Georg Ferdinand Ludwig Phillip Cantor ( – January 6 1918) was a
GermanGermany , officially the Federal Republic of Germany , is a country in Central Europe. It is bordered to the north by the North Sea, Denmark, and the Baltic Sea; to the east by Poland and the Czech Republic; to the south by Austria and Switzerland; and to the west by France, Luxembourg, Belgium,...
mathematicianA mathematician is a person whose primary area of study and/or research is the field of mathematics. Mathematicians are concerned with particular problems related to logic, space, transformations, numbers and more general ideas which encompass these concepts...
, born in
RussiaRussia , officially known as both Russia and the Russian Federation , is a country in northern Eurasia . It is a semi-presidential republic, comprising 83 federal subjects...
. He is best known as the creator of
set theoryThe modern study of set theory was initiated by Cantor and Dedekind in the 1870s. After the discovery of paradoxes in informal set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.The...
, which has become a
fundamental theoryFoundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...
in mathematics. Cantor established the importance of one-to-one correspondence between sets, defined
infiniteIn set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. Some examples are:* the set of all integers, {..., -1, 0, 1, 2, ...}, is a countably infinite set; and...
and
well-ordered setsIn mathematics, a well-order relation on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering.Equivalently, a well-ordering is a well-founded total order....
, and proved that the
real numberIn mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real...
s are "more numerous" than the
natural numberIn mathematics, there are two conventions for the set of natural numbers: it is either the set of positive integers {, , , ...} according to the traditional definition or the set of non-negative integers {, 1, 2, ...} according to...
s. In fact,
Cantor's theoremIn elementary set theory, Cantor's theorem states that, for any set A, the set of all subsets of A has a strictly greater cardinality than A itself...
implies the existence of an "
infinityInfinity refers to several distinct concepts – usually linked to the idea of "without end" – which arise in philosophy, mathematics, and theology...
of infinities". He defined the
cardinalIn mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
and
ordinalIn set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware.
Cantor's theory of
transfinite numberTransfinite numbers are cardinal numbers or ordinal numbers that are larger than all finite numbers, yet not necessarily absolutely infinite. The term transfinite was coined by Georg Cantor, who wished to avoid some of the implications of the word infinite in connection with these objects, which...
s was originally regarded as so counter-intuitive—even shocking—that it encountered
resistanceIn mathematical logic, the theory of infinite sets was first developed by Georg Cantor. Although this work has found near-universal acceptance in the mathematics community, it has been criticized in several areas by mathematicians and philosophers....
from mathematical contemporaries such as
Leopold KroneckerLeopold Kronecker was a German mathematician and logician who argued that arithmetic and analysis must be founded on "whole numbers", saying, "God made the integers; all else is the work of man" . This put Kronecker in bitter opposition to some of the mathematical extensions of Georg Cantor,...
and
Henri PoincaréJules Henri Poincaré was a French mathematician and theoretical physicist, and a philosopher of science...
and later from
Hermann WeylHermann Klaus Hugo Weyl was a German mathematician. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His research has had major...
and
L. E. J. BrouwerLuitzen Egbertus Jan Brouwer [ˈlœyt.sən ɛx.ˈbɛʁ.təs jɑn ˈbʁʌu.əʁ] , usually cited as L. E. J...
, while
Ludwig WittgensteinLudwig Josef Johann Wittgenstein was an Austrian-British philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language....
raised philosophical objections. Some
Christian theologiansChristian theology is discourse concerning Christian faith. Christian theologians use Biblical exegesis, rational analysis and argument to understand, explain, test, critique, defend or promote Christianity...
(particularly
neo-ScholasticsNeo-Scholasticism is the revival and development from the second half of the nineteenth century of medieval scholastic philosophy. It has some times been called neo-Thomism partly because Thomas Aquinas in the 13th century gave to scholasticism a final form, partly because the idea gained ground...
) saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of
GodGod is a deity in theistic and deistic religions and other belief systems, representing either the sole deity in monotheism, or a principal deity in polytheism....
, on one occasion equating the theory of transfinite numbers with
pantheismPantheism is the view that everything is part of an all-encompassing immanent God and that the Universe and God are equivalent...
. The objections to his work were occasionally fierce: Poincaré referred to Cantor's ideas as a "grave disease" infecting the discipline of
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, and Kronecker's public opposition and personal attacks included describing Cantor as a "scientific charlatan", a "renegade" and a "corrupter of youth." 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
depressionMajor depressive disorder is a mental disorder characterized by an all-encompassing low mood accompanied by low self-esteem, and loss of interest or pleasure in normally enjoyable activities...
from 1884 to the end of his life were once blamed on the hostile attitude of many of his contemporaries, but these episodes can now be seen as probable manifestations of a
bipolar disorderBipolar disorder, also known as manic depressive disorder, manic depression or bipolar affective disorder, is a serious mental disorder that describes a category of mood disorders defined by the presence of one or more episodes of abnormally elevated mood clinically referred to as mania or, if...
.
The harsh criticism has been matched by later accolades. In 1904, the
Royal SocietyThe Royal Society of London for the Improvement of Natural Knowledge, known simply as the Royal Society, or even the Royal, is a learned society for science that was founded in 1660 and is considered by most to be the oldest such society still in existence...
awarded Cantor its
Sylvester MedalThe Sylvester Medal is a bronze medal awarded by the Royal Society for the encouragement of mathematical research, and accompanied by a £1,000 prize...
, the highest honor it can confer. Cantor believed his theory of transfinite numbers had been communicated to him by God.
David HilbertDavid Hilbert was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. He discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry...
defended it from its critics by famously declaring: "No one shall expel us from the Paradise that Cantor has created."
Youth and studies
Cantor was born in 1845 in the Western merchant colony in
Saint PetersburgSaint Petersburg is a city and a federal subject of Russia located on the Neva River at the head of the Gulf of Finland on the Baltic Sea. The city's other names were Petrograd and Leningrad...
,
RussiaRussia , officially known as both Russia and the Russian Federation , is a country in northern Eurasia . It is a semi-presidential republic, comprising 83 federal subjects...
, and brought up in the city until he was eleven. Georg, the eldest of six children, was an outstanding
violinThe violin is a bowed string instrument with four strings usually tuned in perfect fifths. It is the smallest and highest-pitched member of the violin family of string instruments, which also includes the viola and cello....
ist, having inherited his parents' considerable musical and artistic talents. Cantor's father had been a member of the Saint Petersburg stock exchange; when he became ill, the family moved to Germany in 1856, first to
WiesbadenWiesbaden is a city in southwestern Germany and the capital of the federal state of Hesse. It has about 275,400 inhabitants, plus approximately 10,000 United States citizens...
then to
FrankfurtFrankfurt am Main , commonly known simply as Frankfurt, is the largest city in the German state of Hesse and the fifth-largest city in Germany, with a 2008 population of 670,000. The urban area had an estimated population of 2.26 million in 2001...
, seeking winters milder than those of Saint Petersburg. In 1860, Cantor graduated with distinction from the Realschule in
DarmstadtDarmstadt is a city in the Bundesland of Hesse in Germany, located in the southern part of the Rhine Main Area.The city of Darmstadt was founded by the Counts of Katzenelnbogen in 1330, though settlement in the area is known to have been present as early as the late 11th century...
