328:"The Motives Behind Cantor's Set TheoryâPhysical, Biological and Philosophical Questions"
3970:
3778:
2135:
Cantor devoted some of his most vituperative correspondence, as well as a portion of the
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Until the 1970s, the chief academic publications on Cantor were two short monographs by
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2009:
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had stated this theorem a bit earlier, but his proof, as well as Cantor's, was flawed.
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2123:"; and as "one of the worst". Bell presents Cantor's relationship with his father as
2057:
1981:
1866:
1801:
1696:
1616:
1604:
1581:(or denumerable sets) as sets which can be put into a 1-to-1 correspondence with the
1203:
850:
590:
519:
431:
416:
380:
5480:. Almost everything that Cantor wrote. Includes excerpts of his correspondence with
4720:
From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics
4362:
4307:
Newstead, Anne (2009). "Cantor on Infinity in Nature, Number, and the Divine Mind".
4092:
2143:
Cholera bacillus of mathematics', which had spread from Germany through the work of
1973:
1854:'s attacks were finitist: he believed that Cantor's diagonal argument conflated the
1843:
1237:
of mathematics". Cantor also published an erroneous "proof" of the inconsistency of
384:
7983:
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7890:
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7844:
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7609:
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6538:
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6302:
6267:
5988:
5857:
5773:
5603:
5537:
5481:
5446:
5409:
5349:
5311:
5278:
5245:
5212:
5179:
5154:
5114:
5000:
4962:
4861:
4834:
4749:
4645:
4636:(1977). "Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite".
4616:
4342:
4289:
4199:
4161:
4157:
4064:
4007:
3304:
3213:
3191:
3129:
3087:
3033:
2920:
2568:
2268:
2177:
2110:
2076:
2053:
2030:
1902:
1839:
1766:
1724:
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1453:
1325:
1297:
1290:
1042:
1038:
998:
862:
739:
570:, whom he had met at Interlaken in Switzerland two years earlier while on holiday.
567:
523:
479:
435:
392:
285:
250:
237:
77:
5649:
Newstead, Anne (2009). "Cantor on Infinity in Nature, Number, and the Divine Mind"
5551:
Labyrinth of Thought: A History of Set Theory and Its Role in Mathematical Thought
3851:
Moore devotes a chapter to this criticism: "Zermelo and His Critics (1904â1908)",
1285:
that he gave in 1891. Cantor's article also contains a new method of constructing
7824:
7809:
7696:
7508:
7446:
7264:
7077:
6875:
6812:
6473:
6348:
6252:
5909:
5872:
5158:
5064:
4714:
is actually from the works of Seneca and has no implication of divine revelation.
3187:
3011:
2238:
2021:
1985:
1977:
1915:
1788:
1759:
1680:
1590:
1416:
1385:
1348:
1258:
1211:
1002:
940:) with a natural number subscript; for the ordinals he employed the Greek letter
905:
854:
827:
685:
446:
427:
accepted it as a valid theory (after Cantor made some important clarifications).
365:
184:
7932:
6570:
5833:
4710:
2004. Note, though, that Cantor's Latin quotation described in this article as
1873:
Cantor also believed that his theory of transfinite numbers ran counter to both
1862:, thus conflating the concept of rules for generating a set with an actual set.
693:
618:
too great a demand! ... But of course I never want to know anything again about
8034:
8022:
7906:
7880:
7855:
7850:
7644:
7441:
7422:
7326:
7311:
7268:
7204:
7146:
6890:
6822:
6793:
6749:
6732:
6715:
6668:
6612:
6597:
6565:
6503:
6292:
5948:
5689:
Chapter 16 illustrates how Cantorian thinking intrigues a leading contemporary
5504:
The Mystery of the Aleph: Mathematics, the Kabbala, and the Search for Infinity
5485:
4839:
3956:
3669:
2946:
2495:
1972:
and defined in terms of previously accepted concepts. He also cites Aristotle,
1940:
1747:
and Cantor even though neither of them believed that they had found paradoxes.
1582:
1381:
1270:
1207:
909:
806:; this showed, for the first time, that there exist infinite sets of different
803:
369:
357:
81:
6834:
4753:
4567:, vol. I, p. 229. In English in the original; italics also as in the original.
4346:
3145:
3133:
3091:
1407:. This established the richness of the hierarchy of infinite sets, and of the
483:
43:
8068:
7963:
7875:
7860:
7649:
7451:
7365:
7360:
6744:
6720:
6590:
6560:
6543:
6508:
6493:
6317:
6277:
6115:
5672:
5587:
5467:
5118:
4633:
4407:
2991:
2248:
2140:
2128:
2038:
1989:
1969:
1947:
1927:
1806:
1620:
1578:
1574:
1238:
1219:
1018:
994:
978:
857:
are everywhere dense, but countable. He also showed that all countable dense
823:
705:
639:
594:
547:
450:
442:
99:
7619:
6282:
4029:
4001:
2704:
2069:
1782:
1668:
1585:, and proved that the rational numbers are denumerable. He also proved that
1415:
that Cantor had defined. His argument is fundamental in the solution of the
7870:
7814:
7594:
7412:
7341:
7299:
7158:
7055:
6989:
6984:
6885:
6622:
6555:
6287:
6257:
6110:
5516:. A popular treatment of infinity, in which Cantor is frequently mentioned.
5098:"Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen"
5047:
Older sources on Cantor's life should be treated with caution. See section
4354:
3268:
3063:
2228:
2218:
2064:
1993:
1911:
1847:
1822:
1728:
1712:
1538:
1468:
1215:
1077:
of zeros of a trigonometric series. Given a trigonometric series f(x) with
1030:
1022:
853:, but has the same cardinality as the set of all real numbers, whereas the
819:
752:
586:
555:
543:
527:
503:
453:
defended it from its critics by declaring, "No one shall expel us from the
388:
345:
245:
5005:
4293:
1707:"every set can be well-ordered" and stated that it is a "law of thought".
1690:
1376:
were a systematic extension of the natural numbers. It begins by defining
680:
After Cantor's 1884 hospitalization there is no record that he was in any
650:); this ultimately resulted in two pamphlets, published in 1896 and 1897.
7819:
7624:
7259:
6950:
6870:
6580:
6575:
5953:
5943:
5837:
5697:
5567:
4084:
4011:
2911:
Johnson, Phillip E. (1972). "The Genesis and Development of Set Theory".
2682:
Math and mathematicians: the history of math discoveries around the world
2025:
1935:
1878:
1874:
1597:
1570:
1534:
1445:
1396:
1096:
858:
838:
807:
799:
756:
353:
270:
202:
5661:
3971:"Cantor's Concept of Infinity: Implications of Infinity for Contingence"
2717:
7778:
7604:
7375:
7031:
6803:
6788:
6783:
6764:
6498:
6018:
5607:
5499:
5450:
5413:
5353:
5315:
5282:
5249:
5216:
5183:
4966:
4657:
4143:"The Three Crises in Mathematics: Logicism, Intuitionism and Formalism"
3842:, pp. 158â160. Moore argues that the latter was his primary motivation.
3316:
3261:
3037:
2932:
2152:
2017:
1377:
1352:
1223:
1006:
868:
Cantor introduced fundamental constructions in set theory, such as the
842:
815:
768:
728:
727:
in 1903, and attending the International Congress of Mathematicians at
681:
511:
360:. Cantor's method of proof of this theorem implies the existence of an
349:
333:
151:
128:
4211:
4076:
4052:
1922:, who once replied by equating the theory of transfinite numbers with
1249:
441:
The harsh criticism has been matched by later accolades. In 1904, the
7901:
7407:
7370:
7321:
7219:
6759:
6705:
6617:
6468:
5993:
4277:
1923:
1855:
1792:
1648:
Cantor was the first to formulate what later came to be known as the
1612:
1563:
1530:
1512:
1369:
1278:
869:
772:
574:
499:
495:
491:
420:
5668:. Brief mention is made of Cantor's learning from F.A.Trendelenburg.
5472:"Gesammelte Abhandlungen mathematischen und philosophischen inhalts"
4649:
3308:
2924:
1336:
Between 1879 and 1884, Cantor published a series of six articles in
490:; when he became ill, the family moved to Germany in 1856, first to
7799:
7691:
6660:
5968:
5023:
From Frege to Godel: A Source Book in Mathematical Logic, 1879â1931
4804:
Zermelo's Axiom of Choice: Its Origins, Development & Influence
4203:
4068:
3753:
whose cardinality is not an aleph. A function from the ordinals to
1954:
1826:
1344:
1301:
1274:
1234:
1012:
834:
792:
361:
115:
8125:
Academic staff of the Martin Luther University of Halle-Wittenberg
5878:
5070:
Contributions to the Founding of the Theory of Transfinite Numbers
4552:
IsmerjĂŒk oket?: zsidĂł szĂĄrmazĂĄsĂș nevezetes magyarok arckĂ©pcsarnoka
3082:
1703:
larger ordinals to add to it. In 1883, Cantor also introduced the
1695:
In 1883, Cantor divided the infinite into the transfinite and the
7717:
6607:
6410:
5746:
4554:, IstvĂĄn RemĂ©nyi Gyenes Ex Libris, (Budapest 1997), pages 132â133
3265:
3061:
2124:
1951:
1719:
1231:
1121:, then he could construct a trigonometric series whose zeros are
784:
605:
in Sweden, and soon began to publish in Mittag-Leffler's journal
554:, where he spent his entire career. He was awarded the requisite
111:
3773:. Therefore, the function maps all the ordinals one-to-one into
2079:, Cantor described his ancestry and self-perception as follows:
1499:
supplied a correct proof in his 1898 PhD thesis; hence the name
1372:. It contained Cantor's reply to his critics and showed how the
470:
411:, believed the theory had been communicated to him by God. Some
7660:
7432:
7254:
6678:
5758:
3350:
2309:
The biographical material in this article is mostly drawn from
1449:
1277:
are not equinumerous. In other words, the real numbers are not
1045:, and Heine himself: the uniqueness of the representation of a
877:
751:
Cantor retired in 1913, and lived in poverty and suffered from
219:
3757:
is constructed by successively choosing different elements of
3194:
suggested that all infinite sets were equinumerous â see
1812:
Debate among mathematicians grew out of opposing views in the
7304:
7064:
7000:
4436:, George Allen and Unwin Ltd., 1971 (London), vol. 1, p. 217.
3345:
Cantor's construction starts with the set of transcendentals
1541:. In an 1877 letter to Richard Dedekind, Cantor proved a far
965:
937:
933:
798:
In one of his earliest papers, Cantor proved that the set of
4003:
Georg Cantor: His Mathematics and Philosophy of the Infinite
3489:
and two countable sets. A one-to-one correspondence between
1517:
303:
8165:
Members of the Göttingen Academy of Sciences and Humanities
6379:
4278:"Cantor on Infinity in Nature, Number, and the Divine Mind"
3257:
3161:
This follows closely the first part of Cantor's 1891 paper.
2421:
2419:
2417:
2347:
Georg Cantor His Mathematics and Philosophy of the Infinite
563:
449:, the highest honor it can confer for work in mathematics.
5435:"BeitrĂ€ge zur BegrĂŒndung der transfiniten Mengenlehre (2)"
5398:"BeitrĂ€ge zur BegrĂŒndung der transfiniten Mengenlehre (1)"
3196:
Moore, A. W. (April 1995). "A brief history of infinity".
1960:
Cantor's 1883 paper reveals that he was well aware of the
566:, Cantor spent much time in mathematical discussions with
430:
The objections to Cantor's work were occasionally fierce:
1783:
Philosophy, religion, literature and Cantor's mathematics
1684:
1434:
Passage of Georg Cantor's article with his set definition
294:
6003:
5847:"Cantor infinities", analysis of Cantor's 1874 article,
5371:"Ueber eine elementare Frage der Mannigfaltigkeitslehre"
5338:"Ueber unendliche, lineare Punktmannichfaltigkeiten (6)"
5300:"Ueber unendliche, lineare Punktmannichfaltigkeiten (5)"
5267:"Ueber unendliche, lineare Punktmannichfaltigkeiten (4)"
5234:"Ueber unendliche, lineare Punktmannichfaltigkeiten (3)"
5201:"Ueber unendliche, lineare Punktmannichfaltigkeiten (2)"
5168:"Ueber unendliche, lineare Punktmannichfaltigkeiten (1)"
2414:
810:. He was also the first to appreciate the importance of
700:. The paper attempted to prove that the basic tenets of
5625:
Husserl or Frege? Meaning, Objectivity, and Mathematics
2569:
ru: The musical encyclopedia (ĐŃĐ·ŃĐșĐ°Đ»ŃĐœĐ°Ń ŃĐœŃĐžĐșĐ»ĐŸĐżĐ”ĐŽĐžŃ)
1691:
Absolute infinite, well-ordering theorem, and paradoxes
1438:
In 1895 and 1897, Cantor published a two-part paper in
5714:
Deals with similar topics to Aczel, but in more depth.
