Georg Cantor

How to use this course

Georg Cantor created set theory. Before him, mathematicians treated the infinite as a process that never ends. Cantor treated infinite collections as finished objects, compared their sizes by one-to-one correspondence, and found that there are infinitely many different sizes of infinity. He then built an arithmetic of these sizes and of infinite orderings. His ideas met fierce resistance, and he spent the last three decades of his life in recurring illness. Within a generation his theory had become the foundation on which almost all of mathematics is written.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. The lessons follow the order in which Cantor found his ideas, from trigonometric series (1870) to the theory of ordinals and cardinals (1883–1897). That order shows how naturally the transfinite grew out of classical analysis.

Timeline

St Petersburg to Darmstadt, 1845–1862

Georg Ferdinand Ludwig Philipp Cantor was born in St Petersburg on 3 March 1845. His father, Georg Waldemar Cantor, was a merchant and stock-exchange broker born in Denmark. His mother, Maria Anna Böhm, was Russian-born and very musical. Music and religion were part of daily life in the household. Cantor himself became an outstanding violinist, and he remained a devout Lutheran all his life.

In 1856, partly for the sake of his father’s health, the family left Russia for Germany. They lived first in Wiesbaden and then in Frankfurt. Georg attended schools in Wiesbaden and Darmstadt, graduating from the Darmstadt Realschule in 1860 with a strong record in mathematics, and then entered the Höhere Gewerbeschule in Darmstadt.

His father wanted him to become an engineer, a respectable and well-paid career. Georg wanted to study mathematics. In 1862 he asked his father’s permission, and his father eventually agreed.

Zürich, Berlin, Göttingen, 1862–1869

Cantor began his university studies at the Polytechnic in Zürich in 1862. His father died in June 1863, and Cantor moved to the University of Berlin. Berlin was then the leading centre of German mathematics. He attended the lectures of Karl Weierstrass, Ernst Kummer and Leopold Kronecker, and became friends with Hermann Amandus Schwarz, a fellow student. He spent the summer semester of 1866 at Göttingen.

His dissertation, De aequationibus secundi gradus indeterminatis (1867), was a piece of number theory in the tradition of Carl Friedrich Gauss, on integer solutions of equations \(ax^2+by^2+cz^2=0\). Kummer and Weierstrass supervised it. For a short time Cantor taught at a girls’ school in Berlin.

In 1869 he habilitated at the University of Halle, again with a thesis on number theory. Halle was a respectable but modest university. Cantor hoped to move to Berlin later; he never did, and he stayed at Halle for the rest of his career.

Halle, Heine and Dedekind, 1869–1874

At Halle the senior mathematician was Eduard Heine. Heine turned Cantor from number theory to analysis by posing an open problem: if a function is represented by a trigonometric series, is the representation unique? Cantor solved it by April 1870 and refined the result over the next two years (Lesson 04). To handle the exceptional sets of points that his proofs allowed, he needed a precise theory of real numbers and of infinite point sets. Both appeared in his paper of 1872 (Lesson 05).

Cantor was promoted to extraordinary professor in 1872. That year, on holiday in Switzerland, he met Richard Dedekind, whose own theory of real numbers by cuts appeared the same year. Their correspondence over the next decade is one of the great documents of modern mathematics. In letters of late 1873 Cantor asked whether the integers and the real numbers could be put in one-to-one correspondence. Dedekind could not answer, but sent a proof that the algebraic numbers are countable. On 7 December 1873 Cantor sent his proof that the real numbers are not countable. The result was published in 1874 (Lesson 06).

On 9 August 1874 Cantor married Vally Guttmann. They had six children. On their honeymoon in Interlaken Cantor spent much of his time in mathematical discussion with Dedekind.

Trigonometric series and derived sets

For the general reader. Set theory began with a question from analysis. If a sum of sines and cosines adds up to zero at every point, must every term be zero? Cantor proved yes. Then he asked how many points he could ignore and still get the same answer. To describe those exceptional points he invented the derived set, and iterating it led him, step by step, to counting beyond the finite.