; his exceptional skills in mathematics,
trigonometryTrigonometry is a branch of mathematics that deals with triangles, particularly those plane triangles in which one angle has 90 degrees...
in particular, were noted. In 1862, Cantor entered the Federal Polytechnic Institute in
ZürichZürich or Zurich is the largest city in Switzerland and the capital of the canton of Zürich. The city is Switzerland's main commercial and cultural centre and sometimes called the Cultural Capital of Switzerland, the political capital of Switzerland being Berne...
, today the
ETH ZurichETH Zurich or Swiss Federal Institute of Technology Zurich is a science and technology university in the City of Zurich, Switzerland...
. After receiving a substantial inheritance upon his father's death in 1863, Cantor shifted his studies to the
University of BerlinThe Humboldt University of Berlin is Berlin's oldest university, founded in 1810 as the University of Berlin by the liberal Prussian educational reformer and linguist Wilhelm von Humboldt, whose university model has strongly influenced other European and Western universities...
, attending lectures by
Leopold KroneckerLeopold Kronecker was a German mathematician and logician who argued that arithmetic and analysis must be founded on "whole numbers", saying, "God made the integers; all else is the work of man" . This put Kronecker in bitter opposition to some of the mathematical extensions of Georg Cantor,...
,
Karl WeierstrassKarl Theodor Wilhelm Weierstrass was a German mathematician who is often cited as the "father of modern analysis".- Biography :...
and
Ernst KummerErnst Eduard Kummer was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a Gymnasium , where he inspired the mathematical career of Leopold Kronecker.Kummer was born in Sorau, Brandenburg...
. He spent the summer of 1866 at the
University of GöttingenThe University of Göttingen , known informally as Georgia Augusta, is a university in the city of Göttingen, Germany.It was founded in 1734 by George II, King of Great Britain and Elector of Hanover, and was then opened in 1737. The University of Göttingen soon grew in size and popularity...
, then and later a very important center for mathematical research. In 1867, Berlin granted him the
PhDDoctor of Philosophy, abbreviated PhD , for the Latin , meaning "teacher of philosophy", or alternatively, DPhil, for the equivalent , is an advanced academic degree awarded by universities...
for a
thesisA dissertation or thesis is a document submitted in support of candidature for a degree or professional qualification presenting the author's research and findings...
on
number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
,
De aequationibus secundi gradus indeterminatis.
Teacher and researcher
After teaching briefly in a Berlin girls' school, Cantor took up a position at the
University of HalleThe Martin Luther University of Halle-Wittenberg , also referred to as MLU, is a public, research-oriented university in the cities of Halle and Wittenberg within Saxony-Anhalt, Germany. MLU offers German and English programmes leading to academic degrees such as B. A., B. Sc., M. A., M. Sc., Ph....
, where he spent his entire career. He was awarded the requisite
habilitationHabilitation is the highest academic qualification a person can achieve by their own pursuit in certain European and Asian countries. Earned after obtaining a research doctorate Habilitation is the highest academic qualification a person can achieve by their own pursuit in certain European and...
for his thesis on number theory.
In 1874, Cantor married Vally Guttmann. They had six children, the last (Rudolph) born in 1886. Cantor was able to support a family despite modest academic pay, thanks to his inheritance from his father. During his honeymoon in the
Harz mountainsThe Harz is a mountain range in central Germany. It is the highest mountain chain in northern Germany occupying parts of the German states of Lower Saxony, Saxony-Anhalt and Thuringia. The name Harz derives from the Middle High German word Hardt or Hart...
, Cantor spent much time in mathematical discussions with
Richard DedekindJulius Wilhelm Richard Dedekind was a German mathematician who did important work in abstract algebra , algebraic number theory and the foundations of the real numbers.-Life:...
, whom he befriended two years earlier while on
SwissSwitzerland , officially the Swiss Confederation , is a federal republic consisting of 26 states named cantons, with Bern as the seat of the federal authorities...
holiday.
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
chairThe meaning of the word professor varies. In some English-speaking countries, it refers to a senior academic who holds a departmental chair, especially as head of the department, or a personal chair awarded specifically to that individual...
at a more prestigious university, in particular at Berlin, then 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 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, fundamentally disagreed with the thrust of Cantor's work. Kronecker, now seen as one of the founders of the
constructive viewpoint in mathematicsIn the philosophy of mathematics, constructivism asserts that it is necessary to find a mathematical object to prove that it exists...
, 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. Cantor came to believe that Kronecker's stance would make it impossible for Cantor to ever leave Halle.
In 1881, Cantor's Halle colleague
Eduard HeineHeinrich Eduard Heine was a German mathematician.Heine was born in Berlin, and became known for results on special functions and in real analysis. In particular, he authored an important treatise on spherical harmonics and Legendre functions . He also investigated basic hypergeometric series...
died, creating a vacant chair. Halle accepted Cantor's suggestion that it be offered to
DedekindJulius Wilhelm Richard Dedekind was a German mathematician who did important work in abstract algebra , algebraic number theory and the foundations of the real numbers.-Life:...
, Heinrich M. Weber and
Franz MertensFranz Mertens was a German mathematician. He was born in Środa in the Grand Duchy of Poznań, Kingdom of Prussia and died in Vienna, Austria....
, in that order, but each declined the chair after being offered it. Friedrich Wangerin was eventually appointed, but he was never close to Cantor.
In 1882 the mathematical correspondence between Cantor and Dedekind came to an end, apparently as a result of Dedekind's refusal to accept the chair at Halle. Cantor also began another important correspondence, with
Gösta Mittag-LefflerMagnus Gustaf Mittag-Leffler was a Swedish mathematician.Mittag-Leffler was born in Stockholm, son of the school principal John Olof Leffler and Gustava Wilhelmina Mittag; he later added his mother's maiden name to his paternal surname. His sister was the writer Anne Charlotte Leffler...
in Sweden, and soon began to publish in Mittag-Leffler's journal
Acta Mathematica. But in 1885, Mittag-Leffler was concerned about the philosophical nature and new terminology in a paper Cantor had submitted to
Acta. He asked Cantor to withdraw the paper from
Acta while it was in proof, writing that it was "... about one hundred years too soon." Cantor complied, but wrote to a third party:
Cantor then sharply curtailed his relationship and correspondence with Mittag-Leffler, displaying a tendency to interpret well-intentioned criticism as a deeply personal affront.
Cantor suffered his first known bout of depression in 1884. Criticism of his work weighed on his mind: every one of the fifty-two letters he wrote to Mittag-Leffler in 1884 attacked Kronecker. A passage from one of these letters is revealing of the damage to Cantor's self-confidence:
This emotional crisis led him to apply to lecture on
philosophyPhilosophy is the study of general and fundamental problems concerning matters such as existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing these questions by its critical, generally systematic approach and its reliance on reasoned...
rather than mathematics. He also began an intense study of
Elizabethan literatureThe term Elizabethan literature refers to the English literature produced during the reign of Queen Elizabeth I .The Elizabethan era saw a great flourishing of literature, especially in the field of drama...
in an attempt to prove that
Francis BaconFrancis Bacon,1st Viscount St Alban KC , son of Nicholas Bacon by his second wife Anne Bacon, was an English philosopher, statesman, scientist, lawyer, jurist, and author. He served both as Attorney General and Lord Chancellor of England...
wrote the plays attributed to
ShakespeareWilliam Shakespeare was an English poet and playwright, widely regarded as the greatest writer in the English language and the world's preeminent dramatist. He is often called England's national poet and the "Bard of Avon"...