4951:"Untersuchungen ĂŒber die Grundlagen der Mengenlehre I"
4850:"Burali-Forti's Paradox: A Reappraisal of Its Origins"
4691:
Georg Cantor and the Battle for Transfinite Set Theory
4053:"Georg Cantor: The Personal Matrix of His Mathematics"
2139:, to attacking what he described at one point as the '
1456:
and ordinal numbers. Cantor attempts to prove that if
7994:
2456:
2454:
2351:. princeton university press. pp. introduction.
1777:
models of set theory that satisfy von Neumann's axiom
946:
918:
896:
is an infinite set; this result soon became known as
767:
Cantor's work between 1874 and 1884 is the origin of
340:
in mathematics. Cantor established the importance of
312:
297:
291:
5378:
Jahresbericht der Deutschen Mathematiker-Vereinigung
3470:
is the sequence of real algebraic numbers. So both
2167:
1950:, describing them as both an "abomination" and "the
1758:. Zermelo had proved this theorem in 1904 using the
1361:
Grundlagen einer allgemeinen Mannigfaltigkeitslehre"
884:. He later proved that the size of the power set of
672:, which took place in ZĂŒrich, Switzerland, in 1897.
300:
5488:
Cantor biography (p. 452â483) in the appendix.
5330:
Grundlagen einer allgemeinen Mannigfaltigkeitslehre
3627:. New York: W. W. Norton and Company. p.
1798:
Grundlagen einer allgemeinen Mannigfaltigkeitslehre
1289:. Transcendental numbers were first constructed by
325:[ËÉĄeËÉÊkËfÉÊdinantËluËtvÉȘçËfiËlÉȘpËkantoËÉÌŻ]
288:
5622:
4848:Moore, Gregory H.; Garciadiego, Alejandro (1981).
4666:
3624:Everything and More: A Compact History of Infinity
2451:
2344:
1988:on infinity. Instead, he always strongly rejected
1946:Meanwhile, Cantor himself was fiercely opposed to
1900:. Trendelenburg was also the examiner of Cantor's
1556:. About this discovery Cantor wrote to Dedekind: "
1230:, describing them as both "an abomination" and "a
952:
924:
5800:
5778:
3245:A propos de l'existence des nombres transcendants
1930:himself, and addressed several pamphlets to him.
1164:,... formed a limit set, which we would now call
664:In 1889, Cantor was instrumental in founding the
8066:
5726:. Metaphysics Research Lab, Stanford University.
5506:. New York: Four Walls Eight Windows Publishing.
5056:
4827:Russell: The Journal of Bertrand Russell Studies
3781:is an inconsistent submultiplicity contained in
3769:is an aleph, contradicting the assumption about
3057:
3055:
2906:
2904:
1013:Number theory, trigonometric series and ordinals
5147:Journal fĂŒr die Reine und Angewandte Mathematik
5106:Journal fĂŒr die Reine und Angewandte Mathematik
5089:
4904:Purkert, Walter; Ilgauds, Hans Joachim (1985).
4875:Purkert, Walter (1989). "Cantor's Views on the
4740:(1971). "Towards a Biography of Georg Cantor".
2718:O'Connor, John J; Robertson, Edmund F. (1998).
861:without end points are order-isomorphic to the
5637:Three chapters and 18 index entries on Cantor.
2679:Bruno, Leonard C.; Baker, Lawrence W. (1999).
2020:and fled to Russia from the disruption of the
1858:of a set of cardinal or real numbers with its
1529:paper was the first to invoke the notion of a
1366:Foundations of a General Theory of Aggregates"
1180:would also have to have a set of limit points
226:De aequationibus secundi gradus indeterminatis
7676:
7016:
6395:
5894:
4879:". In Rowe, David E.; McCleary, John (eds.).
3052:
2901:
2478:, p. 280: "... the tradition made popular by
1885:hold to deterministic philosophical beliefs.
738:in Scotland. Cantor attended, hoping to meet
613:. He asked Cantor to withdraw the paper from
8145:Baltic-German people from the Russian Empire
5854:(for English version, click 'à télécharger')
5368:
5335:
5264:
5231:
5198:
4766:The Search for Mathematical Roots: 1870â1940
1557:
1506:
5640:
5623:Hill, C. O.; Rosado Haddock, G. E. (2000).
4882:The History of Modern Mathematics, Volume 1
4785:Cantorian Set Theory and Limitation of Size
4231:
4229:
3975:Perspectives on Science and Christian Faith
3112:
2533:
2531:
2404:
2402:
1403:is strictly larger than the cardinality of
1025:, the Professor at Halle, Cantor turned to
7683:
7669:
7023:
7009:
6402:
6388:
5901:
5887:
5820:Mainly devoted to Cantor's accomplishment.
5720:"Wittgenstein's Philosophy of Mathematics"
3280:For more details on Cantor's article, see
2870:, pp. 248â250. For Cantor's reaction, see
2833:, pp. 376â377. Letter dated June 21, 1884.
2810:
2808:
2678:
2506:
2504:
2471:
2469:
1918:, as well as theologians such as Cardinal
42:
5654:American Catholic Philosophical Quarterly
5548:
5004:
4865:
4838:
4325:
4309:American Catholic Philosophical Quarterly
4282:American Catholic Philosophical Quarterly
4189:
3684:, pp. 587â588; English translation:
3288:"Georg Cantor and Transcendental Numbers"
3186:For example, geometric problems posed by
3081:
2016:Cantor's paternal grandparents were from
1021:, his thesis topic. At the suggestion of
977:, introduced by Cantor, was presented by
537:
502:; his exceptional skills in mathematics,
344:between the members of two sets, defined
5143:"Ein Beitrag zur Mannigfaltigkeitslehre"
4306:
4275:
4226:
3256:The real algebraic numbers are the real
2913:The Two-Year College Mathematics Journal
2528:
2399:
2385:
2383:
2370:
2368:
2003:
1850:stance against Cantor's work. Finally,
1516:
1429:
1248:
987:International Congress of Mathematicians
968:). This notation is still in use today.
900:. Cantor developed an entire theory and
698:International Congress of Mathematicians
675:
670:International Congress of Mathematicians
638:, thinking there might be evidence that
469:
8160:Technische UniversitÀt Darmstadt alumni
6956:List of fractals by Hausdorff dimension
5866:Georg Cantor and Transcendental Numbers
5812:MacTutor History of Mathematics Archive
5790:MacTutor History of Mathematics Archive
5724:The Stanford Encyclopedia of Philosophy
5671:
5586:
5492:
4513:Memoires Scientifique 13 Correspondance
4458:
4456:
4454:
4372:from the original on 21 September 2020.
4257:
4140:
4122:
4110:
3922:
3765:. This implies that the cardinality of
3620:
3570:
3282:Georg Cantor's first set theory article
2910:
2889:
2848:
2805:
2793:
2781:
2726:
2720:"Georg Ferdinand Ludwig Philipp Cantor"
2522:
2501:
2485:
2466:
2279:List of things named after Georg Cantor
1637:
833:Cantor developed important concepts in
14:
8067:
5717:
5696:
5566:
5519:
5465:
5432:
5395:
5297:
5165:
5140:
5095:
4413:
4247:
4050:
3999:
3968:
3681:
3604:
3600:
3588:
3564:
3170:
2979:
2945:
2866:For a discussion of König's paper see
2738:
2460:
2342:
2033:born in Saint Petersburg and baptized
8175:Mathematicians from the German Empire
7664:
7004:
6383:
5882:
5662:https://doi.org/10.5840/acpq200983444
5498:
4787:. New York: Oxford University Press.
4722:. New York: Oxford University Press.
4704:from the original on 23 January 2018.
4539:Modern Jews and their musical agendas
4487:
4434:The Autobiography of Bertrand Russell
4410:, see Hill and Rosado Haddock (2000).
3195:
3026:Archive for History of Exact Sciences
3023:
2674:
2672:
2670:
2668:
2666:
2664:
2662:
2660:
2634:
2632:
2630:
2628:
2626:
2544:. New York: Springer-Verlag. p.
2380:
2365:
580:constructive viewpoint in mathematics
530:. He spent the summer of 1866 at the
323:
280:Georg Ferdinand Ludwig Philipp Cantor
64:Georg Ferdinand Ludwig Philipp Cantor
7733:Hilbert's paradox of the Grand Hotel
5391:from the original on 1 January 2018.
5136:from the original on 7 October 2017.
4673:. Boston: Harvard University Press.
4451:
3955:, p. 13. Compare to the writings of
3871:, pp. 263â264; English translation:
3749:, starts by assuming there is a set
3285:
2537:
1999:
1559:Je le vois, mais je ne le crois pas!
1421:Gödel's first incompleteness theorem
888:is strictly larger than the size of
762:
465:
5908:
5842:The Early Development of Set Theory
5830:Stanford Encyclopedia of Philosophy
5735:. Editrice Pitagora, Bologna, 2008.
3014:Mathematical Association of America
1787:The concept of the existence of an
1029:. Heine proposed that Cantor solve
876:, which is the set of all possible
802:is "more numerous" than the set of
102:, Province of Saxony, German Empire
24:
8110:20th-century German mathematicians
8105:19th-century German mathematicians
5574:. New York & Berlin: Springer.
5542:10.1038/scientificamerican0683-122
5474:. Berlin: Springer. Archived from
5021:van Heijenoort, Jean (1967).
5016:from the original on 28 June 2004.
4515:, Gauthier-Villars, Paris, p. 306.
3243:Liouville, Joseph (May 13, 1844).
3218:10.1038/scientificamerican0495-112
2722:. MacTutor History of Mathematics.
2657:
2623:
1017:Cantor's first ten papers were on
919:
841:. For example, he showed that the
585:In 1881, Cantor's Halle colleague
25:
8186:
8170:Emigrants from the Russian Empire
7928:Differential geometry of surfaces
6938:How Long Is the Coast of Britain?
5862:Cantor's first set theory article
5739:
5553:. Basel, Switzerland: BirkhÀuser.
4760:
4736:
4712:a familiar passage from the Bible
4588:
4576:
4479:
2830:
2775:
2685:. Detroit, Mich.: U X L. p.
2314:
2297:
2233:Deutsche Mathematiker-Vereinigung
1996:who knew so little mathematics."
1683:(the combination referred to as "
1545:result: for any positive integer
1501:CantorâBernsteinâSchröder theorem
1358:The fifth paper in this series, "
826:(or countably infinite) sets and
724:Deutsche Mathematiker-Vereinigung
648:Shakespearean authorship question
8120:20th-century German philosophers
8115:19th-century German philosophers
8095:19th-century German male writers
8052:
8040:
8028:
8016:
8004:
7723:Controversy over Cantor's theory
7690:
7054:
6002:
5757:
5745:
5073:. New York: Dover Publications.
5048:
4981:
4948:
4874:
4823:"The Roots of Russell's Paradox"
4609:
4598:
4582:
4570:
4557:
4542:, Oxford University Press, p. 9.
3912:
3868:
3827:
2644:www-history.mcs.st-andrews.ac.uk
2198:
2184:
2170:
1943:, whom Cantor had met in Halle.
1743:. Russell named paradoxes after
508:Technische UniversitÀt Darmstadt
455:paradise that Cantor has created
405:Controversy over Cantor's theory
284:
269:
27:German mathematician (1845â1918)
7784:Synthetic differential geometry
5062:
5040:
5020:
4718:Ewald, William B., ed. (1996).
4687:
4664:
4638:Journal of the History of Ideas
4632:
4604:
4592:
4591:(quotation from p. 350, note),
4545:
4530:
4524:
4518:
4505:
4493:
4475:
4468:
4439:
4426:
4420:
4400:
4394:
4388:
4382:
4376:
4319:
4300:
4269:
4263:
4251:
4241:
4235:
4183:
4134:
4128:
4116:
4044:
3993:
3962:
3946:
3940:
3934:
3928:
3906:
3894:
3878:
3872:
3858:
3845:
3833:
3817:
3801:
3739:
3727:
3715:
3703:
3691:
3675:
3661:
3651:
3645:
3614:
3608:
3594:
3582:
3576:
3339:
3274:
3250:
3237:
3224:
3180:
3164:
3155:
3106:
3017:
2998:
2984:
2973:
2939:
2895:
2883:
2877:
2871:
2867:
2860:
2854:
2842:
2836:
2826:
2820:
2814:
2799:
2787:
2771:
2765:
2756:
2750:
2744:
2732:
2711:
2599:
2574:
2562:
2516:
2510:
2491:
2475:
2441:
2437:
2425:
2408:
2393:
2389:
2374:
2330:
2310:
1752:his axiom system for set theory
1603:, as does a countably infinite
779:has come to play the role of a
488:Saint Petersburg stock exchange
375:Originally, Cantor's theory of
170:
7030:
6962:The Fractal Geometry of Nature
6369:Tractatus Logico-Philosophicus
5974:Problem of multiple generality
5770:Works by or about Georg Cantor
4847:
4820:
4801:
4768:. Princeton University Press.
4706:Internet version published in
4536:Mendelsohn, Ezra (ed.) (1993)
4250:, p. 404. Translation in
4162:10.1080/0025570X.1979.11976784
4051:Dauben, Joseph Warren (1978).