Proof sketch

Step 1 (Cantor–Lebesgue). If the series converges on an interval, then \(a_n\cos nx+b_n\sin nx\to0\) there, and this forces \(a_n,b_n\to0\). In particular the coefficients are bounded.

Step 2 (Riemann’s function). Integrate twice formally:

The series converges uniformly, so \(F\) is continuous. Riemann had shown that wherever the original series converges to \(s\), the second symmetric derivative \(\lim_{h\to0}\frac{F(x+h)+F(x-h)-2F(x)}{h^2}\) equals \(s\). Here it is \(0\) everywhere.

Step 3 (Schwarz’s lemma). A continuous function whose second symmetric derivative vanishes everywhere is linear. The lemma is due to his friend Schwarz. So \(F(x)=cx+d\). The sum in \(F\) is bounded and periodic, so \(a_0=0\) and \(c=0\). Then \(\sum (a_n\cos nx+b_n\sin nx)/n^2=-d\) uniformly, and computing Fourier coefficients term by term gives \(a_n=b_n=0\) for \(n\ge1\). \(\square\)

Exceptional sets

In 1871 Cantor allowed finitely many points where nothing is assumed. In 1872 he went much further. For \(P\subseteq\mathbb R\) let \(P'\) be its derived set, the set of limit points of \(P\). Put \(P^{(1)}=P'\) and \(P^{(n+1)}=(P^{(n)})'\).

The proof inducts on \(n\): on each open interval avoiding \(P^{(n-1)}\), the exceptional points are isolated, and the linearity of \(F\) extends across them. But some sets satisfy \(P^{(n)}\ne\varnothing\) for every \(n\). Then one can form \(P^{(\infty)}=\bigcap_n P^{(n)}\), and its derived set, and so on. In 1883 Cantor wrote these indices as \(\omega,\omega+1,\dots\): the transfinite ordinals of Lesson 08.

Constructing the real numbers

For the general reader. What is \(\sqrt2\)? It is not a fraction, yet we can approximate it by fractions as closely as we like. In 1872 Cantor defined a real number as the common target of a family of rational approximations. Dedekind published a different construction the same year. Both made the real line, which analysts had used for centuries, a precisely defined object built from the rationals.

Fundamental sequences

A sequence \((a_n)\) of rationals is fundamental (today: Cauchy) if for every rational \(\varepsilon>0\) there is \(N\) with \(|a_m-a_n|<\varepsilon\) for all \(m,n\ge N\). Two fundamental sequences are equivalent if \(a_n-b_n\to0\). A real number is an equivalence class. Sums and products are defined termwise, and \((a_n)>0\) means that \(a_n\ge\delta\) for some rational \(\delta>0\) and all large \(n\).

Proof idea for completeness. Given a Cauchy sequence \((x_k)\) of reals, choose rationals \(q_k\) with \(|x_k-q_k|<1/k\), possible because \(\mathbb Q\) is dense. Then \((q_k)\) is a fundamental sequence of rationals, and its class \(x\) satisfies \(x_k\to x\). Cantor noted that iterating the construction gives nothing new; it is this stability that makes \(\mathbb R\) the right setting for analysis. Charles Méray had published a similar construction in 1869, and Heine adopted Cantor’s approach in 1872.

Dedekind’s cuts

Dedekind defined a real number as a partition of \(\mathbb Q\) into a lower set \(A\) and an upper set \(B\), with every element of \(A\) less than every element of \(B\). Completeness then becomes the statement that every cut of \(\mathbb R\) is produced by a single number. Both constructions treat an infinite set of rationals as a single completed object, the step Kronecker rejected.

Countable and uncountable

For the general reader. Two collections have the same size if their members can be paired off exactly. By this test the even numbers are as numerous as all whole numbers, and so are the fractions. Cantor proved that even the algebraic numbers can be listed one by one. The real numbers cannot. However you try to list them, some number is missing. This was the first proof that infinity comes in more than one size.

The 1874 paper

The paper “Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen” appeared in Crelle’s Journal in 1874. Weierstrass advised Cantor to keep the uncountability result in the background, so the title mentions only algebraic numbers.