(see Shakespearean authorship question); this ultimately resulted in two pamphlets, published in 1896 and 1897.
Cantor recovered soon thereafter, and subsequently made further important contributions, including his famous
diagonal argumentCantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers...
and
theoremIn elementary set theory, Cantor's theorem states that, for any set A, the set of all subsets of A has a strictly greater cardinality than A itself...
. However, he never again attained the high level of his remarkable papers of 1874–1884. He eventually sought a reconciliation with Kronecker, which Kronecker graciously accepted. Nevertheless, the philosophical disagreements and difficulties dividing them persisted. It was once thought that Cantor's recurring bouts of depression were triggered by the opposition his work met at the hands of Kronecker. While Cantor's mathematical worries and his difficulties dealing with certain people were greatly magnified by his depression, it is doubtful that they were its cause. Rather, his posthumous diagnosis of
bipolarityBipolar disorder, also known as manic depressive disorder, manic depression or bipolar affective disorder, is a serious mental disorder that describes a category of mood disorders defined by the presence of one or more episodes of abnormally elevated mood clinically referred to as mania or, if...
has been accepted as the
root causeA root cause is an initiating cause of a causal chain which leads to an outcome or effect of interest. Commonly, root cause is used to describe the depth in the causal chain where an intervention could reasonably be implemented to change performance and prevent an undesirable outcome.The term root...
of his erratic mood.
In 1890, Cantor was instrumental in founding the
Deutsche Mathematiker-Vereinigung and chaired its first meeting in Halle in 1891, where he first introduced his
diagonalIn mathematics, the term diagonalization may refer to:* A diagonalizable matrix, which can be put into a form with nonzero entries only on the main diagonal.* The diagonal lemma, used to create self-referential sentences in formal logic....
argument* In logic, an argument is a set of one or more meaningful declarative sentences known as the premises along with another meaningful declarative sentence known as the conclusion...
; 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 he felt towards Kronecker, Cantor invited him to address the meeting, but Kronecker was unable to do so because his spouse was dying from a skiing accident at the time.
Late years
After Cantor's 1884 hospitalization, there is no record that he was in any
sanatoriumA sanatorium is a medical facility for long-term illness, typically tuberculosis. A distinction is sometimes made between "sanitarium" and "sanatorium" .-History:The rationale for sanatoria was that before antibiotic treatments existed, a regimen of rest and good...
again until 1899. Soon after that second hospitalization, Cantor's youngest son Rudolph died suddenly (while Cantor was delivering a lecture on his views on
Baconian theoryThe Baconian theory of Shakespearean authorship holds that Sir Francis Bacon wrote the plays conventionally attributed to William Shakespeare....
and
William ShakespeareWilliam Shakespeare was an English poet and playwright, widely regarded as the greatest writer in the English language and the world's preeminent dramatist. He is often called England's national poet and the "Bard of Avon"...
), 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
Julius KönigJulius König was a Hungarian mathematician. He was born in in Győr, Hungary and died in Budapest.- Biography :König's name in Hungarian was Kőnig Gyula or in the more common European name order Gyula Kőnig, but when König contributed to German mathematical journals he called himself "Julius...
at the Third
International Congress of MathematiciansThe International Congress of Mathematicians is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union ....
. The paper attempted to prove that the basic tenets of transfinite set theory were false. Since it had been read in front of his daughters and colleagues, Cantor perceived himself as having been publicly humiliated. Although
Ernst ZermeloErnst Friedrich Ferdinand Zermelo was a German mathematician, whose work has major implications for the foundations of mathematics and hence on philosophy.-Life:...
demonstrated less than a day later that König's proof had failed, Cantor remained shaken, even 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 in various sanatoria. The 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 (
Burali-Forti paradoxIn set theory, a field of mathematics, the Burali-Forti paradox demonstrates that naively constructing "the set of all ordinal numbers" leads to a contradiction and therefore shows an antinomy in a system that allows its construction...
,
Cantor's paradoxIn set theory, Cantor's paradox is derivable from the theorem that there is no greatest cardinal number, so that the collection of "infinite sizes" is itself infinite...
, and
Russell's paradoxIn the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory of Frege leads to a contradiction....
) to a meeting of the
Deutsche Mathematiker–Vereinigung in 1903, and attending the International Congress of Mathematicians at Heidelberg in 1904.
In 1911, Cantor was one of the distinguished foreign scholars invited to attend the 500th anniversary of the founding of the University of St. Andrews in
ScotlandScotland is a country that is part of the United Kingdom. Occupying the northern third of the island of Great Britain, it shares a border with England to the south and is bounded by the North Sea to the east, the Atlantic Ocean to the north and west, and the North Channel and Irish Sea to the...
. Cantor attended, hoping to meet
Bertrand RussellBertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was an English philosopher, logician, mathematician, historian, and social critic. Although he spent the majority of his life in England, he was born in Wales, where he also died.Russell led the British "revolt against idealism" in the...
, whose newly published
Principia MathematicaThe Principia Mathematica is a 3-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912 & 1913...
repeatedly cited Cantor's work, but this did not come about. The following year, St. Andrews awarded Cantor an
honorary doctorateAn honorary degree or a degree honoris causa is an academic degree for which a university has waived the usual requirements...
, but illness precluded his receiving the degree in person.
Cantor retired in 1913, and suffered from poverty, even malnourishment, during
World War IWorld War I , also known as the First World War, the Great War, and the War to End All Wars, was a global military conflict which involved most of the world's great powers, assembled in two opposing alliances: the Triple Entente and the Triple Alliance...
. The public celebration of his 70th birthday was canceled because of the war. He died on January 6 1918 in the sanatorium where he had spent the final year of his life.
Mathematical work
Cantor's work between 1874 and 1884 is the origin of
set theoryThe modern study of set theory was initiated by Cantor and Dedekind in the 1870s. After the discovery of paradoxes in informal set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.The...
. Prior to this work, the concept of a set was a rather elementary one that had been used implicitly since the beginnings of mathematics, dating back to the ideas of
AristotleAristotle was a Greek philosopher, a student of Plato and teacher of Alexander the Great. He wrote on many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, politics, government, ethics, biology, and zoology.Together with Plato and Socrates , Aristotle is one of...
. 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. Set theory has come to play the role of a
foundational theoryFoundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...
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
algebraAlgebra is the branch of mathematics concerning the study of the rules of operations and the things which can be constructed from them, including terms, polynomials, equations and algebraic structures...
,
analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of pure mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function...
and
topologyTopology is a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects, for example deformations that involve stretching, but no tearing or gluing...
) 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.