4006:. Princeton University Press.
4000:Dauben, Joseph Warren (1979).
3864:
3852:
3839:
3823:
3811:
3807:
3721:
3709:
3621:Wallace, David Foster (2003).
2431:
2343:Dauben, Joseph Warren (1979).
2336:
2324:
2321:are useful additional sources.
2303:
2291:
2100:
1623:and the unit square was not a
1112:is the set of limit points of
1035:Peter Gustav Lejeune Dirichlet
822:, subdividing the latter into
642:wrote the plays attributed to
478:Georg Cantor, born in 1845 in
364:of infinities. He defined the
13:
1:
6359:The Principles of Mathematics
5704:. Princeton University Press.
5641:Meschkowski, Herbert (1983).
5057:Primary literature in English
4626:
3607:. The English translation is
3296:American Mathematical Monthly
2498:as "cyclic manic-depression".
2259:Epsilon numbers (mathematics)
2131:'s biography. Writes Dauben:
1281:. His proof differs from the
1244:
830:(uncountably infinite sets).
510:. In 1862 Cantor entered the
49:
8140:People with bipolar disorder
7830:Cardinality of the continuum
6409:
6055:Commutativity of conjunction
5722:. In Edward N. Zalta (ed.).
5159:10.1515/crelle-1878-18788413
5090:Primary literature in German
5025:. Harvard University Press.
4903:
4867:10.1016/0315-0860(81)90070-7
4782:
4483:
4462:
3952:
3900:
3888:
3884:
3795:
3733:
3697:
3005:The Cantor Set Before Cantor
2318:
2224:Cardinality of the continuum
1735:(which was just mentioned),
1656:aleph-one, rather than just
460:
7:
8100:20th-century German writers
6978:Chaos: Making a New Science
4926:
4885:. Academic Press. pp.
4474:For more information, see:
3745:Cantor's proof, which is a
3231:
2586:www-groups.dcs.st-and.ac.uk
2163:
1750:In 1908, Zermelo published
1677:ZermeloâFraenkel set theory
1173:, and then he noticed that
1088:as its set of zeros, where
985:in his address at the 1900
902:arithmetic of infinite sets
666:German Mathematical Society
356:are more numerous than the
10:
8191:
7793:Formalizations of infinity
7521:von NeumannâBernaysâGödel
6075:Monotonicity of entailment
4877:Foundations of Mathematics
4840:10.15173/russell.v8i1.1732
4821:Moore, Gregory H. (1988).
4802:Moore, Gregory H. (1982).
4717:
4665:Dauben, Joseph W. (1979).
3916:
3824:Moore and Garciadiego 1981
3812:Moore and Garciadiego 1981
3685:
3497:is given by the function:
3174:
2582:"Georg Cantor (1845-1918)"
1840:L. E. J. Brouwer
1641:
1596:has the same power as the
1510:
1479:equivalent to a subset of
1399:of the power set of a set
1255:Cantor's diagonal argument
983:twenty-three open problems
812:one-to-one correspondences
393:L. E. J. Brouwer
8080:Scientists from Darmstadt
7969:Gottfried Wilhelm Leibniz
7951:
7920:
7792:
7756:
7705:
7585:
7548:
7460:
7350:
7322:One-to-one correspondence
7238:
7179:
7063:
7052:
7038:
6929:
6853:
6802:
6773:
6689:
6659:
6641:
6482:
6417:
6341:
6245:
6200:
6154:
6088:
6047:
6011:
6000:
5964:Idempotency of entailment
5916:
5807:"A history of set theory"
5328:Published separately as:
4927:Suppes, Patrick (1972) .
4783:Hallett, Michael (1986).
4754:10.1080/00033797100203837
4482:, pp. 350â352 and notes;
4347:10.1017/S0269889704000055
3134:10.1007/s00407-005-0102-4
3092:10.1007/s10699-011-9223-1
2254:Derived set (mathematics)
1814:philosophy of mathematics
1537:and the points of a unit
1507:One-to-one correspondence
736:University of St. Andrews
512:Swiss Federal Polytechnic
342:one-to-one correspondence
268:
263:
259:
236:
218:
208:
198:
191:
180:
157:
147:
129:Swiss Federal Polytechnic
121:
107:
88:
59:
41:
34:
5817:University of St Andrews
5795:University of St Andrews
5549:Ferreirós, José (2007).
5119:10.1515/crll.1874.77.258
4688:Dauben, Joseph (2004) .
4484:Purkert and Ilgauds 1985
4463:Purkert and Ilgauds 1985
4406:On Cantor, Husserl, and
4326:Ferreiros, Jose (2004).
3349:and removes a countable
2538:Reid, Constance (1996).
2480:Arthur Moritz Schönflies
2377:, pp. 8, 11, 12â13.
2319:Purkert and Ilgauds 1985
2284:
2235:in honor of Georg Cantor
2107:Arthur Moritz Schönflies
2052:According to biographer
1920:Johann Baptist Franzelin
847:Henry John Stephen Smith
742:, whose newly published
518:, attending lectures by
425:Johann Baptist Franzelin
401:philosophical objections
7974:August Ferdinand Möbius
7757:Branches of mathematics
7748:Paradoxes of set theory
6323:Willard Van Orman Quine
5871:21 January 2022 at the
5718:Rodych, Victor (2007).
5645:. Vieweg, Braunschweig.
5063:Cantor, Georg (1955) .
4992:Fundamenta Mathematicae
4982:Zermelo, Ernst (1930).
4949:Zermelo, Ernst (1908).
4906:Georg Cantor: 1845â1918
4276:Newstead, Anne (2009).
4141:Snapper, Ernst (1979).
3915:; English translation:
3478:are the union of three
3173:. English translation:
2264:Factorial number system
2063:In a letter written to
1821:and its two offshoots,
1705:well-ordering principle
1573:" (a term he took from
953:{\displaystyle \omega }
925:{\displaystyle \aleph }
532:University of Göttingen
139:University of Göttingen
7280:Constructible universe
7100:Constructibility (V=L)
6970:The Beauty of Fractals
6298:Charles Sanders Peirce
6141:Hypothetical syllogism
5750:Quotations related to
5627:. Chicago: Open Court.
5484:(p. 443â451) and
5466:Cantor, Georg (1932).
5433:Cantor, Georg (1897).
5396:Cantor, Georg (1895).
5141:Cantor, Georg (1878).
5096:Cantor, Georg (1874).
4762:Grattan-Guinness, Ivor
4738:Grattan-Guinness, Ivor
3969:Hedman, Bruce (1993).
3747:proof by contradiction
3069:Foundations of Science
3010:29 August 2022 at the
2607:Georg Cantor 1845-1918
2161:
2121:history of mathematics
2098:
2089:
2047:
2013:
1831:nonconstructive proofs
1558:
1522:
1435:
1423:. Cantor wrote on the
1287:transcendental numbers
1262:
1222:of his contemporaries
991:Charles Sanders Peirce
954:
926:
837:and their relation to
721:) to a meeting of the
702:transfinite set theory
636:Elizabethan literature
632:
538:Teacher and researcher
475:
352:, and proved that the
7938:Möbius transformation
7835:Dedekind-infinite set
7743:Paradoxes of infinity
7738:Infinity (philosophy)
7503:Principia Mathematica
7337:Transfinite induction
7196:(i.e. set difference)
6364:Principia Mathematica
6136:Disjunctive syllogism
6121:modus ponendo tollens
5702:Infinity and the Mind
5691:theoretical physicist
5596:Mathematische Annalen
5592:"Ăber das Unendliche"
5439:Mathematische Annalen
5402:Mathematische Annalen
5369:Georg Cantor (1891).
5342:Mathematische Annalen
5336:Georg Cantor (1884).
5304:Mathematische Annalen
5298:Georg Cantor (1883).
5271:Mathematische Annalen
5265:Georg Cantor (1883).
5238:Mathematische Annalen
5232:Georg Cantor (1882).
5205:Mathematische Annalen
5199:Georg Cantor (1880).
5172:Mathematische Annalen
5166:Georg Cantor (1879).
5006:10.4064/fm-16-1-29-47
4955:Mathematische Annalen
4589:Grattan-Guinness 1971
4577:Grattan-Guinness 1971
4511:Tannery, Paul (1934)
4486:; the letter is from
4480:Grattan-Guinness 1971
4294:10.5840/acpq200983444
3794:arbitrary choices." (
3286:Gray, Robert (1994).
3151:on February 15, 2013.
3122:Arch. Hist. Exact Sci
2831:Grattan-Guinness 1971
2776:Grattan-Guinness 1971
2315:Grattan-Guinness 1971
2298:Grattan-Guinness 2000
2231: â award by the
2133:
2093:
2081:
2043:
2007:
1756:well-ordering theorem
1615:, stressing that his
1531:1-to-1 correspondence
1520:
1441:Mathematische Annalen
1433:
1339:Mathematische Annalen
1257:for the existence of
1252:
955:
927:
745:Principia Mathematica
676:Later years and death
628:
542:Cantor submitted his
473:
413:Christian theologians
336:, which has become a
7774:Nonstandard analysis
7577:Burali-Forti paradox
7332:Set-builder notation
7285:Continuum hypothesis
7225:Symmetric difference
6916:Lewis Fry Richardson
6911:Hamid Naderi Yeganeh
6701:Burning Ship fractal
6633:Weierstrass function
6354:Function and Concept
6126:Constructive dilemma
6101:Material implication
5803:Robertson, Edmund F.
5781:Robertson, Edmund F.
5766:at Wikimedia Commons
5733:L'infinito di Cantor
5493:Secondary literature
5478:on February 3, 2014.
4930:Axiomatic Set Theory
4854:Historia Mathematica
4150:Mathematics Magazine
4012:10.2307/j.ctv10crfh1
3688:, pp. 916–917.
3427:}. The set of reals
3204:(4): 112â116 (114).
3113:Ehrlich, P. (2006).
2955:. Dover. p. 1.
2952:Axiomatic Set Theory
2609:. Birkhauser. 1985.
2396:, pp. 120, 143.
2244:Continuum hypothesis
2085:regular just Germain
1903:Habilitationsschrift
1733:Burali-Forti paradox
1650:continuum hypothesis
1644:Continuum hypothesis
1638:Continuum hypothesis
1521:A bijective function
1355:during this period.
1228:Paul du Bois-Reymond
1208:convergent sequences
1199: + 2, ...
1051:trigonometric series
981:as the first of his
974:Continuum hypothesis
944:
916:
711:Burali-Forti paradox
603:Gösta Mittag-Leffler
516:University of Berlin
328:; 3 March [
134:University of Berlin
7943:Riemannian manifold
7912:Transfinite numbers
7769:Internal set theory
7538:TarskiâGrothendieck
6674:Space-filling curve
6651:Multifractal system
6534:Space-filling curve
6519:Sierpinski triangle
6328:Ludwig Wittgenstein
6131:Destructive dilemma
5959:Well-formed formula
5801:O'Connor, John J.;
5779:O'Connor, John J.;
5677:The Road to Reality
5534:1983SciAm.248f.122D
5522:Scientific American
4933:. New York: Dover.
4708:Journal of the ACMS
4563:Russell, Bertrand.
3873:van Heijenoort 1967
3210:1995SciAm.272d.112M
3198:Scientific American
3062:Katz, Karin Usadi;
2037:; she converted to
2012:, Saint-Petersburg.
1894:F. A. Trendelenburg
1835:constructive proofs
1745:Cesare Burali-Forti
1425:Goldbach conjecture
1374:transfinite numbers
1267:Cantor's 1874 paper
1253:An illustration of
1130:. Because the sets
828:nondenumerable sets
781:foundational theory
690:William Shakespeare
644:William Shakespeare
552:University of Halle
474:Cantor, around 1870
445:awarded Cantor its
407:. Cantor, a devout
397:Ludwig Wittgenstein
377:transfinite numbers
213:University of Halle
8047:History of science
7896:Sphere at infinity
7847:(Complex infinity)
7127:Limitation of size
6901:Aleksandr Lyapunov
6881:Desmond Paul Henry
6845:Self-avoiding walk
6840:Percolation theory
6484:Iterated function
6425:Fractal dimensions
6273:Augustus De Morgan
5844:by José Ferreirós.
5679:. Alfred A. Knopf.