Each algebraic number is a root of a unique irreducible equation \(a_0x^n+a_1x^{n-1}+\cdots+a_n=0\) with integer coefficients, \(a_0>0\) and \(\gcd(a_0,\dots,a_n)=1\). Cantor defined its height as \(N=n-1+|a_0|+|a_1|+\cdots+|a_n|\). For each \(N\) there are finitely many such equations, each with at most \(n\) roots. List the numbers of height 1, then 2, and so on.

Proof (nested intervals). Let \(a_1<b_1\) be the first two terms of the sequence inside \((a,b)\); let \(a_2<b_2\) be the next two terms inside \((a_1,b_1)\); and so on. If the process stops, some point of the last interval is missed. Otherwise \(a_k\) increases to \(a_\infty\) and \(b_k\) decreases to \(b_\infty\ge a_\infty\). Any point of \([a_\infty,b_\infty]\) differs from every \(x_k\): each term either lies outside some interval \((a_j,b_j)\) or is one of the endpoints \(a_j,b_j\), which lie strictly outside \([a_\infty,b_\infty]\). \(\square\)

Joseph Liouville had exhibited transcendental numbers in 1844. Cantor’s argument proves that almost all real numbers are transcendental, without exhibiting one.

The diagonal argument (1891)

In “Über eine elementare Frage der Mannigfaltigkeitslehre” (Jahresbericht der DMV, 1891) Cantor gave a simpler proof, which works for any set.

Proof. Let \(D=\{s\in S: s\notin f(s)\}\). If \(D=f(t)\), then \(t\in D\iff t\notin f(t)=D\), a contradiction. \(\square\) Cantor stated it for the set of binary sequences: given a list \(e_1,e_2,\dots\), the sequence whose \(k\)-th entry differs from the \(k\)-th entry of \(e_k\) is not in the list.

Dimension and one-to-one correspondence

For the general reader. A square seems to contain far more points than one of its sides. In 1877 Cantor showed that this is false: the points of a square can be paired exactly with the points of a segment. He wrote to Dedekind in French that he saw it but did not believe it. The discovery forced mathematicians to say precisely what dimension means.

First idea. Interleave digits: send \((0.x_1x_2x_3\ldots,\ 0.y_1y_2y_3\ldots)\) to \(0.x_1y_1x_2y_2\ldots\). This is injective once one fixes expansions without trailing 9s, but not surjective (Exercise 7.2). Dedekind pointed out this gap in Cantor’s first attempt. Cantor repaired it by working with continued fractions of irrational numbers, whose expansions are unique, and then dealing separately with the countable set of rationals. The cleanest modern repair uses the next theorem.

Proof. Call \(a\in A\) a B-stopper if, tracing back \(a\leftarrow b\leftarrow a'\leftarrow\cdots\) through \(g^{-1}\) and \(f^{-1}\), the chain ends at an element of \(B\) not in \(f(A)\). Define \(h(a)=g^{-1}(a)\) for B-stoppers and \(h(a)=f(a)\) otherwise. Each chain is handled by one rule, and \(h\) is a bijection. Cantor stated the theorem but proved it only with the help of the well-ordering principle. Dedekind found a proof in 1887 that he did not publish, and Felix Bernstein, then a student in Cantor’s seminar, proved it in 1897.

Dimension

Dedekind replied that any such bijection must be discontinuous, and that dimension should be invariant only under continuous bijections. The conjecture that \(\mathbb R^m\) and \(\mathbb R^n\) are not homeomorphic for \(m\ne n\) was proved by L. E. J. Brouwer in 1911. Space-filling curves of Giuseppe Peano (1890) and Hilbert (1891) show that a continuous surjection from a segment onto a square does exist; it cannot be injective.

Ordinal numbers

For the general reader. Counting has two jobs: saying how many, and saying in what order. For finite collections the two agree. For infinite ones they come apart. Cantor’s ordinal numbers keep count of order: after all the finite numbers comes \(\omega\), then \(\omega+1\), and the process never stops. His 1883 Grundlagen made these numbers a subject in their own right.