In one of his earliest papers, Cantor proved that the set of
real numberIn mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real...
s is "more numerous" than the set of
natural numberIn mathematics, there are two conventions for the set of natural numbers: it is either the set of positive integers {, , , ...} according to the traditional definition or the set of non-negative integers {, 1, 2, ...} according to...
s; this showed, for the first time, that there exist infinite sets of different
sizesIn mathematics, the cardinality of a set is a measure of the "number of elements of the set". For example, the set A = {2, 4, 6} contains 3 elements, and therefore A has a cardinality of 3...
. He was also the first to appreciate the importance of one-to-one correspondences (hereinafter denoted "1-to-1") in set theory. He used this concept to define
finiteIn mathematics, finite set is a set that has a finite number of elements. For example,is a finite set with five elements. The number of elements of a finite set is a natural number , and is called the cardinality of the set. A set that is not finite is called infinite...
and
infinite setIn set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. Some examples are:* the set of all integers, {..., -1, 0, 1, 2, ...}, is a countably infinite set; and...
s, subdividing the latter into
denumerableIn mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable...
(or countably infinite) sets and
uncountable setIn mathematics, an uncountable set is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the natural numbers.-Characterizations:There are many...
s (nondenumerable infinite sets).
Cantor introduced fundamental constructions in set theory, such as the
power setIn mathematics, given a set S, the power set of S, written , P, ℘ or 2
S, is the set of all subsets of S. In axiomatic set theory In mathematics, given a set S, the power set (or powerset) of S, written , P(S), ℘(S) or 2
S, is the set of all subsets of S. In...
of a set
A, which is the set of all possible
subsetIn mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide...
s of
A. He later proved that the size of the power set of
A is strictly larger than the size of
A, even when
A is an infinite set; this result soon became known as
Cantor's theoremIn elementary set theory, Cantor's theorem states that, for any set A, the set of all subsets of A has a strictly greater cardinality than A itself...
. Cantor developed an entire theory and
arithmetic of infinite setsIn the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an explicit well-ordered set which represents the...
, called
cardinalIn mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
s and
ordinalIn set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
s, which extended the arithmetic of the natural numbers. His notation for the cardinal numbers was the Hebrew letter (
alephIn set theory, the aleph numbers are a sequence of numbers used to represent the cardinality of infinite sets. They are named after the symbol used to denote them, the Hebrew letter aleph ....
) with a natural number subscript; for the ordinals he employed the Greek letter ω (
omegaOmega is the 24th and last letter of the Greek alphabet. In the Greek numeric system, it has a value of 800. The word literally means "great O" , as opposed to Omicron, which means "little O"...
). This notation is still in use today.
The
Continuum hypothesisIn mathematics, the continuum hypothesis is a hypothesis, advanced by Georg Cantor in 1877, about the possible sizes of infinite sets. It states:Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's twenty-three problems presented in the year 1900...
, introduced by Cantor, was presented by
David HilbertDavid Hilbert was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. He discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry...
as the first of his
twenty-three open problemsHilbert's problems are a list of twenty-three problems in mathematics published by German mathematician David Hilbert during 1900. The problems were all unsolved at the time, and several of them were very influential for 20th century mathematics...
in his famous address at the 1900
International Congress of MathematiciansThe International Congress of Mathematicians is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union ....
in
ParisParis is the capital of France and the country's most populous city. It is situated on the river Seine, in northern France, at the heart of the Île-de-France region...
. Cantor's work also attracted favorable notice beyond Hilbert's celebrated encomium. The US philosopher Charles Sanders Peirce praised Cantor's set theory, and, following public lectures delivered by Cantor at the first International Congress of Mathematicians, held in Zurich in 1897,
HurwitzAdolf Hurwitz , was a German mathematician, and was described by Jean-Pierre Serre as "one of the most important figures in mathematics in the second half of the nineteenth century".-Early life:...
and
HadamardJacques Salomon Hadamard was a French mathematician who made major contributions in number theory, complex function theory, differential geometry and partial differential equations.-Biography:...
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
Philip JourdainPhilip Edward Bertrand Jourdain was a British logician and follower of Bertrand Russell.He was born in Ashbourne in Derbyshire one of a large family belonging to Emily Clay and his father Francis Jourdain . He was partly disabled by Friedreich's ataxia...
on the history of
set theoryThe modern study of set theory was initiated by Cantor and Dedekind in the 1870s. After the discovery of paradoxes in informal set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.The...
and on Cantor's religious ideas. This was later published, as were several of his expository works.
Number theory and function theory
Cantor's first ten papers were on
number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
, his thesis topic. At the suggestion of
Eduard HeineHeinrich Eduard Heine was a German mathematician.Heine was born in Berlin, and became known for results on special functions and in real analysis. In particular, he authored an important treatise on spherical harmonics and Legendre functions . He also investigated basic hypergeometric series...
, the Professor at Halle, Cantor turned to
analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of pure mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function...
. Heine proposed that Cantor solve
an open problemIn science and mathematics, an open problem or an open question is a known problem that can be accurately stated, and has not yet been solved...
that had eluded
DirichletJohann Peter Gustav Lejeune Dirichlet was a German mathematician credited with the modern formal definition of a function.- Biography :...
,
LipschitzRudolf Otto Sigismund Lipschitz was a German mathematician and professor at the University of Bonn from 1864. Peter Gustav Dirichlet was his teacher. He supervised the early work of Felix Klein....
,
Bernhard Riemannwas an influential German mathematician who made contributions to analysis and differential geometry, some of them enabling the later development of general relativity.-Early life:...
, and Heine himself: the uniqueness of the representation of a
functionIn mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...
by
trigonometric seriesIn mathematics, a trigonometric series is any series of the form:It is called a Fourier series when the terms and have the form:where is an integrable function....
. Cantor solved this difficult problem in 1869. Between 1870 and 1872, Cantor published more papers on trigonometric series, including one defining
irrational numberIn mathematics, an irrational number is any real number that is not a rational number—that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero. Informally, this means numbers that cannot be represented as simple fractions...
s as
convergent sequencesIn functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers...
of
rational numberIn mathematics, a rational number is any number that can be expressed as the quotient a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer corresponds to a rational number. The set of all rational numbers is usually denoted .Formally each rational...
s. 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
Dedekind cutIn mathematics, a Dedekind cut, named after Richard Dedekind, is a partition of the rational numbers into two non-empty parts A and B, such that all elements of A are less than all elements of B and A contains no greatest element. The cut itself is, conceptually, the "gap" defined between A and B...
s.
Set theory
The beginning of set theory as a branch of mathematics is often marked by the publication of Cantor's 1874 paper, "Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" ("On a Characteristic Property of All Real Algebraic Numbers"). The paper, published in
Crelle's JournalCrelle's Journal, or just Crelle, is the common name for a leading mathematical journal, the Journal für die reine und angewandte Mathematik .- History :...
thanks to Dedekind's support (and despite Kronecker's opposition), was the first to formulate a mathematically rigorous proof that there was more than one kind of infinity. This demonstration is a centerpiece of his legacy as a mathematician, helping lay the groundwork for both calculus and the analysis of the continuum of real numbers. Previously, all infinite collections had been implicitly assumed to be
equinumerousIn mathematics, two sets are equinumerous if they have the same cardinality, i.e., if there exists a bijection f : A → B for sets A and B. This is usually denoted or ....