5608:10.1007/BF01206605
5451:10.1007/bf01444205
5414:10.1007/bf02124929
5354:10.1007/BF01446598
5316:10.1007/bf01446819
5283:10.1007/bf01442612
5250:10.1007/bf01443330
5217:10.1007/bf01446232
5184:10.1007/bf01444101
4967:10.1007/bf01449999
4502:, 1937, E. T. Bell
4478:, p. 1 and notes;
4432:Russell, Bertrand
4335:Science in Context
4171:on August 15, 2012
3326:on 21 January 2022
3038:10.1007/BF01886630
2640:"Cantor biography"
2274:Transfinite number
2206:Mathematics portal
2116:Men of Mathematics
2014:
2010:Vasilievsky Island
1671:and a 1963 one by
1554:-dimensional space
1523:
1436:
1413:ordinal arithmetic
1263:
1204:irrational numbers
1005:on the history of
950:
922:
476:
409:Lutheran Christian
338:fundamental theory
8130:ETH Zurich alumni
7992:
7991:
7886:Point at infinity
7866:Hyperreal numbers
7840:Directed infinity
7805:Absolute infinite
7728:Galileo's paradox
7713:Ananta (infinite)
7658:
7657:
7567:Russell's paradox
7516:ZermeloâFraenkel
7417:Dedekind-infinite
7290:Diagonal argument
7189:Cartesian product
7046:Set (mathematics)
6998:
6997:
6944:Coastline paradox
6921:WacĆaw SierpiĆski
6906:Benoit Mandelbrot
6830:Fractal landscape
6738:Misiurewicz point
6643:Strange attractor
6524:Apollonian gasket
6514:Sierpinski carpet
6377:
6376:
6241:
6240:
5855:
5762:Media related to
5731:Leonida Lazzari,
5666:Pantheismusstreit
5428:on 23 April 2014.
5080:978-0-486-60045-1
5032:978-0-674-32449-7
4940:978-0-486-61630-8
4919:978-0-8176-1770-7
4896:978-0-12-599662-4
4813:978-1-4613-9480-8
4794:978-0-19-853283-5
4775:978-0-691-05858-0
4742:Annals of Science
4729:978-0-19-853271-2
4680:978-0-691-02447-9
4634:Dauben, Joseph W.
3777:. The function's
3638:978-0-393-00338-3
3480:pairwise disjoint
3374:). Call this set
2555:978-0-387-04999-1
2214:Absolute infinite
2192:Philosophy portal
2058:Annals of Science
2000:Cantor's ancestry
1982:Gottfried Leibniz
1957:of mathematics".
1802:absolute infinite
1741:Russell's paradox
1454:well-ordered sets
1419:and the proof of
1298:algebraic numbers
1283:diagonal argument
763:Mathematical work
719:Russell's paradox
655:diagonal argument
591:Heinrich M. Weber
520:Leopold Kronecker
466:Youth and studies
432:Leopold Kronecker
381:Leopold Kronecker
350:well-ordered sets
277:
276:
193:Scientific career
16:(Redirected from
8182:
8135:German Lutherans
8085:German logicians
8057:
8056:
8055:
8045:
8044:
8043:
8033:
8032:
8031:
8021:
8020:
8019:
8009:
8008:
8000:
7984:Abraham Robinson
7979:Bernhard Riemann
7898:(Kleinian group)
7891:Regular cardinal
7845:Division by zero
7825:Cardinal numbers
7764:Complex analysis
7699:
7685:
7678:
7671:
7662:
7661:
7640:Bertrand Russell
7630:John von Neumann
7615:Abraham Fraenkel
7610:Richard Dedekind
7572:Suslin's problem
7483:Cantor's theorem
7200:De Morgan's laws
7058:
7025:
7018:
7011:
7002:
7001:
6861:Michael Barnsley
6728:Lyapunov fractal
6586:SierpiĆski curve
6539:Blancmange curve
6404:
6397:
6390:
6381:
6380:
6313:Henry M. Sheffer
6303:Bertrand Russell
6268:Richard Dedekind
6152:
6151:
6096:De Morgan's laws
6070:Noncontradiction
6012:Classical logics
6006:
5903:
5896:
5889:
5880:
5879:
5858:non-constructive
5853:
5826:, britannica.com
5819:
5797:
5774:Internet Archive
5761:
5749:
5727:
5705:
5680:
5646:
5628:
5619:
5575:
5572:Naive Set Theory
5554:
5545:
5507:
5479:
5462:
5429:
5424:. Archived from
5392:
5390:
5375:
5365:
5327:
5294:
5261:
5228:
5195:
5162:
5137:
5135:
5102:
5084:
5036:
5017:
5015:
5008:
4988:
4978:
4944:
4923:
4900:
4871:
4869:
4844:
4842:
4817:
4798:
4779:
4757:
4733:
4705:
4703:
4696:
4684:
4672:
4661:
4620:
4617:Grattan-Guinness
4613:
4607:
4602:
4596:
4586:
4580:
4574:
4568:
4561:
4555:
4549:
4543:
4534:
4528:
4522:
4516:
4509:
4503:
4497:
4491:
4472:
4466:
4460:
4449:
4443:
4437:
4430:
4424:
4417:
4411:
4404:
4398:
4392:
4386:
4380:
4374:
4373:
4371:
4332:
4323:
4317:
4316:
4304:
4298:
4297:
4273:
4267:
4261:
4255:
4245:
4239:
4233:
4224:
4223:
4187:
4181:
4180:
4178:
4176:
4170:
4164:. Archived from
4147:
4138:
4132:
4126:
4120:
4114:
4108:
4107:
4101:
4099:
4048:
4042:
4041:
3997:
3991:
3990:
3988:
3986:
3966:
3960:
3950:
3944:
3938:
3932:
3926:
3920:
3919:, pp. 1208â1233.
3910:
3904:
3898:
3892:
3882:
3876:
3862:
3856:
3849:
3843:
3837:
3831:
3826:, pp. 331, 343;
3821:
3815:
3805:
3799:
3743:
3737:
3731:
3725:
3719:
3713:
3707:
3701:
3695:
3689:
3679:
3673:
3665:
3659:
3649:
3643:
3642:
3618:
3612:
3598:
3592:
3586:
3580:
3574:
3568:
3360:} (for example,
3343:
3337:
3335:
3333:
3331:
3325:
3319:. Archived from
3292:
3278:
3272:
3254:
3248:
3241:
3235:
3228:
3222:
3221:
3192:John Duns Scotus
3184:
3178:
3168:
3162:
3159:
3153:
3152:
3150:
3144:. Archived from
3119:
3110:
3104:
3103:
3085:
3064:Katz, Mikhail G.
3059:
3050:
3049:
3021:
3015:
3002:
2996:
2988:
2982:
2977:
2971:
2970:
2943:
2937:
2936:
2908:
2899:
2893:
2887:
2881:
2875:
2864:
2858:
2852:
2846:
2840:
2834:
2824:
2818:
2812:
2803:
2797:
2791:
2785:
2779:
2769:
2763:
2754:
2748:
2742:
2736:
2730:
2724:
2723:
2715:
2709:
2708:
2676:
2655:
2654:
2652:
2650:
2636:
2621:
2620:
2603:
2597:
2596:
2594:
2592:
2578:
2572:
2566:
2560:
2559:
2535:
2526:
2520:
2514:
2508:
2499:
2489:
2483:
2473:
2464:
2458:
2449:
2435:
2429:
2423:
2412:
2406:
2397:
2387:
2378:
2372:
2363:
2362:
2350:
2340:
2334:
2328:
2322:
2307:
2301:
2295:
2269:Pairing function
2208:
2203:
2202:
2194:
2189:
2188:
2187:
2180:
2178:Biography portal
2175:
2174:
2173:
2111:Eric Temple Bell
2077:Bertrand Russell
2054:Eric Temple Bell
2031:Austro-Hungarian
1767:John von Neumann
1737:Cantor's paradox
1725:Bertrand Russell
1632:Karl Weierstrass
1561:
1491:are equivalent.
1393:Cantor's theorem
1326:nested intervals
1291:Joseph Liouville
1259:uncountable sets
1212:rational numbers
1195: + 1,
1043:Bernhard Riemann
1039:Rudolf Lipschitz
1033:that had eluded
999:Jacques Hadamard
963:
959:
957:
956:
951:
931:
929:
928:
923:
898:Cantor's theorem
863:rational numbers
845:, discovered by
740:Bertrand Russell
715:Cantor's paradox
620:Acta Mathematica
607:Acta Mathematica
568:Richard Dedekind
524:Karl Weierstrass
480:Saint Petersburg
436:bipolar disorder
327:
322:
315:
310:
309:
306:
305:
302:
299:
296:
293:
290:
273:
251:Karl Weierstrass
238:Doctoral advisor
232:
174:
172:
95:
78:Saint Petersburg
73:
71:
54:
51:
46:
32:
31:
21:
8190:
8189:
8185:
8184:
8183:
8181:
8180:
8179:
8065:
8064:
8063:
8053:
8051:
8041:
8039:
8029:
8027:
8017:
8015:
8003:
7995:
7993:
7988:
7947:
7916:
7907:Surreal numbers
7881:Ordinal numbers
7810:Actual infinity
7788:
7752:
7701:
7695:
7689:
7659:
7654:
7581:
7560:
7544:
7509:New Foundations
7456:
7346:
7265:Cardinal number
7248:
7234:
7175:
7059:
7050:
7034:
7029:
6999:
6994:
6925:
6876:Felix Hausdorff
6849:
6813:Brownian motion
6798:
6769:
6692:
6685:
6655:
6637:
6628:Pythagoras tree
6485:
6478:
6474:Self-similarity
6418:Characteristics
6413:
6408:
6378:
6373:
6349:Begriffsschrift
6337:
6333:Jan Ćukasiewicz
6253:Bernard Bolzano
6237:
6208:Double negation
6196:
6167:Double negation
6150:
6084:
6060:Excluded middle
6043:
6007:
5998:
5912:
5910:Classical logic
5907:
5873:Wayback Machine
5742:
5495:
5388:
5373:
5153:(84): 242â258.
5133:
5113:(77): 258â262.
5100:
5092:
5081:
5065:Philip Jourdain
5059:
5043:
5033:
5013:
4986:
4941:
4920:
4897:
4814:
4795:
4776:
4730:
4701:
4694:
4681:
4650:10.2307/2708842
4629:
4624:
4623:
4614:
4610:
4603:
4599:
4587:
4583:
4575:
4571:
4562:
4558:
4550:
4546:
4535:
4531:
4523:
4519:
4510:
4506:
4498:
4494:
4473:
4469:
4461:
4452:
4444:
4440:
4431:
4427:
4418:
4414:
4405:
4401:
4393:
4389:
4381:
4377:
4369:
4330:
4324:
4320:
4305:
4301:
4274:
4270:
4262:
4258:
4246:
4242:
4234:
4227:
4188:
4184:
4174:
4172:
4168:
4145:
4139:
4135:
4127:
4123:
4115:
4111:
4097:
4095:
4049:
4045:
4022:
3998:
3994:
3984:
3982:
3967:
3963:
3951:
3947:
3939:
3935:
3927:
3923:
3911:
3907:
3899:
3895:
3883:
3879:
3867:, pp. 158â160.
3863:
3859:
3850:
3846:
3838:
3834:
3822:
3818:
3806:
3802:
3798:, pp. 169â170.)
3744:
3740:
3732:
3728:
3720:
3716:
3708:
3704:
3696:
3692:
3680:
3676:
3666:
3662:
3650:
3646:
3639:
3619:
3615:
3599:
3595:
3587:
3583:
3575:
3571:
3561:
3555:
3540:
3534:
3519:
3488:
3468:
3461:
3456:} âȘ {
3454:
3448:
3440:
3426:
3417:} âȘ {
3416:
3405:
3397:
3391:
3380:
3365:
3358:
3344:
3340:
3329:
3327:
3323:
3309:10.2307/2975129
3290:
3279:
3275:
3264:equations with
3255:
3251:
3242:
3238:
3229:
3225:
3185:
3181:
3169:
3165:
3160:
3156:
3148:
3117:
3111:
3107:
3060:
3053:
3022:
3018:
3012:Wayback Machine
3003:
2999:
2989:
2985:
2978:
2974:
2963:
2947:Suppes, Patrick
2944:
2940:
2925:10.2307/3026799
2909:
2902:
2894:
2890:
2882:
2878:
2874:, pp. 248, 283.