The Grundlagen (1883)

Grundlagen einer allgemeinen Mannigfaltigkeitslehre was the fifth of six papers on linear point sets in Mathematische Annalen (1879–1884). It also appeared as a monograph with a philosophical defence of the actual infinite. Cantor generated ordinals by two principles: add one to any number already formed, and take the limit of any increasing sequence already formed. A third principle, of limitation, grouped them into number classes.

In modern terms, a well-ordering of a set is a linear order in which every non-empty subset has a least element. Two well-orderings have the same ordinal if there is an order-preserving bijection between them. Any two well-orderings are comparable: one is isomorphic to an initial segment of the other.

These operations are associative but not commutative: \(1+\omega=\omega\ne\omega+1\), and \(2\cdot\omega=\omega\ne\omega\cdot2\).

Cantor also introduced the epsilon numbers, solutions of \(\omega^\varepsilon=\varepsilon\). The least is \(\varepsilon_0=\sup\{\omega,\omega^\omega,\omega^{\omega^\omega},\dots\}\). Gerhard Gentzen used induction up to \(\varepsilon_0\) in 1936 to prove the consistency of arithmetic.

Back to derived sets

Cardinals and the continuum hypothesis

For the general reader. Having found two sizes of infinity, the countable and the continuum, Cantor wanted a list of all sizes. He named them with the Hebrew letter aleph: \(\aleph_0,\aleph_1,\aleph_2,\dots\). He believed that the continuum is the next size after the countable, and he tried for years to prove it. The question became the first of Hilbert’s problems in 1900. In the twentieth century it turned out that the usual axioms of mathematics cannot settle it.

Cardinal arithmetic

In the Beiträge zur Begründung der transfiniten Mengenlehre (1895, 1897) Cantor defined, for sets \(A,B\), \(|A|+|B|=|A\sqcup B|\), \(|A|\cdot|B|=|A\times B|\) and \(|A|^{|B|}=|A^B|\), the set of functions \(B\to A\). The usual laws hold, such as \((\kappa^\lambda)^\mu=\kappa^{\lambda\mu}\). He wrote \(\aleph_0\) for \(|\mathbb N|\) and \(\aleph_1\) for the cardinality of the set of countable ordinals, his second number class. Then \(|\mathbb R|=2^{\aleph_0}\), often written \(\mathfrak c\).

The equivalence uses the axiom of choice. Cantor proved the hypothesis for closed sets (Lesson 10), but not in general. Kurt Gödel showed in 1940 that it cannot be disproved from the Zermelo–Fraenkel axioms with choice (ZFC), by constructing a model in which it holds. Paul Cohen showed in 1963, with his method of forcing, that it cannot be proved either.

Well-ordering and the paradoxes

Cantor took for granted that every set can be well-ordered, which makes all cardinals comparable and alephs. Ernst Zermelo proved this in 1904 from the axiom of choice. Around 1895–1897 Cantor saw that the collection of all ordinals cannot be a set: it would be well-ordered and hence have an ordinal larger than every ordinal. Cesare Burali-Forti published a version of this paradox in 1897. Cantor explained to Hilbert in 1896 and Dedekind in 1899 that such “inconsistent multiplicities” are too large to be treated as single objects. Bertrand Russell’s paradox of 1901 showed the same problem with the set of all sets that do not contain themselves. Zermelo’s axioms of 1908 were written to avoid them.

Perfect sets and the Cantor set

For the general reader. Start with a segment and remove its open middle third. Remove the middle third of each remaining piece, and repeat forever. What is left has total length zero, yet it contains as many points as the whole segment. Cantor used this set to show how strange closed sets can be, and he proved that every closed set is a countable dust plus a part of this perfect kind.