(that is, of "the same size" or having the same number of elements). He then proved that the real numbers were
not countableGeorg Cantor's first uncountability proof demonstrates that the set of all real numbers is uncountable. Cantor formulated the proof in December 1873 and published it in 1874 in Crelle's Journal, more formally known as the Journal für die Reine und Angewandte Mathematik...
, albeit employing a proof more complex than the remarkably elegant and justly celebrated
diagonal argumentCantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers...
he set out in 1891. Prior to this, he had already proven that the set of
rational numberIn mathematics, a rational number is any number that can be expressed as the quotient a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer corresponds to a rational number. The set of all rational numbers is usually denoted .Formally each rational...
s is countable.
Joseph LiouvilleJoseph Liouville was a French mathematician.- Life and work :Liouville graduated from the École Polytechnique in 1827. After some years as an assistant at various institutions including the Ecole Centrale Paris, he was appointed as professor at the École Polytechnique in 1838...
had established the existence of
transcendental numberIn mathematics, a transcendental number is a number that is not algebraic, that is, not a solution of a non-constant polynomial equation with rational coefficients....
s in 1844, and Cantor's paper established that the set of transcendental numbers is uncountable. The logic is as follows: Cantor had shown that the union of two countable sets must be countable. The set of all real numbers is equal to the union of the set of algebraic numbers with the set of transcendental numbers (that is, every real number must be either algebraic or transcendental). The 1874 paper showed that the
algebraic numberIn mathematics, an algebraic number is a complex number that is a root of a non-zero polynomial in one variable with rational coefficients...
s (that is, the
rootIn vascular plants, the root is the organ of a plant that typically lies below the surface of the soil. This is not always the case, however, since a root can also be aerial or aerating . Furthermore, a stem normally occurring below ground is not exceptional either...
s of
polynomialIn mathematics, a polynomial is a finite length expression constructed from variables and constants, by using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents...
equations with
integerThe integers are natural numbers including 0 and their negatives . They are numbers that can be written without a fractional or decimal component, and fall within the set {.....
coefficientIn mathematics, a coefficient is a constant multiplicative factor of a specific object. For example, in the expression 9x2, the coefficient of x2 is 9.The object can be such things as a variable, a vector, a function, etc...
s), were countable. In contrast, Cantor had also just shown that the
real numberIn mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real...
s were
not countable. If transcendental numbers were countable, then the result of their union with algebraic numbers would also be countable. Since their union (which equals the set of all real numbers) is
uncountable, it logically follows that the transcendentals must be uncountable. The transcendentals have the same "power" (see below) as the reals, and "almost all" real numbers must be transcendental. Cantor remarked that he had effectively reproved a theorem, due to
LiouvilleJoseph Liouville was a French mathematician.- Life and work :Liouville graduated from the École Polytechnique in 1827. After some years as an assistant at various institutions including the Ecole Centrale Paris, he was appointed as professor at the École Polytechnique in 1838...
, to the effect that there are infinitely many transcendental numbers in each interval.
Between 1879 and 1884, Cantor published a series of six articles in
Mathematische AnnalenThe Mathematische Annalen is a German mathematical research journal published by Springer Science+Business Media...
that together formed an introduction to his set theory. At the same time, there was growing opposition to Cantor's ideas, led by Kronecker, who admitted mathematical concepts only if they could be constructed in a
finiteIn the philosophy of mathematics, finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps...
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
actual infinityActual infinity is the idea that numbers, or some other type of mathematical object, can form an actual, completed totality; namely, a set. Hence, in the philosophy of mathematics, the abstraction of actual infinity involves the acceptance of infinite entities, such as the set of all natural...
would open the door to paradoxes which would challenge the validity of mathematics as a whole. Cantor also discovered the
Cantor setIn mathematics, the Cantor set, introduced by German mathematician Georg Cantor in 1883 , is a set of points lying on a single line segment that has a number of remarkable and deep properties. Through consideration of it, Cantor and others helped lay the foundations of modern general topology...
during this period.
The fifth paper in this series, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre" ("Foundations of a General Theory of Aggregates"), published in 1883, was the most important of the six and was also published as a separate
monographA monograph is a work of writing upon a single subject, usually also by a single author. It is often a scholarly essay or learned treatise, and may be released in the manner of a book or journal article. It is by definition a single document that forms a complete text in itself...
. It contained Cantor's reply to his critics and showed how the
transfinite numberTransfinite numbers are cardinal numbers or ordinal numbers that are larger than all finite numbers, yet not necessarily absolutely infinite. The term transfinite was coined by Georg Cantor, who wished to avoid some of the implications of the word infinite in connection with these objects, which...
s were a systematic extension of the natural numbers. It begins by defining
well-orderIn mathematics, a well-order relation on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering.Equivalently, a well-ordering is a well-founded total order....
ed sets.
Ordinal numberIn set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
s are then introduced as the order types of well-ordered sets. Cantor then defines the addition and multiplication of the
cardinalIn mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
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.
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
Cantor's theoremIn elementary set theory, Cantor's theorem states that, for any set A, the set of all subsets of A has a strictly greater cardinality than A itself...
: the
cardinalityIn mathematics, the cardinality of a set is a measure of the "number of elements of the set". For example, the set A = {2, 4, 6} contains 3 elements, and therefore A has a cardinality of 3...
of the power set of a set
A is strictly larger than the cardinality of
A. This established the richness of the hierarchy of infinite sets, and of the cardinal and
ordinal arithmeticIn the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an explicit well-ordered set which represents the...
that Cantor had defined. His argument is fundamental in the solution of the
Halting problemIn computability theory, the halting problem is a decision problem which can be stated as follows: given a description of a program, decide whether the program finishes running or will run forever...
and the proof of Gödel's first incompleteness theorem.
In 1895 and 1897, Cantor published a two-part paper in
Mathematische AnnalenThe Mathematische Annalen is a German mathematical research journal published by Springer Science+Business Media...
under
Felix KleinFelix Christian Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...
's editorship; these were his last significant papers on set theory. The first paper begins by defining set,
subsetIn mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide...
, 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 well-ordered sets and ordinal numbers. Cantor attempts to prove that if
A and
B are sets with
A equivalent to a subset of
B and
B equivalent to a subset of
A, then
A and
B are equivalent.
Ernst SchröderErnst Schröder was a German mathematician mainly known for his work on algebraic logic. He is a major figure in the history of mathematical logic , by virtue of summarizing and extending the work of George Boole, Augustus De Morgan, Hugh MacColl, and especially Charles Peirce...
had stated this theorem a bit earlier, but his proof, as well as Cantor's, was flawed.
Felix BernsteinFelix Bernstein was a German mathematician known for developing a theorem of the equivalence of sets in 1897, and less well known for demonstrating the correct blood group inheritance pattern of multiple alleles at one locus in 1924 through statistical analysis...
supplied a correct proof in his 1898 PhD thesis; hence the name
Cantor–Bernstein–Schroeder theoremIn set theory, the Cantor–Bernstein–Schroeder theorem, named after Georg Cantor, Felix Bernstein, and Ernst Schröder, states that, if there exist injective functions f : A → B and g : B → A between the sets A and B, then there exists a bijective function h : A → B...