2865:
2861:
2853:
2849:
2841:
2837:
2825:
2821:
2813:
2806:
2798:
2794:
2786:
2782:
2770:
2766:
2755:
2751:
2743:
2739:
2731:
2727:
2716:
2712:
2697:
2677:
2658:
2648:
2646:
2638:
2637:
2624:
2617:
2605:
2604:
2600:
2590:
2588:
2580:
2579:
2575:
2567:
2563:
2556:
2536:
2529:
2521:
2517:
2509:
2502:
2490:
2486:
2474:
2467:
2459:
2452:
2436:
2432:
2424:
2415:
2407:
2400:
2388:
2381:
2373:
2366:
2359:
2341:
2337:
2329:
2325:
2308:
2304:
2296:
2292:
2287:
2239:Cardinal number
2204:
2197:
2190:
2185:
2183:
2176:
2171:
2169:
2166:
2149:du Bois Reymond
2103:
2075:In a letter to
2022:Napoleonic Wars
2002:
1986:Bernard Bolzano
1978:George Berkeley
1916:Joseph Hontheim
1842:and especially
1789:actual infinity
1785:
1760:axiom of choice
1718:Cantor avoided
1693:
1681:axiom of choice
1646:
1640:
1591:Euclidean space
1583:natural numbers
1515:
1509:
1497:Felix Bernstein
1471:to a subset of
1417:Halting problem
1382:Ordinal numbers
1349:actual infinity
1323:
1316:
1309:
1247:
1186:
1179:
1172:
1163:
1156:
1149:
1138:
1129:
1120:
1111:
1094:
1087:
1072:
1031:an open problem
1015:
1003:Philip Jourdain
961:
945:
942:
941:
917:
914:
913:
804:natural numbers
765:
686:Baconian theory
678:
540:
468:
463:
447:Sylvester Medal
417:neo-Scholastics
387:and later from
358:natural numbers
320:
313:
287:
283:
255:
230:
185:Sylvester Medal
176:
173: 1874)
168:
164:
143:
122:Alma mater
103:
97:
93:
84:
75:
69:
67:
66:
65:
55:
52:
37:
28:
23:
22:
15:
12:
11:
5:
8188:
8178:
8177:
8172:
8167:
8162:
8157:
8152:
8147:
8142:
8137:
8132:
8127:
8122:
8117:
8112:
8107:
8102:
8097:
8092:
8087:
8082:
8077:
8062:
8061:
8049:
8037:
8025:
8013:
7990:
7989:
7987:
7986:
7981:
7976:
7971:
7966:
7961:
7955:
7953:
7952:Mathematicians
7949:
7948:
7946:
7945:
7940:
7935:
7930:
7924:
7922:
7918:
7917:
7915:
7914:
7909:
7904:
7899:
7893:
7888:
7883:
7878:
7873:
7868:
7863:
7858:
7856:Gimel function
7853:
7851:Epsilon number
7848:
7842:
7837:
7832:
7827:
7822:
7817:
7812:
7807:
7802:
7796:
7794:
7790:
7789:
7787:
7786:
7781:
7776:
7771:
7766:
7760:
7758:
7754:
7753:
7751:
7750:
7745:
7740:
7735:
7730:
7725:
7720:
7715:
7709:
7707:
7703:
7702:
7688:
7687:
7680:
7673:
7665:
7656:
7655:
7653:
7652:
7647:
7645:Thoralf Skolem
7642:
7637:
7632:
7627:
7622:
7617:
7612:
7607:
7602:
7597:
7591:
7589:
7583:
7582:
7580:
7579:
7574:
7569:
7563:
7561:
7559:
7558:
7555:
7549:
7546:
7545:
7543:
7542:
7541:
7540:
7535:
7530:
7529:
7528:
7513:
7512:
7511:
7499:
7498:
7497:
7486:
7485:
7480:
7475:
7470:
7464:
7462:
7458:
7457:
7455:
7454:
7449:
7444:
7439:
7430:
7425:
7420:
7410:
7405:
7404:
7403:
7398:
7393:
7383:
7373:
7368:
7363:
7357:
7355:
7348:
7347:
7345:
7344:
7339:
7334:
7329:
7327:Ordinal number
7324:
7319:
7314:
7309:
7308:
7307:
7302:
7292:
7287:
7282:
7277:
7272:
7262:
7257:
7251:
7249:
7247:
7246:
7243:
7239:
7236:
7235:
7233:
7232:
7227:
7222:
7217:
7212:
7207:
7205:Disjoint union
7202:
7197:
7191:
7185:
7183:
7177:
7176:
7174:
7173:
7172:
7171:
7166:
7155:
7154:
7152:Martin's axiom
7149:
7144:
7139:
7134:
7129:
7124:
7119:
7117:Extensionality
7114:
7113:
7112:
7102:
7097:
7096:
7095:
7090:
7085:
7075:
7069:
7067:
7061:
7060:
7053:
7051:
7049:
7048:
7042:
7040:
7036:
7035:
7028:
7027:
7020:
7013:
7005:
6996:
6995:
6993:
6992:
6987:
6982:
6974:
6966:
6958:
6953:
6948:
6947:
6946:
6933:
6931:
6927:
6926:
6924:
6923:
6918:
6913:
6908:
6903:
6898:
6893:
6891:Helge von Koch
6888:
6883:
6878:
6873:
6868:
6863:
6857:
6855:
6851:
6850:
6848:
6847:
6842:
6837:
6832:
6827:
6826:
6825:
6823:Brownian motor
6820:
6809:
6807:
6800:
6799:
6797:
6796:
6794:Pickover stalk
6791:
6786:
6780:
6778:
6771:
6770:
6768:
6767:
6762:
6757:
6752:
6750:Newton fractal
6747:
6742:
6741:
6740:
6733:Mandelbrot set
6730:
6725:
6724:
6723:
6718:
6716:Newton fractal
6713:
6703:
6697:
6695:
6687:
6686:
6684:
6683:
6682:
6681:
6671:
6669:Fractal canopy
6665:
6663:
6657:
6656:
6654:
6653:
6647:
6645:
6639:
6638:
6636:
6635:
6630:
6625:
6620:
6615:
6613:Vicsek fractal
6610:
6605:
6600:
6595:
6594:
6593:
6588:
6583:
6578:
6573:
6568:
6563:
6558:
6553:
6552:
6551:
6541:
6531:
6529:Fibonacci word
6526:
6521:
6516:
6511:
6506:
6504:Koch snowflake
6501:
6496:
6490:
6488:
6480:
6479:
6477:
6476:
6471:
6466:
6465:
6464:
6459:
6454:
6449:
6444:
6443:
6442:
6432:
6421:
6419:
6415:
6414:
6407:
6406:
6399:
6392:
6384:
6375:
6374:
6372:
6371:
6366:
6361:
6356:
6351:
6345:
6343:
6339:
6338:
6336:
6335:
6330:
6325:
6320:
6315:
6310:
6308:Ernst Schröder
6305:
6300:
6295:
6293:Giuseppe Peano
6290:
6285:
6280:
6275:
6270:
6265:
6260:
6255:
6249:
6247:
6243:
6242:
6239:
6238:
6236:
6235:
6230:
6225:
6220:
6215:
6210:
6204:
6202:
6198:
6197:
6195:
6194:
6189:
6184:
6179:
6174:
6169:
6164:
6158:
6156:
6149:
6148:
6143:
6138:
6133:
6128:
6123:
6118:
6113:
6108:
6103:
6098:
6092:
6090:
6086:
6085:
6083:
6082:
6077:
6072:
6067:
6062:
6057:
6051:
6049:
6045:
6044:
6042:
6041:
6036:
6031:
6026:
6021:
6015:
6013:
6009:
6008:
6001:
5999:
5997:
5996:
5991:
5986:
5981:
5976:
5971:
5966:
5961:
5956:
5951:
5949:Truth function
5946:
5941:
5936:
5931:
5926:
5920:
5918:
5914:
5913:
5906:
5905:
5898:
5891:
5883:
5877:
5876:
5845:
5827:
5821:
5798:
5785:"Georg Cantor"
5776:
5767:
5755:
5741:
5740:External links
5738:
5737:
5736:
5729:
5715:
5694:
5673:Penrose, Roger
5669:
5660:(4): 532â553,
5647:
5638:
5620:
5588:Hilbert, David
5584:
5564:
5546:
5528:(6): 122â131.
5517:
5500:Aczel, Amir D.
5494:
5491:
5490:
5489:
5463:
5445:(2): 207â246.
5430:
5408:(4): 481â512.
5393:
5366:
5348:(4): 453â488.
5333:
5310:(4): 545â591.
5295:
5262:
5244:(1): 113â121.
5229:
5211:(3): 355â358.
5196:
5163:
5138:
5091:
5088:
5087:
5086:
5079:
5058:
5055:
5054:
5053:
5042:
5039:
5038:
5037:
5031:
5018:
4979:
4961:(2): 261â281.
4946:
4939:
4924:
4918:
4901:
4895:
4872:
4860:(3): 319â350.
4845:
4818:
4812:
4799:
4793:
4780:
4774:
4758:
4748:(4): 345â391.
4734:
4728:
4715:
4685:
4679:
4662:
4628:
4625:
4622:
4621:
4608:
4597:
4581:
4569:
4556:
4544:
4529:
4517:
4504:
4492:
4467:
4450:
4438:
4425:
4412:
4399:
4387:
4375:
4341:(1â2): 49â83.
4318:
4299:
4288:(4): 533â553.
4268:
4256:
4240:
4225:
4204:10.1086/383692
4198:(2): 263â295.
4182:
4156:(4): 207â216.
4133:
4121:
4109:
4069:10.1086/352113
4043:
4020:
3992:
3961:
3957:Thomas Aquinas
3945:
3933:
3921:
3905:
3903:, pp. 291â292.
3893:
3877:
3857:
3844:
3832:
3816:
3814:, pp. 330â331.
3800:
3738:
3736:, pp. 166â169.
3726:
3714:
3702:
3690:
3674:
3670:W. Hugh Woodin
3660:
3654:, pp. 69, 324
3644:
3637:
3613:
3593:
3581:
3569:
3559:
3556:) =
3550:
3538:
3535:) =
3528:
3517:
3505:) =
3486:
3466:
3459:
3452:
3449: âȘ {
3446:
3438:
3435: âȘ {
3421:
3410:
3406: âȘ {
3403:
3395:
3392: âȘ {
3389:
3378:
3372: / n
3363:
3356:
3338:
3303:(9): 819â832.
3273:
3249:
3236:
3223:
3179:
3177:, pp. 840â843.
3163:
3154:
3105:
3051:
3016:
2997:
2983:
2972:
2961:
2938:
2900:
2888:
2886:, pp. 283â284.
2876:
2859:
2847:
2845:, pp. 281â283.
2835:
2819:
2804:
2792:
2780:
2778:, pp. 354â355.