.
One-to-one correspondence
Cantor's 1874 Crelle paper was the first to invoke the notion of a
1-to-1In mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property that, for every y in Y, there is exactly one x in X such that f = y and no unmapped element remains in both X and Y.Alternatively, f is bijective if it is a one-to-one correspondence...
correspondence, though he did not use that phrase. He then began looking for a 1-to-1 correspondence between the points of the
unit squareIn mathematics, a unit square is a square whose sides have length 1. Often, “the” unit square refers specifically to the square in the Cartesian plane with corners at , , , and .-In the real plane:...
and the points of a unit
line segmentIn geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment...
. In an 1877 letter to Dedekind, Cantor proved a far stronger result: for any positive integer
n, there exists a 1-to-1 correspondence between the points on the unit line segment and all of the points in an
n-dimensional spaceIn mathematics, an n-dimensional space is a topological space whose dimension is n . The archetypical example is n-dimensional Euclidean space, which describes Euclidean geometry in n dimensions....
. About this discovery Cantor famously wrote to Dedekind: "
Je le vois, mais je ne le crois pas!" ("I see it, but I don't believe it!") The result that he found so astonishing has implications for geometry and the notion of
dimensionIn mathematics and physics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify each point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
.
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 "
powerIn mathematics, the cardinality of a set is a measure of the "number of elements of the set". For example, the set A = {2, 4, 6} contains 3 elements, and therefore A has a cardinality of 3...
" (a term he took from
Jakob SteinerJakob Steiner was a Swiss mathematician.He was born in the village of Utzenstorf, Canton of Bern. At eighteen he became a pupil of Heinrich Pestalozzi, and afterwards studied at Heidelberg. Thence he went to Berlin, earning a livelihood there, as in Heidelberg, by tutoring. Here he became...
) or "equivalence" of sets: two sets are
equivalentIn mathematics, an equivalence relation is, loosely, a binary relation on a set that specifies how to split up the set into subsets such that every element of the larger set is in exactly one of the subsets...
(have the same power) if there exists a 1-to-1 correspondence between them. Cantor defined
countable setIn mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable...
s (or denumerable sets) as sets which can be put into a 1-to-1 correspondence with the
natural numberIn mathematics, there are two conventions for the set of natural numbers: it is either the set of positive integers {, , , ...} according to the traditional definition or the set of non-negative integers {, 1, 2, ...} according to...
s, and proved that the rational numbers are denumerable. He also proved that
n-dimensional
Euclidean spaceIn mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions higher dimensions...
Rn has the same power as the
real numberIn mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real...
s
R, as does a countably infinite
productIn mathematics, a Cartesian product is the direct product of two sets. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to this concept....
of copies of
R. 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
dimensionIn mathematics and physics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify each point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
, stressing that his
mappingIn mathematics and related technical fields, the term map or mapping is often a synonym for function. Thus, for example, a partial map is a partial function, and a total map is a total function. Related terms like domain, codomain, injective, continuous, etc...
between the
unit intervalIn mathematics, the unit interval is the closed interval [0,1], that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1...
and the unit square was not a
continuousIn mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous. A continuous function with a continuous inverse function is called bicontinuous...
one.
This paper, like the 1874 paper, displeased Kronecker, and Cantor wanted to withdraw it; however, Dedekind persuaded him not to do so and
WeierstrassKarl Theodor Wilhelm Weierstrass was a German mathematician who is often cited as the "father of modern analysis".- Biography :...
also supported its publication. Nevertheless, Cantor never again submitted anything to Crelle.
Continuum hypothesis
Cantor was the first to formulate what later came to be known as the
continuum hypothesisIn mathematics, the continuum hypothesis is a hypothesis, advanced by Georg Cantor in 1877, about the possible sizes of infinite sets. It states:Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's twenty-three problems presented in the year 1900...
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
exactly aleph-one, rather than just
at least aleph-one). Cantor believed the continuum hypothesis to be true and tried for many years to
proveIn mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
it, in vain. His inability to prove the continuum hypothesis caused him considerable anxiety.
The difficulty Cantor had in proving the continuum hypothesis has been underscored by later developments in the field of mathematics: a 1940 result by
GödelKurt Gödel was an Austrian-American logician, mathematician and philosopher. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when many, such as Bertrand Russell, A. N...
and a 1963 one by
Paul CohenPaul Joseph Cohen was an American mathematician best known for his proof of the independence of the continuum hypothesis and the axiom of choice from Zermelo–Fraenkel set theory, the most widely accepted axiomatization of set theory.- Early years :Paul J. Cohen was born in Long Branch, New Jersey...
together imply that the continuum hypothesis can neither be proved nor disproved using standard
Zermelo–Fraenkel set theoryZermelo–Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics...
plus the
axiom of choiceIn mathematics, the axiom of choice, or AC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin, even if there are infinitely many bins and...
(the combination referred to as "ZFC").
Paradoxes of set theory
Discussions of set-theoretic
paradoxA paradox is a statement or group of statements that leads to a contradiction or a situation which defies intuition. The term is also used for an apparent contradiction that actually expresses a non-dual truth...
es began to appear around the end of the nineteenth century. Some of these implied fundamental problems with Cantor's set theory program. In an 1897 paper on an unrelated topic,
Cesare Burali-FortiCesare Burali-Forti was an Italian mathematician.He was born in Arezzo, and was an assistant of Giuseppe Peano in Turin from 1894 to 1896, during which time he discovered what came to be called the Burali-Forti paradox of Cantorian set theory. He died in Turin.-Books by C. Burali-Forti:* with R....
set out the first such paradox, the
Burali-Forti paradoxIn set theory, a field of mathematics, the Burali-Forti paradox demonstrates that naively constructing "the set of all ordinal numbers" leads to a contradiction and therefore shows an antinomy in a system that allows its construction...
: the
ordinal numberIn set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
of the set of all ordinals must be an ordinal and this leads to a contradiction. Cantor discovered this paradox in 1895, and described it in an 1896 letter to
HilbertDavid Hilbert was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. He discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry...
. Criticism mounted to the point where Cantor launched counter-arguments in 1903, intended to defend the basic tenets of his set theory.
In 1899, Cantor discovered his eponymous
paradoxIn set theory, Cantor's paradox is derivable from the theorem that there is no greatest cardinal number, so that the collection of "infinite sizes" is itself infinite...
: what is the cardinal number of the set of all sets? Clearly it must be the greatest possible cardinal. Yet for any set
A, the cardinal number of the power set of
A is strictly larger than the cardinal number of
A (this fact is now known as
Cantor's theoremIn elementary set theory, Cantor's theorem states that, for any set A, the set of all subsets of A has a strictly greater cardinality than A itself...
). This paradox, together with Burali-Forti's, led Cantor to formulate a concept called
limitation of sizeIn the philosophy of mathematics, specifically the philosophical foundations of set theory, limitation of size is a concept developed by Philip Jourdain and/or Georg Cantor to avoid Cantor's paradox. It identifies certain "inconsistent multiplicities", in Cantor's terminology, that cannot be sets...
, according to which the collection of all ordinals, or of all sets, was an "inconsistent multiplicity" that was "too large" to be a set. Such collections later became known as
proper classesIn set theory and its applications throughout mathematics, a class is a collection of sets which can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context...
.
One common view among mathematicians is that these paradoxes, together with
Russell's paradoxIn the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory of Frege leads to a contradiction....
, demonstrate that it is not possible to take a "naive", or non-axiomatic, approach to set theory without risking contradiction, and it is certain that they were among the motivations for
ZermeloErnst Friedrich Ferdinand Zermelo was a German mathematician, whose work has major implications for the foundations of mathematics and hence on philosophy.-Life:...
and others to produce axiomatizations of set theory. Others note, however, that the paradoxes do not obtain in an informal view motivated by the
iterative hierarchyIn set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted V, is the class of well-founded sets...
, which can be seen as explaining the idea of limitation of size. Some also question whether the
FregeanFriedrich Ludwig Gottlob Frege was a German mathematician who became a logician and philosopher. He was one of the founders of modern logic, and made major contributions to the foundations of mathematics. As a philosopher, he is generally considered to be the father of analytic philosophy, for his...
formulation of
naive set theoryNaive set theory is one of several theories of sets used in the discussion of the foundations of mathematics. The informal content of this naive set theory supports both the aspects of mathematical sets familiar in discrete mathematics , and the everyday usage of set theory concepts in most...
(which was the system directly refuted by the Russell paradox) is really a faithful interpretation of the Cantorian conception.
Philosophy, religion and Cantor's mathematics
The concept of the existence of an
actual infinityActual infinity is the idea that numbers, or some other type of mathematical object, can form an actual, completed totality; namely, a set. Hence, in the philosophy of mathematics, the abstraction of actual infinity involves the acceptance of infinite entities, such as the set of all natural...
was an important shared concern within the realms of mathematics, philosophy and religion. Preserving the
orthodoxyThe word orthodox, from Greek orthodoxos "having the right opinion", from orthos + doxa , is typically used to mean adhering to the accepted or traditional and established faith, especially in religion.The term did not conventionally exist with any degree of formality The word orthodox, from Greek...
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
Grundlagen einer allgemeinen Mannigfaltigkeitslehre, 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—he identified the
Absolute InfiniteThe Absolute Infinite is mathematician Georg Cantor's concept of an "infinity" that transcended the transfinite numbers. Cantor equated the Absolute Infinite with God...
with
GodGod is a deity in theistic and deistic religions and other belief systems, representing either the sole deity in monotheism, or a principal deity in polytheism....
, 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.
Debate among mathematicians grew out of opposing views in the
philosophy of mathematicsThe philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of...
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 (
constructivismIn the philosophy of mathematics, constructivism asserts that it is necessary to find a mathematical object to prove that it exists...
and its two offshoots,
intuitionismIn the philosophy of mathematics, intuitionism, or neointuitionism , is an approach to mathematics as the constructive mental activity of humans. That is, mathematics does not consist of analytic activities wherein deep properties of existence are revealed and applied...
and
finitismIn the philosophy of mathematics, finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps...
) 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 nonconstructive proofs such as Cantor's diagonal argument are sufficient proof that something exists, holding instead that
constructive proofIn mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object with certain properties by creating or providing a method for creating such an object...
s 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 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
BrouwerLuitzen Egbertus Jan Brouwer [ˈlœyt.sən ɛx.ˈbɛʁ.təs jɑn ˈbʁʌu.əʁ] , usually cited as L. E. J...
and especially
PoincaréJules Henri Poincaré was a French mathematician and theoretical physicist, and a philosopher of science...
adopted an
intuitionistIn the philosophy of mathematics, intuitionism, or neointuitionism , is an approach to mathematics as the constructive mental activity of humans. That is, mathematics does not consist of analytic activities wherein deep properties of existence are revealed and applied...
stance against Cantor's work. Citing the paradoxes of set theory as an example of its fundamentally flawed nature, Poincaré held that "most of the ideas of Cantorian set theory should be banished from mathematics once and for all." Finally,
WittgensteinLudwig Josef Johann Wittgenstein was an Austrian-British philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language....
's attacks were finitist: he believed that Cantor's diagonal argument conflated the
intensionIn linguistics, logic, philosophy, and other fields, an intension is any property or quality connoted by a word, phrase or other symbol. In the case of a word, it is often implied by the word's definition...
of a set of cardinal or real numbers with its
extensionIn any of several studies that treat the use of signs, for example in linguistics, logic, mathematics, semantics, and semiotics, the extension of a concept, idea, or sign consists of the things to which it applies, in contrast with its comprehension or intension, which consists very roughly of the...
, thus conflating the concept of rules for generating a set with an actual set.
Christian theologians saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God. In particular,
Neo-ThomistNeo-Scholasticism is the revival and development from the second half of the nineteenth century of medieval scholastic philosophy. It has some times been called neo-Thomism partly because Thomas Aquinas in the 13th century gave to scholasticism a final form, partly because the idea gained ground...
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:
Cantor also believed that his theory of transfinite numbers ran counter to both
materialismThe philosophy of materialism holds that the only thing that exists is matter; that all things are composed of material and all phenomena are the result of material interactions. In other words, matter is the only substance. As a theory, materialism is a form of physicalism and belongs to the...
and
determinismDeterminism is the view that every event, including human cognition, behavior, decision, and action, is causally determined by an unbroken chain of prior occurrences. With numerous historical debates, many varieties and philosophical positions on the subject of determinism exist from...
—and was shocked when he realized that he was the only faculty member at Halle who did
not hold to deterministic philosophical beliefs.
In 1888, Cantor published his correspondence with several philosophers on the philosophical implications of his set theory. In an extensive attempt to persuade Christian thinkers and authorities to adopt his views, Cantor had corresponded with Christian philosophers such as
Tilman PeschTilman Pesch was a German Jesuit philosopher.-Life:...
and Joseph Hontheim, as well as theologians such as Cardinal Johannes Franzelin, who once replied by equating the theory of transfinite numbers with
pantheismPantheism is the view that everything is part of an all-encompassing immanent God and that the Universe and God are equivalent...
. Cantor even sent one letter directly to
Pope Leo XIIIPope Leo XIII , born Count Vincenzo Gioacchino Raffaele Luigi Pecci, was the 257th Pope of the Roman Catholic Church, reigning from 1878 to 1903, succeeding Pope Pius IX. Reigning until the age of 93, he was the oldest pope, and had the third longest pontificate, behind Pius IX and John Paul II...
himself, and addressed several pamphlets to him.
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
metaphysicalMetaphysics investigates principles of reality transcending those of any particular science. Cosmology and ontology are traditional branches of metaphysics. It is concerned with explaining the fundamental nature of being and the world...
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 famous assertion that "the essence of mathematics is its freedom." These ideas parallel those of
Edmund HusserlEdmund Gustav Albrecht Husserl was a philosopher who is deemed the founder of phenomenology...
.