2764:
2749:
2737:
2725:
2710:
2696:978-0787638139
2695:
2656:
2622:
2616:978-3764317706
2615:
2598:
2573:
2561:
2554:
2527:
2515:
2500:
2496:mental illness
2484:
2465:
2450:
2430:
2413:
2398:
2379:
2364:
2357:
2335:
2323:
2302:
2289:
2288:
2286:
2283:
2282:
2281:
2276:
2271:
2266:
2261:
2256:
2251:
2246:
2241:
2236:
2226:
2221:
2216:
2210:
2209:
2195:
2181:
2165:
2162:
2102:
2099:
2035:Roman Catholic
2001:
1998:
1974:René Descartes
1948:infinitesimals
1941:Edmund Husserl
1844:Henri Poincaré
1819:constructivism
1784:
1781:
1692:
1689:
1642:Main article:
1639:
1636:
1579:countable sets
1525:Cantor's 1874
1511:Main article:
1508:
1505:
1493:Ernst Schröder
1464:are sets with
1321:
1314:
1307:
1246:
1243:
1239:infinitesimals
1220:infinitesimals
1184:
1177:
1168:
1161:
1154:
1147:
1134:
1125:
1116:
1107:
1095:is the set of
1092:
1085:
1068:
1014:
1011:
949:
921:
764:
761:
753:malnourishment
677:
674:
564:Harz mountains
539:
536:
467:
464:
462:
459:
415:(particularly
385:Henri Poincaré
275:
274:
266:
265:
261:
260:
257:
256:
254:
253:
248:
242:
240:
234:
233:
222:
216:
215:
210:
206:
205:
200:
196:
195:
189:
188:
182:
178:
177:
166:
163:Vally Guttmann
162:
161:
159:
155:
154:
149:
148:Known for
145:
144:
142:
141:
136:
131:
125:
123:
119:
118:
109:
105:
104:
98:
96:(aged 72)
92:6 January 1918
90:
86:
85:
82:Russian Empire
76:
63:
61:
57:
56:
47:
39:
38:
35:
26:
9:
6:
4:
3:
2:
8187:
8176:
8173:
8171:
8168:
8166:
8163:
8161:
8158:
8156:
8153:
8151:
8148:
8146:
8143:
8141:
8138:
8136:
8133:
8131:
8128:
8126:
8123:
8121:
8118:
8116:
8113:
8111:
8108:
8106:
8103:
8101:
8098:
8096:
8093:
8091:
8090:Set theorists
8088:
8086:
8083:
8081:
8078:
8076:
8073:
8072:
8070:
8060:
8050:
8048:
8038:
8036:
8026:
8024:
8014:
8012:
8007:
8002:
8001:
7998:
7985:
7982:
7980:
7977:
7975:
7972:
7970:
7967:
7965:
7964:David Hilbert
7962:
7960:
7957:
7956:
7954:
7950:
7944:
7941:
7939:
7936:
7934:
7931:
7929:
7926:
7925:
7923:
7919:
7913:
7910:
7908:
7905:
7903:
7900:
7897:
7894:
7892:
7889:
7887:
7884:
7882:
7879:
7877:
7876:Infinitesimal
7874:
7872:
7869:
7867:
7864:
7862:
7861:Hilbert space
7859:
7857:
7854:
7852:
7849:
7846:
7843:
7841:
7838:
7836:
7833:
7831:
7828:
7826:
7823:
7821:
7818:
7816:
7813:
7811:
7808:
7806:
7803:
7801:
7798:
7797:
7795:
7791:
7785:
7782:
7780:
7777:
7775:
7772:
7770:
7767:
7765:
7762:
7761:
7759:
7755:
7749:
7746:
7744:
7741:
7739:
7736:
7734:
7731:
7729:
7726:
7724:
7721:
7719:
7716:
7714:
7711:
7710:
7708:
7704:
7698:
7693:
7686:
7681:
7679:
7674:
7672:
7667:
7666:
7663:
7651:
7650:Ernst Zermelo
7648:
7646:
7643:
7641:
7638:
7636:
7635:Willard Quine
7633:
7631:
7628:
7626:
7623:
7621:
7618:
7616:
7613:
7611:
7608:
7606:
7603:
7601:
7598:
7596:
7593:
7592:
7590:
7588:
7587:Set theorists
7584:
7578:
7575:
7573:
7570:
7568:
7565:
7564:
7562:
7556:
7554:
7551:
7550:
7547:
7539:
7536:
7534:
7533:KripkeâPlatek
7531:
7527:
7524:
7523:
7522:
7519:
7518:
7517:
7514:
7510:
7507:
7506:
7505:
7504:
7500:
7496:
7493:
7492:
7491:
7488:
7487:
7484:
7481:
7479:
7476:
7474:
7471:
7469:
7466:
7465:
7463:
7459:
7453:
7450:
7448:
7445:
7443:
7440:
7438:
7436:
7431:
7429:
7426:
7424:
7421:
7418:
7414:
7411:
7409:
7406:
7402:
7399:
7397:
7394:
7392:
7389:
7388:
7387:
7384:
7381:
7377:
7374:
7372:
7369:
7367:
7364:
7362:
7359:
7358:
7356:
7353:
7349:
7343:
7340:
7338:
7335:
7333:
7330:
7328:
7325:
7323:
7320:
7318:
7315:
7313:
7310:
7306:
7303:
7301:
7298:
7297:
7296:
7293:
7291:
7288:
7286:
7283:
7281:
7278:
7276:
7273:
7270:
7266:
7263:
7261:
7258:
7256:
7253:
7252:
7250:
7244:
7241:
7240:
7237:
7231:
7228:
7226:
7223:
7221:
7218:
7216:
7213:
7211:
7208:
7206:
7203:
7201:
7198:
7195:
7192:
7190:
7187:
7186:
7184:
7182:
7178:
7170:
7169:specification
7167:
7165:
7162:
7161:
7160:
7157:
7156:
7153:
7150:
7148:
7145:
7143:
7140:
7138:
7135:
7133:
7130:
7128:
7125:
7123:
7120:
7118:
7115:
7111:
7108:
7107:
7106:
7103:
7101:
7098:
7094:
7091:
7089:
7086:
7084:
7081:
7080:
7079:
7076:
7074:
7071:
7070:
7068:
7066:
7062:
7057:
7047:
7044:
7043:
7041:
7037:
7033:
7026:
7021:
7019:
7014:
7012:
7007:
7006:
7003:
6991:
6988:
6986:
6983:
6980:
6979:
6975:
6972:
6971:
6967:
6964:
6963:
6959:
6957:
6954:
6952:
6949:
6945:
6942:
6941:
6939:
6935:
6934:
6932:
6928:
6922:
6919:
6917:
6914:
6912:
6909:
6907:
6904:
6902:
6899:
6897:
6894:
6892:
6889:
6887:
6884:
6882:
6879:
6877:
6874:
6872:
6869:
6867:
6864:
6862:
6859:
6858:
6856:
6852:
6846:
6843:
6841:
6838:
6836:
6833:
6831:
6828:
6824:
6821:
6819:
6818:Brownian tree
6816:
6815:
6814:
6811:
6810:
6808:
6805:
6801:
6795:
6792:
6790:
6787:
6785:
6782:
6781:
6779:
6776:
6772:
6766:
6763:
6761:
6758:
6756:
6753:
6751:
6748:
6746:
6745:Multibrot set
6743:
6739:
6736:
6735:
6734:
6731:
6729:
6726:
6722:
6721:Douady rabbit
6719:
6717:
6714:
6712:
6709:
6708:
6707:
6704:
6702:
6699:
6698:
6696:
6694:
6688:
6680:
6677:
6676:
6675:
6672:
6670:
6667:
6666:
6664:
6662:
6658:
6652:
6649:
6648:
6646:
6644:
6640:
6634:
6631:
6629:
6626:
6624:
6621:
6619:
6616:
6614:
6611:
6609:
6606:
6604:
6601:
6599:
6596:
6592:
6591:Z-order curve
6589:
6587:
6584:
6582:
6579:
6577:
6574:
6572:
6569:
6567:
6564:
6562:
6561:Hilbert curve
6559:
6557:
6554:
6550:
6547:
6546:
6545:
6544:De Rham curve
6542:
6540:
6537:
6536:
6535:
6532:
6530:
6527:
6525:
6522:
6520:
6517:
6515:
6512:
6510:
6509:Menger sponge
6507:
6505:
6502:
6500:
6497:
6495:
6494:Barnsley fern
6492:
6491:
6489:
6487:
6481:
6475:
6472:
6470:
6467:
6463:
6460:
6458:
6455:
6453:
6450:
6448:
6445:
6441:
6438:
6437:
6436:
6433:
6431:
6428:
6427:
6426:
6423:
6422:
6420:
6416:
6412:
6405:
6400:
6398:
6393:
6391:
6386:
6385:
6382:
6370:
6367:
6365:
6362:
6360:
6357:
6355:
6352:
6350:
6347:
6346:
6344:
6340:
6334:
6331:
6329:
6326:
6324:
6321:
6319:
6318:Alfred Tarski
6316:
6314:
6311:
6309:
6306:
6304:
6301:
6299:
6296:
6294:
6291:
6289:
6286:
6284:
6281:
6279:
6278:Gottlob Frege
6276:
6274:
6271:
6269:
6266:
6264:
6261:
6259:
6256:
6254:
6251:
6250:
6248:
6244:
6234:
6231:
6229:
6226:
6224:
6223:Biconditional
6221:
6219:
6216:
6214:
6211:
6209:
6206:
6205:
6203:
6199:
6193:
6190:
6188:
6185:
6183:
6182:Biconditional
6180:
6178:
6175:
6173:
6170:
6168:
6165:
6163:
6160:
6159:
6157:
6153:
6147:
6144:
6142:
6139:
6137:
6134:
6132:
6129:
6127:
6124:
6122:
6119:
6117:
6116:modus tollens
6114:
6112:
6109:
6107:
6106:Transposition
6104:
6102:
6099:
6097:
6094:
6093:
6091:
6087:
6081:
6078:
6076:
6073:
6071:
6068:
6066:
6063:
6061:
6058:
6056:
6053:
6052:
6050:
6046:
6040:
6037:
6035:
6032:
6030:
6027:
6025:
6024:Propositional
6022:
6020:
6017:
6016:
6014:
6010:
6005:
5995:
5992:
5990:
5987:
5985:
5982:
5980:
5979:Associativity
5977:
5975:
5972:
5970:
5967:
5965:
5962:
5960:
5957:
5955:
5952:
5950:
5947:
5945:
5942:
5940:
5937:
5935:
5932:
5930:
5927:
5925:
5922:
5921:
5919:
5915:
5911:
5904:
5899:
5897:
5892:
5890:
5885:
5884:
5881:
5874:
5870:
5867:
5863:
5859:
5852:
5851:
5846:
5843:
5839:
5835:
5831:
5828:
5825:
5822:
5818:
5814:
5813:
5808:
5804:
5799:
5796:
5792:
5791:
5786:
5782:
5777:
5775:
5771:
5768:
5765:
5760:
5756:
5753:
5748:
5744:
5743:
5734:
5730:
5725:
5721:
5716:
5713:
5712:0-553-25531-2
5709:
5703:
5699:
5695:
5692:
5688:
5687:0-679-77631-1
5684:
5678:
5674:
5670:
5667:
5663:
5659:
5655:
5651:
5648:
5644:
5639:
5636:
5635:0-8126-9538-0
5632:
5626:
5621:
5617:
5613:
5609:
5605:
5601:
5597:
5593:
5589:
5585:
5583:
5582:3-540-90092-6
5579:
5573:
5569:
5565:
5562:
5561:3-7643-8349-6
5558:
5552:
5547:
5543:
5539:
5535:
5531:
5527:
5523:
5518:
5515:
5514:0-7607-7778-0
5511:
5505:
5501:
5497:
5496:
5487:
5483:
5477:
5473:
5469:
5468:Ernst Zermelo
5464:
5460:
5456:
5452:
5448:
5444:
5440:
5436:
5431:
5427:
5423:
5419:
5415:
5411:
5407:
5403:
5399:
5394:
5387:
5383:
5379:
5372:
5367:
5363:
5359:
5355:
5351:
5347:
5343:
5339:
5334:
5331:
5325:
5321:
5317:
5313:
5309:
5305:
5301:
5296:
5292:
5288:
5284:
5280:
5276:
5272:
5268:
5263:
5259:
5255:
5251:
5247:
5243:
5239:
5235:
5230:
5226:
5222:
5218:
5214:
5210:
5206:
5202:
5197:
5193:
5189:
5185:
5181:
5177:
5173:
5169:
5164:
5160:
5156:
5152:
5148:
5144:
5139:
5132:
5128:
5124:
5120:
5116:
5112:
5108:
5107:
5099:
5094:
5093:
5082:
5076:
5072:
5071:
5066:
5061:
5060:
5052:
5050:
5049:§ Biographies
5045:
5044:
5034:
5028:
5024:
5019:
5012:
5007:
5002:
4998:
4994:
4993:
4985:
4980:
4976:
4972:
4968:
4964:
4960:
4956:
4952:
4947:
4942:
4936:
4932:
4931:
4925:
4921:
4915:
4911:
4907:
4902:
4898:
4892:
4888:
4884:
4883:
4878:
4873:
4868:
4863:
4859:
4855:
4851:
4846:
4841:
4836:
4832:
4828:
4824:
4819:
4815:
4809:
4805:
4800:
4796:
4790:
4786:
4781:
4777:
4771:
4767:
4763:
4759:
4755:
4751:
4747:
4743:
4739:
4735:
4731:
4725:
4721:
4716:
4713:
4709:
4700:
4693:
4692:
4686:
4682:
4676:
4671:
4670:
4663:
4659:
4655:
4651:
4647:
4644:(1): 85â108.
4643:
4639:
4635:
4631:
4630:
4618:
4612:
4606:
4601:
4594:
4590:
4585:
4578:
4573:
4566:
4565:Autobiography
4560:
4553:
4548:
4541:
4540:
4533:
4526:
4521:
4514:
4508:
4501:
4496:
4489:
4485:
4481:
4477:
4471:
4464:
4459:
4457:
4455:
4447:
4442:
4435:
4429:
4422:
4416:
4409:
4408:Gottlob Frege
4403:
4396:
4391:
4384:
4379:
4368:
4364:
4360:
4356:
4352:
4348:
4344:
4340:
4336:
4329:
4322:
4314:
4310:
4303:
4295:
4291:
4287:
4283:
4279:
4272:
4265:
4260:
4253:
4249:
4244:
4237:
4232:
4230:
4221:
4217:
4213:
4209:
4205:
4201:
4197:
4193:
4186:
4167:
4163:
4159:
4155:
4151:
4144:
4137:
4130:
4125:
4118:
4113:
4106:
4094:
4090:
4086:
4082:
4078:
4074:
4070:
4066:
4062:
4058:
4054:
4047:
4039:
4035:
4031:
4027:
4023:
4021:9780691024479
4017:
4013:
4009:
4005:
4004:
3996:
3980:
3976:
3972:
3965:
3958:
3954:
3949:
3942:
3937:
3930:
3925:
3918:
3914:
3909:
3902:
3897:
3890:
3886:
3881:
3874:
3870:
3866:
3861:
3855:, pp. 85â141.
3854:
3848:
3841:
3836:
3829:
3825:
3820:
3813:
3810:, pp. 52â53;
3809:
3804:
3797:
3793:
3788:
3785:, so the set
3784:
3780:
3776:
3772:
3768:
3764:
3760:
3756:
3752:
3748:
3742:
3735:
3730:
3723:
3718:
3711:
3706:
3699:
3694:
3687:
3683:
3678:
3671:
3664:
3657:
3653:
3648:
3640:
3634:
3630:
3626:
3625:
3617:
3610:
3606:
3605:Cantor (1897)
3602:
3601:Cantor (1895)
3597:
3590:
3585:
3578:
3573:
3567:, p. 4).