Cantor's 1883 paper reveals that he was well aware of the
oppositionIn mathematical logic, the theory of infinite sets was first developed by Georg Cantor. Although this work has found near-universal acceptance in the mathematics community, it has been criticized in several areas by mathematicians and philosophers....
his ideas were encountering:
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
contradictionIn classical logic, a contradiction consists of a logical incompatibility between two or more propositions. It occurs when the propositions, taken together, yield two conclusions which form the logical inversions of each other...
and defined in terms of previously accepted concepts. He also cites
AristotleAristotle was a Greek philosopher, a student of Plato and teacher of Alexander the Great. He wrote on many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, politics, government, ethics, biology, and zoology.Together with Plato and Socrates , Aristotle is one of...
,
DescartesRené Descartes , , also known as Renatus Cartesius , was a French philosopher, mathematician, physicist, and writer who spent most of his adult life in the Dutch Republic...
,
BerkeleyGeorge Berkeley , also known as Bishop Berkeley, was an Anglo-Irish philosopher whose primary achievement was the advancement of a theory he called "immaterialism" . This theory contends that individuals can only know directly sensations and ideas of objects, not abstractions such as "matter"...
,
LeibnizGottfried Wilhelm Leibniz was a German philosopher, polymath and mathematician who wrote primarily in Latin and French....
, and
BolzanoBernhard Placidus Johann Nepomuk Bolzano , Bernard Bolzano in English, was a Bohemian mathematician, theologian, philosopher, logician and antimilitarist of German mother tongue.-Family:...
on infinity.
Cantor's ancestry
Cantor was frequently described as Jewish in his lifetime. Cantor's paternal grandparents were from
CopenhagenCopenhagen ; ) is the capital and largest city of Denmark, with an urban area with a population of 1,167,569 and a metropolitan area with a population of 1,875,179...
, and fled to Russia from the disruption of the
Napoleonic WarsThe Napoleonic Wars were a series of conflicts declared against Napoleon's French Empire and changing sets of European allies by opposing coalitions that ran from 1803 to 1815. As a continuation of the wars sparked by the French Revolution of 1789, they revolutionized European armies and played...
. In his letters, Cantor referred to them as "Israelites". However, there is no direct evidence on whether his grandparents practiced
JudaismJudaism is a set of beliefs and practices originating in the Hebrew Bible , as later further explored and explained in the Talmud and other texts...
; there is very little direct information on them of any kind. Jakob Cantor, Cantor's grandfather, gave his children
ChristianChristianity is a monotheistic religion based on the life and teachings of Jesus of Nazareth as presented by the revelations in the New Testament....
saintSaints, individuals of exceptional holiness, are significant in many religions, particularly Christianity.-General characteristics :Though the term is mostly used for Christians considered holy or virtuous, many religions use similar concepts to elevate people worthy of respect, e.g. see Hindu...
s' names. Further, several of his grandmother's relatives were in the Czarist civil service, which would not welcome Jews, unless they, or their ancestors, converted to Orthodox Christianity. Cantor's father, Georg Woldemar Cantor, was educated in the
LutheranLutheranism is a major branch of Western Christianity that identifies with the teachings of the 16th century German reformer Martin Luther. Luther's efforts to reform the theology and practice of the church launched the Protestant Reformation...
mission in Saint Petersburg, and his correspondence with his son shows both of them as devout Lutherans. His mother, Maria Anna Böhm, was an
AustriaAustria , officially the Republic of Austria , is a landlocked country of roughly 8.3 million people in Central Europe. It borders both Germany and the Czech Republic to the north, Slovakia and Hungary to the east, Slovenia and Italy to the south, and Switzerland and Liechtenstein to the west...
n born in Saint Petersburg and baptized
Roman CatholicThe Catholic Church, also known as the Roman Catholic Church, is the world's largest Christian church. With more than a billion members, over half of all Christians and more than one-sixth of the world's population, the Catholic Church is a communion of the Western, or Latin Rite Church, and...
; she converted to
ProtestantismProtestantism is a branch within Christianity, containing many denominations with some differing practices and doctrines, that principally originated in the sixteenth-century Protestant Reformation. It is considered to be one of the major divisions within Christianity, together with the Roman...
upon marriage. However, there is a letter from Cantor's brother Louis to their mother, saying
which could be read to imply that she was of Jewish ancestry.
Thus Cantor did not follow
JudaismJudaism is a set of beliefs and practices originating in the Hebrew Bible , as later further explored and explained in the Talmud and other texts...
himself, but has nevertheless been called variously German, Jewish, Russian, and Danish.
Historiography
Until the 1970s, the chief academic publications on Cantor were two short monographs by
SchönfliesArthur Moritz Schönflies was a German mathematician, known for his contributions to the application of group theory to crystallography, and for work in topology....
(1927)—largely the correspondence with Mittag-Leffler—and Fraenkel (1930). Both were at second and third hand; neither had much on his personal life. The gap was largely filled by
Eric Temple BellEric Temple Bell was a mathematician and science fiction author born in Scotland who lived in the U.S. for most of his life...
's
Men of MathematicsMen of Mathematics is a well-known book on the history of mathematics written in 1937 by the mathematician E.T. Bell. After a brief chapter on three ancient mathematicians, the remainder of the book is devoted to the lives of about thirty famous mathematicians who worked in the seventeenth,...
(1937), which one of Cantor's modern biographers describes as "perhaps the most widely read modern book on the
history of mathematicsThe area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past....
"; and as "one of the worst". Bell presents Cantor's relationship with his father as Oedipal, 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, and fills the picture with stereotypes. 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—including one that labels Cantor's father a foundling, shipped to Saint Petersburg by unknown parents.
See also
- Cantor cube
In mathematics, a Cantor cube is a topological group of the form {0, 1}A for some index set A. Its algebraic and topological structures are the group direct product and product topology over the cyclic group of order 2 .If A is a countably infinite set, the corresponding Cantor cube is a...
- Cantor space
In mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it is homeomorphic to the Cantor set. In set theory, the topological space 2ω is called "the" Cantor space.- Examples :The...
- Cantor's back-and-forth method
In mathematical logic, especially set theory and model theory, the back-and-forth method is a method for showing isomorphism between countably infinite structures satisfying specified conditions...
- Cantor function
In mathematics, the Cantor function, named after Georg Cantor, is an example of a function that is continuous, but not absolutely continuous. It is also referred to as the Devil's staircase.-Definition:...
- Heine–Cantor theorem
In mathematics, the Heine–Cantor theorem, named after Eduard Heine and Georg Cantor, states that if M is a compact metric space, then every continuous functionwhere N is a metric space, is uniformly continuous....
- Cantor medal
The Cantor medal of the Deutsche Mathematiker-Vereinigung is named in honor of Georg Cantor. It is awarded at most every second year during the yearly meetings of the society...
—award by the Deutsche Mathematiker-Vereinigung in honor of Georg Cantor.
- Controversy over Cantor's theory
In mathematical logic, the theory of infinite sets was first developed by Georg Cantor. Although this work has found near-universal acceptance in the mathematics community, it has been criticized in several areas by mathematicians and philosophers....
External links
Mainly devoted to Cantor's accomplishment.