3566:
3562:
3554:
3549:
3545:
3541:
3532:
3527:
3523:
3516:
3513: â
3512:
3508:
3504:
3500:
3496:
3492:
3485:
3481:
3477:
3473:
3469:
3462:
3455:
3445:
3441:
3434:
3430:
3425:
3420:
3414:
3409:
3402:
3398:
3388:
3384:
3377:
3373:
3371:
3367: =
3366:
3359:
3352:
3348:
3342:
3322:
3318:
3314:
3310:
3306:
3302:
3298:
3297:
3289:
3283:
3277:
3270:
3267:
3263:
3259:
3253:
3246:
3240:
3233:
3227:
3219:
3215:
3211:
3207:
3203:
3199:
3193:
3189:
3183:
3176:
3172:
3167:
3158:
3147:
3143:
3139:
3135:
3131:
3127:
3123:
3116:
3109:
3101:
3097:
3093:
3089:
3084:
3079:
3075:
3071:
3070:
3065:
3058:
3056:
3047:
3043:
3039:
3035:
3031:
3027:
3020:
3013:
3009:
3006:
3001:
2993:
2992:countable set
2987:
2981:
2976:
2969:
2964:
2962:9780486616308
2958:
2954:
2953:
2948:
2942:
2934:
2930:
2926:
2922:
2918:
2914:
2907:
2905:
2897:
2892:
2885:
2880:
2873:
2869:
2863:
2856:
2851:
2844:
2839:
2832:
2828:
2823:
2816:
2811:
2809:
2801:
2796:
2789:
2784:
2777:
2773:
2768:
2762:
2758:
2753:
2746:
2741:
2734:
2729:
2721:
2714:
2706:
2702:
2698:
2692:
2688:
2684:
2683:
2675:
2673:
2671:
2669:
2667:
2665:
2663:
2661:
2645:
2641:
2635:
2633:
2631:
2629:
2627:
2618:
2612:
2608:
2602:
2587:
2583:
2577:
2570:
2565:
2557:
2551:
2547:
2543:
2542:
2534:
2532:
2524:
2523:Hilbert (1926
2519:
2512:
2507:
2505:
2497:
2493:
2488:
2481:
2477:
2472:
2470:
2462:
2457:
2455:
2447:
2443:
2439:
2434:
2427:
2422:
2420:
2418:
2410:
2405:
2403:
2395:
2391:
2386:
2384:
2376:
2371:
2369:
2360:
2358:9780691024479
2354:
2349:
2348:
2339:
2332:
2327:
2320:
2316:
2312:
2306:
2299:
2294:
2290:
2280:
2277:
2275:
2272:
2270:
2267:
2265:
2262:
2260:
2257:
2255:
2252:
2250:
2249:Countable set
2247:
2245:
2242:
2240:
2237:
2234:
2230:
2227:
2225:
2222:
2220:
2217:
2215:
2212:
2211:
2207:
2201:
2196:
2193:
2182:
2179:
2168:
2160:
2158:
2154:
2150:
2146:
2142:
2141:infinitesimal
2138:
2132:
2130:
2129:Joseph Dauben
2126:
2122:
2118:
2117:
2112:
2108:
2097:
2092:
2088:
2086:
2080:
2078:
2073:
2071:
2066:
2061:
2059:
2055:
2050:
2046:
2042:
2040:
2039:Protestantism
2036:
2032:
2027:
2023:
2019:
2011:
2006:
1997:
1995:
1991:
1990:Immanuel Kant
1987:
1983:
1979:
1975:
1971:
1970:contradiction
1966:
1963:
1958:
1956:
1953:
1949:
1944:
1942:
1937:
1931:
1929:
1928:Pope Leo XIII
1925:
1921:
1917:
1913:
1907:
1905:
1904:
1899:
1895:
1891:
1886:
1884:
1880:
1876:
1871:
1868:
1863:
1861:
1857:
1853:
1849:
1845:
1841:
1836:
1832:
1828:
1824:
1820:
1815:
1810:
1808:
1807:Joseph Dauben
1803:
1799:
1794:
1790:
1780:
1778:
1773:
1768:
1763:
1761:
1757:
1753:
1748:
1746:
1742:
1738:
1734:
1730:
1726:
1721:
1716:
1714:
1708:
1706:
1700:
1698:
1688:
1686:
1682:
1678:
1674:
1670:
1665:
1663:
1659:
1655:
1651:
1645:
1635:
1633:
1628:
1626:
1622:
1621:unit interval
1618:
1614:
1610:
1607:of copies of
1606:
1602:
1599:
1595:
1592:
1589:-dimensional
1588:
1584:
1580:
1576:
1575:Jakob Steiner
1572:
1567:
1565:
1560:
1555:
1553:
1548:
1544:
1540:
1536:
1532:
1528:
1519:
1514:
1504:
1502:
1498:
1494:
1490:
1486:
1482:
1478:
1474:
1470:
1467:
1463:
1459:
1455:
1451:
1447:
1443:
1442:
1432:
1428:
1426:
1422:
1418:
1414:
1410:
1406:
1402:
1398:
1394:
1389:
1387:
1383:
1379:
1375:
1371:
1367:
1363:
1362:
1356:
1354:
1350:
1346:
1341:
1340:
1334:
1331:
1327:
1320:
1313:
1306:
1303:
1299:
1294:
1292:
1288:
1284:
1280:
1276:
1272:
1268:
1260:
1256:
1251:
1242:
1240:
1236:
1233:
1229:
1225:
1221:
1217:
1216:Dedekind cuts
1213:
1209:
1205:
1200:
1198:
1194:
1190:
1183:
1176:
1171:
1167:
1160:
1153:
1146:
1142:
1137:
1133:
1128:
1124:
1119:
1115:
1110:
1106:
1102:
1098:
1091:
1084:
1080:
1076:
1071:
1067:
1064:
1060:
1056:
1052:
1048:
1044:
1040:
1036:
1032:
1028:
1024:
1020:
1019:number theory
1010:
1008:
1004:
1000:
996:
995:Adolf Hurwitz
992:
988:
984:
980:
979:David Hilbert
976:
975:
969:
967:
947:
939:
935:
911:
907:
903:
899:
895:
891:
887:
883:
879:
875:
871:
866:
864:
860:
859:linear orders
856:
852:
851:nowhere dense
848:
844:
840:
836:
831:
829:
825:
821:
820:infinite sets
817:
813:
809:
805:
801:
796:
794:
790:
786:
782:
778:
774:
770:
760:
758:
754:
749:
747:
746:
741:
737:
732:
730:
726:
725:
720:
716:
712:
707:
706:Ernst Zermelo
703:
699:
696:at the Third
695:
691:
687:
683:
673:
671:
667:
662:
660:
656:
651:
649:
645:
641:
640:Francis Bacon
637:
631:
627:
623:
621:
616:
612:
608:
604:
598:
596:
595:Franz Mertens
592:
588:
583:
581:
576:
571:
569:
565:
559:
557:
553:
549:
548:number theory
545:
535:
533:
529:
525:
521:
517:
513:
509:
505:
501:
497:
493:
489:
485:
481:
472:
458:
456:
452:
451:David Hilbert
448:
444:
443:Royal Society
439:
437:
433:
428:
426:
422:
418:
414:
410:
406:
402:
398:
394:
390:
386:
382:
378:
373:
371:
367:
363:
359:
355:
351:
347:
343:
339:
335:
331:
326:
318:
317:
308:
281:
272:
267:
262:
258:
252:
249:
247:
244:
243:
241:
239:
235:
228:
227:
223:
221:
217:
214:
211:
207:
204:
201:
197:
194:
190:
186:
183:
179:
160:
156:
153:
150:
146:
140:
137:
135:
132:
130:
127:
126:
124:
120:
117:
113:
110:
106:
101:
91:
87:
83:
79:
62:
58:
45:
40:
33:
30:
19:
8075:Georg Cantor
7959:Georg Cantor
7958:
7933:Möbius plane
7871:Infinite set
7815:Aleph number
7600:Georg Cantor
7599:
7595:Paul Bernays
7526:MorseâKelley
7501:
7434:
7433:Subset
7380:hereditarily
7342:Venn diagram
7300:ordered pair
7215:Intersection
7159:Axiom schema
6990:Chaos theory
6985:Kaleidoscope
6976:
6968:
6960:
6886:Gaston Julia
6866:Georg Cantor
6865:
6691:Escape-time
6623:Gosper curve
6571:LĂ©vy C curve
6556:Dragon curve
6435:Box-counting
6288:Hugh MacColl
6263:Georg Cantor
6262:
6258:George Boole
6155:Introduction
6111:modus ponens
6039:Higher-order
6034:Second-order
5984:Distribution
5944:Truth tables
5848:
5829:
5824:Georg Cantor
5810:
5788:
5764:Georg Cantor
5754:at Wikiquote
5752:Georg Cantor
5732:
5723:
5701:
5698:Rucker, Rudy
5676:
5665:
5657:
5653:
5642:
5624:
5599:
5595:
5571:
5568:Halmos, Paul
5550:
5525:
5521:
5503:
5476:the original
5442:
5438:
5426:the original
5405:
5401:
5381:
5377:
5345:
5341:
5329:
5307:
5303:
5277:(1): 51â58.
5274:
5270:
5241:
5237:
5208:
5204:
5175:
5171:
5150:
5146:
5110:
5104:
5069:
5046:
5041:Bibliography
5022:
4996:
4990:
4958:
4954:
4929:
4905:
4881:
4857:
4853:
4830:
4826:
4806:. Springer.
4803:
4784:
4765:
4745:
4741:
4719:
4711:
4707:
4690:
4668:
4641:
4637:
4611:
4600:
4584:
4572:
4564:
4559:
4551:
4547:
4538:
4532:
4520:
4512:
4507:
4495:
4470:
4445:
4441:
4433:
4428:
4415:
4402:
4397:, pp. 91â93.
4390:
4378:
4338:
4334:
4321:
4312:
4308:
4302:
4285:
4281:
4271:
4259:
4243:
4195:
4191:
4185:
4173:. Retrieved
4166:the original
4153:
4149:
4136:
4124:
4112:
4103:
4096:. Retrieved
4060:
4056:
4046:
4030:j.ctv10crfh1
4002:
3995:
3983:. Retrieved
3978:
3974:
3964:
3953:Hallett 1986
3948:
3936:
3924:
3913:Zermelo 1930
3908:
3901:Hallett 1986
3896:
3889:Hallett 1986
3885:Hallett 1986
3880:
3869:Zermelo 1908
3860:
3847:
3835:
3828:Purkert 1989
3819:
3803:
3796:Hallett 1986
3791:
3786:
3782:
3774:
3770:
3766:
3762:
3758:
3754:
3750:
3741:
3734:Hallett 1986
3729:
3717:
3705:
3700:, pp. 41â42.
3698:Hallett 1986
3693:
3677:
3663:
3655:
3647:
3623:
3616:
3596:
3584:
3572:
3557:
3552:
3547:
3543:
3536:
3530:
3525:
3521:
3514:
3510:
3506:
3502:
3498:
3494:
3490:
3483:
3475:
3471:
3464:
3457:
3450:
3443:
3436:
3432:
3428:
3423:
3418:
3412:
3407:
3400:
3393:
3386:
3382:
3375:
3368:
3361:
3354:
3346:
3341:
3328:. Retrieved
3321:the original
3300:
3294:
3276:
3269:coefficients
3252:
3239:
3226:
3201:
3197:
3182:
3166:
3157:
3146:the original
3128:(1): 1â121.
3125:
3121:
3108:
3076:(1): 51â89.
3073:
3067:
3029:
3025:
3019:
3000:
2995:"countable".
2986:
2975:
2966:
2951:
2941:
2919:(1): 55â62.
2916:
2912:
2891:
2879:
2862:
2850:
2838:
2822:
2795:
2783:
2767:
2760:
2752:
2740:
2728:
2713:
2681:
2647:. Retrieved
2643:
2606:
2601:
2591:14 September
2589:. Retrieved
2585:
2576:
2564:
2540:
2518:
2487:
2445:
2433:
2346:
2338:
2326:
2305:
2293:
2229:Cantor medal
2219:Aleph number
2136:
2134:
2114:
2104:
2094:
2090:
2084:
2082:
2074:
2065:Paul Tannery
2062:
2051:
2048:
2044:
2015:
1967:
1959:
1945:
1936:metaphysical
1932:
1912:Tilman Pesch
1908:
1901:
1897:
1889:
1887:
1882:
1872:
1864:
1852:Wittgenstein
1848:intuitionist
1823:intuitionism
1811:
1797:
1786:
1764:
1749:
1729:inconsistent
1717:
1709:
1701:
1694:
1666:
1657:
1653:
1647:
1629:
1619:between the
1608:
1600:
1598:real numbers
1593:
1586:
1568:
1551:
1546:
1539:line segment
1524:
1488:
1484:
1480:
1476:
1472:
1465:
1461:
1457:
1439:
1437:
1404:
1400:
1390:
1378:well-ordered
1365:
1360:
1359:
1357:
1337:
1335:
1330:intersection
1318:
1311:
1304:
1295:
1271:equinumerous
1264:
1201:
1196:
1192:
1188:
1181:
1174:
1169:
1165:
1158:
1151:
1144:
1140:
1135:
1131:
1126:
1122:
1117:
1113:
1108:
1104:
1100:
1097:limit points
1089:
1082:
1078:
1074:
1069:
1065:
1058:
1054:
1023:Eduard Heine
1016:
972:
970:
893:
892:, even when
889:
885:
881:
873:
867:
849:in 1875, is
832:
800:real numbers
797:
766:
750:
743:
733:
723:
694:Julius König
679:
663:
652:
633:
629:
624:
619:
614:
610:
606:
599:
587:Eduard Heine
584:
572:
560:
556:habilitation
544:dissertation
541:
528:Ernst Kummer
504:trigonometry
477:
440:
429:
389:Hermann Weyl
374:
354:real numbers
279:
278:
246:Ernst Kummer
224:
209:Institutions
192:
94:(1918-01-06)
74:3 March 1845
36:Georg Cantor
29:
8155:1918 deaths
8150:1845 births
8011:Mathematics
7820:Beth number
7625:Thomas Jech
7468:Alternative
7447:Uncountable
7401:Ultrafilter
7260:Cardinality
7164:replacement
7105:Determinacy
6981:(1987 book)
6973:(1986 book)
6965:(1982 book)
6951:Fractal art
6871:Bill Gosper
6835:LĂ©vy flight
6581:Peano curve
6576:Moore curve
6462:Topological
6447:Correlation
6233:Disjunction
6228:Conjunction
6213:Existential
6201:Elimination
6192:Disjunction
6187:Conjunction
6172:Existential
6029:First-order
5954:Truth value
5924:Quantifiers
5838:Thomas Jech
5602:: 161â190.
4605:Dauben 1979
4593:Dauben 1979
4525:Dauben 1979
4476:Dauben 1979
4421:Dauben 1979
4395:Dauben 1977
4383:Dauben 1979
4264:Dauben 1979
4252:Dauben 1977
4248:Cantor 1932
4236:Dauben 1977
4129:Dauben 1979
4117:Dauben 1979
3941:Dauben 1979
3929:Dauben 1979
3682:Cantor 1883
3652:Dauben 1979
3609:Cantor 1955
3589:Cantor 1883
3577:Dauben 1977
3565:Cantor 1879
3232:Suppes 1972
3171:Cantor 1874
2980:Cantor 1874
2896:Dauben 1979
2884:Dauben 1979
2872:Dauben 1979
2868:Dauben 1979
2855:Dauben 1979
2843:Dauben 1979
2827:Dauben 1979
2815:Dauben 1979
2800:Dauben 1979
2788:Dauben 1979
2774:, pp. 2â3;
2772:Dauben 1979
2757:Dauben 1977
2745:Dauben 1979
2733:Dauben 1979
2511:Dauben 1979
2492:Dauben 2004
2476:Dauben 1979
2461:Rodych 2007
2442:Dauben 1977
2438:Dauben 2004
2426:Dauben 1979
2409:Dauben 1977
2394:Dauben 1979
2390:Dauben 1977
2375:Dauben 2004
2331:Dauben 2004
2311:Dauben 1979
2101:Biographies
1879:determinism
1875:materialism
1867:neo-Thomist
1846:adopted an
1535:unit square
1446:Felix Klein
1397:cardinality
1063:derived set
839:cardinality
824:denumerable
757:World War I
484:Joseph Böhm
203:Mathematics
108:Nationality
53: 1910
8069:Categories
8059:Philosophy
7921:Geometries
7779:Set theory
7620:Kurt Gödel
7605:Paul Cohen
7442:Transitive
7210:Identities
7194:Complement
7181:Operations
7142:Regularity
7110:projective
7073:Adjunction
7032:Set theory
6789:Orbit trap
6784:Buddhabrot
6777:techniques
6765:Mandelbulb
6566:Koch curve
6499:Cantor set
6283:Kurt Gödel
6146:Absorption
6048:Principles
5934:Connective
5834:Set theory
5486:Fraenkel's
5178:(1): 1â7.
4910:BirkhÀuser
4627:References
4488:Aczel 2000
4063:(4): 548.
3917:Ewald 1996
3891:, p. 286.)
3865:Moore 1982
3853:Moore 1982
3840:Moore 1982
3808:Moore 1988
3792:successive
3722:Moore 1982
3710:Moore 1982
3686:Ewald 1996
3330:6 December
3262:polynomial
3175:Ewald 1996
3032:(4): 281.
2829:, p. 136;
2428:, chpt. 6.
2070:Josef Böhm
2018:Copenhagen
1994:Philistine
1962:opposition
1890:Grundlagen
1673:Paul Cohen
1669:Kurt Gödel
1625:continuous
1469:equivalent
1353:Cantor set
1245:Set theory
1224:Otto Stolz
1007:set theory
843:Cantor set
777:Set theory
769:set theory
729:Heidelberg
682:sanatorium
494:, then to
334:set theory
152:Set theory
70:1845-03-03
7902:Supertask
7553:Paradoxes
7473:Axiomatic
7452:Universal
7428:Singleton
7423:Recursive
7366:Countable
7361:Amorphous
7220:Power set
7137:Power set
7088:dependent
7083:countable
6896:Paul LĂ©vy
6775:Rendering
6760:Mandelbox
6706:Julia set
6618:Hexaflake
6549:Minkowski
6469:Recursion
6452:Hausdorff
6218:Universal
6177:Universal
6080:Explosion
6065:Bivalence
5994:Soundness
5939:Tautology
5929:Predicate
5700:(2005) .
5616:121888793
5570:(1998) .
5459:121665994
5422:177801164
5384:: 75â78.
5362:179178052
5324:121930608
5291:179177480
5258:177809016
5225:179177438
5192:179177510
5127:199545885
4999:: 29â47.
4975:120085563
4833:: 46â56.
4579:, p. 350.
4527:, p. 274.
4385:, p. 144.
4315:(3): 535.
4266:, p. 296.
4220:154486558
4131:, p. 266.
4038:241372960
3981:(1): 8â16
3943:, p. 120.
3931:, p. 295.
3875:, p. 202.
3442:} =
3399:} =
3142:123157068
3100:119250310
3083:1104.0375
3046:122744778
2898:, p. 284.
2857:, p. 283.
2817:, p. 282.
2802:, p. 139.
2790:, p. 138.
2735:, p. 163.
2649:6 October
2513:, p. 248.
2411:, p. 102.
2392:, p. 86;
2300:, p. 351.
1924:pantheism
1860:extension
1856:intension
1793:orthodoxy
1765:In 1923,
1720:paradoxes
1679:plus the
1613:dimension
1564:dimension
1513:Bijection
1427:in 1894.
1370:monograph
1293:in 1844.
1279:countable
1073:of a set
948:ω
920:ℵ
906:cardinals
904:, called
872:of a set
870:power set
855:rationals
773:Aristotle
731:in 1904.
500:Darmstadt
496:Frankfurt
492:Wiesbaden
461:Biography
421:pantheism
264:Signature
18:Cantorian
7800:0.999...
7692:Infinity
7557:Problems
7461:Theories
7437:Superset
7413:Infinite
7242:Concepts
7122:Infinity
7039:Overview
6806:fractals
6693:fractals
6661:L-system
6603:T-square
6411:Fractals
6162:Negation
5989:Validity
5969:Logicism
5869:Archived
5675:(2004).
5590:(1926).
5502:(2000).
5482:Dedekind
5386:Archived
5131:Archived
5011:Archived
4764:(2000).
4699:Archived
4465:, p. 15.
4423:, p. 96.
4367:Archived
4363:19040786
4355:15359485
4254:, p. 95.
4238:, p. 85.
4175:April 2,
4119:, p. 225
4093:26155985
3830:, p. 56.
3712:, p. 42.
3579:, p. 89.
3463:} where
3431: =
3385: =
3008:Archived
2949:(1972).
2759:, p. 89
2747:, p. 34.
2705:41497065
2444:, p. 89
2440:, p. 1;
2164:See also
2157:Veronese
2137:BeitrÀge
2026:Lutheran
1955:bacillus
1827:finitism
1697:absolute
1658:at least
1543:stronger
1409:cardinal
1386:cardinal
1302:sequence
1275:integers
1235:bacillus
1047:function
1027:analysis
910:ordinals
835:topology
793:topology
789:analysis
395:, while
366:cardinal
362:infinity
346:infinite
48:Cantor,
8035:Germany
8023:Judaism
7997:Portals
7718:Apeiron
7706:History
7495:General
7490:Zermelo
7396:subbase
7378: (
7317:Forcing
7295:Element
7267: (
7245:Methods
7132:Pairing
6755:Tricorn
6608:n-flake
6457:Packing
6440:Higuchi
6430:Assouad
5917:General
5772:at the
5530:Bibcode
5470:(ed.).
5067:(ed.).
4658:2708842
4098:5 March
3985:5 March
3381:. Then
3317:2975129
3266:integer
3206:Bibcode
3188:Galileo
2968:theory.
2933:3026799
2541:Hilbert
2333:, p. 1.
2125:Oedipal
1952:cholera
1654:exactly
1617:mapping
1605:product
1483:, then
1232:cholera
1057:in the
878:subsets
785:algebra
755:during
659:theorem
399:raised
370:ordinal
321:German:
175:
167:
116:Russian
7386:Filter
7376:Finite
7312:Family
7255:Almost
7093:global
7078:Choice
7065:Axioms
6854:People
6804:Random
6711:Filled
6679:H tree
6598:String
6486:system
6246:People
5850:BibNum
5710:
5685:
5633:
5614:
5580:
5559:
5512:
5457:
5420:
5360:
5322:
5289:
5256:
5223:
5190:
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5077:
5051:above.
5029:
4973:
4937:
4916:
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4791:
4772:
4726:
4677:
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4075:
4036:
4028:
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3351:subset
3315:
3140:
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2317:, and
2145:Thomae
1984:, and
1898:Ethica
1739:, and
1527:Crelle
1450:subset
1444:under
1395:: the
1380:sets.
1345:finite
1328:whose
816:finite
791:, and
717:, and
403:; see
231:(1867)
229:
220:Thesis
199:Fields
187:(1904)
181:Awards
158:Spouse
112:German
7478:Naive
7408:Fuzzy
7371:Empty
7354:types
7305:tuple
7275:Class
7269:large
7230:Union
7147:Union
6930:Other
6342:Works
6089:Rules
5612:S2CID
5455:S2CID
5418:S2CID
5389:(PDF)
5374:(PDF)
5358:S2CID
5320:S2CID
5287:S2CID
5254:S2CID
5221:S2CID
5188:S2CID
5134:(PDF)
5123:S2CID
5101:(PDF)
5014:(PDF)
4987:(PDF)
4971:S2CID
4887:49â65
4702:(PDF)
4695:(PDF)
4654:JSTOR
4370:(PDF)
4359:S2CID
4331:(PDF)
4216:S2CID
4208:JSTOR
4169:(PDF)
4146:(PDF)
4089:S2CID
4073:JSTOR
4034:S2CID
4026:JSTOR
3779:image
3324:(PDF)
3313:JSTOR
3291:(PDF)
3258:roots
3149:(PDF)
3138:S2CID
3118:(PDF)
3096:S2CID
3078:arXiv
3042:S2CID
2929:JSTOR
2285:Notes
2153:Stolz
1772:class
1713:aleph
1662:prove
1627:one.
1571:power
1300:as a
1103:. If
966:omega
938:aleph
808:sizes
646:(see
575:chair
169:(
165:
100:Halle
7391:base
6019:Term
5708:ISBN
5683:ISBN
5631:ISBN
5578:ISBN
5557:ISBN
5510:ISBN
5151:1878
5111:1874
5075:ISBN
5027:ISBN
4935:ISBN
4914:ISBN
4891:ISBN
4808:ISBN
4789:ISBN
4770:ISBN
4724:ISBN
4675:ISBN
4446:E.g.
4351:PMID
4192:Isis
4177:2013
4100:2020
4081:PMID
4057:Isis
4016:ISBN
3987:2020
3633:ISBN
3493:and
3474:and
3332:2013
3284:and
3190:and
2957:ISBN
2761:15n.
2701:OCLC
2691:ISBN
2651:2017
2611:ISBN
2593:2019
2550:ISBN
2353:ISBN
2151:and
1914:and
1877:and
1825:and
1687:").
1487:and
1475:and
1460:and
1411:and
1226:and
997:and
971:The
908:and
818:and
688:and
657:and
615:Acta
611:Acta
593:and
526:and
391:and
383:and
368:and
348:and
330:O.S.
316:-tor
89:Died
60:Born
7352:Set
5864:or
5836:by
5604:doi
5538:doi
5526:248
5447:doi
5410:doi
5350:doi
5312:doi
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4963:doi
4862:doi
4835:doi
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4290:doi
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4065:doi
4008:doi
3656:63n
3629:259
3509:if
3305:doi
3301:101
3260:of
3214:doi
3202:272
3130:doi
3088:doi
3034:doi
2921:doi
2546:177
2446:15n
2113:'s
1883:not
1685:ZFC
1210:of
1206:as
1185:Ï+1
1127:k+1
1109:k+1
1099:of
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1049:by
880:of
622:."
546:on
457:."
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932:(
894:A
890:A
886:A
882:A
874:A
307:/
301:t
298:n
295:ĂŠ
292:k
289:Ë
286:/
282:(
114:-
72:)
68:(
20:)
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