The middle-thirds set

Let \(C_0=[0,1]\), and let \(C_{n+1}\) be obtained from \(C_n\) by removing the open middle third of each of its \(2^n\) intervals. The Cantor set is \(C=\bigcap_n C_n\). Equivalently, \(C\) is the set of \(x\in[0,1]\) having a ternary expansion \(x=\sum_{k\ge1}d_k3^{-k}\) with every \(d_k\in\{0,2\}\). Henry John Stephen Smith had described similar sets in 1874; Cantor gave this example in 1883 as a perfect set that is dense in no interval.

A set is perfect if it is closed and equal to its own derived set: it has no isolated points.

Proof sketch. Inside a perfect set \(P\) choose two disjoint closed intervals, each meeting \(P\) in a perfect set. Inside each, choose two more, and so on, with lengths tending to zero. Each infinite sequence of left/right choices determines exactly one point of \(P\), and different sequences give different points. So \(P\) contains a copy of \(\{0,1\}^{\mathbb N}\).

The perfect part is the stable value \(F^{(\alpha)}\) of the derived-set sequence of Lesson 08. Ivar Bendixson proved the decomposition in 1883, answering a question of Cantor, and Cantor gave a proof in 1884.

Opposition and illness, 1877–1899

Cantor’s ideas were radical, and he paid a price. Leopold Kronecker, his former teacher and an editor of Crelle’s Journal, held that only objects constructed in finitely many steps from the integers exist. In 1877 Cantor’s paper on dimension was held up at the journal and appeared in 1878 only after Dedekind intervened. Cantor never sent another paper to Crelle. He believed that Kronecker’s influence in Berlin blocked any chance of the appointment there that he wanted, and he came to see Kronecker as a personal enemy. Henri Poincaré and other mathematicians were also sceptical of transfinite numbers.

Support came from elsewhere. Gösta Mittag-Leffler published French translations of Cantor’s papers in his new journal Acta Mathematica in 1883. But in 1885 Mittag-Leffler persuaded him to withdraw a paper on order types, judging it far ahead of its time. Cantor was hurt, and their correspondence nearly stopped.

In May 1884 Cantor suffered his first recorded attack of depression. After a stay in the Harz mountains he sought reconciliation with Kronecker, who accepted, though their philosophical disagreement remained. He turned for a time to philosophy and theology, corresponding with philosophers and theologians about the nature of the infinite, and to literature: he published pamphlets in 1896 and 1897 arguing that Francis Bacon wrote the plays of Shakespeare. Cantor saw his transfinite numbers as consistent with, and in a sense revealed by, God’s absolute infinity.

Building institutions

Despite everything, he worked for the mathematical community. He was the driving force behind the founding of the Deutsche Mathematiker-Vereinigung in 1890, served as its first president from 1891 to 1893, and chaired its first meeting at Halle in September 1891. He invited Kronecker to speak there; Kronecker did not come, because his wife had just died. Cantor also helped organize the first International Congress of Mathematicians at Zürich in 1897, where Adolf Hurwitz and Jacques Hadamard spoke with admiration of its uses in analysis.

Paradoxes, last years, legacy

From 1899 Cantor’s hospital stays became frequent. His youngest son, Rudolph, died on 16 December 1899. He took leave from teaching in several winter semesters. Recognition came late: the London Mathematical Society made him an honorary member in 1901, the Royal Society awarded him its Sylvester Medal in 1904, and he was elected to the Royal Society of Edinburgh in 1905. In 1911 Cantor visited St Andrews for the university’s quincentenary, and in 1912 St Andrews gave him an honorary doctorate, which illness prevented him from receiving in person.

He retired in 1913. During the First World War food was scarce, and his health declined further. He entered a sanatorium in Halle in June 1917 for the last time and died there of a heart attack on 6 January 1918, aged 72.

Legacy

• Foundations: sets, functions and cardinalities became the common language of mathematics. Zermelo’s axioms of 1908 and their extension by Abraham Fraenkel turned Cantor’s naive theory into ZFC. Hilbert wrote in 1926 that “no one shall expel us from the paradise that Cantor has created.”
• Analysis and topology: derived sets, perfect sets, closed and dense sets, and the Cantor set are the beginnings of point-set topology and measure theory, developed by Felix Hausdorff, Émile Borel and Henri Lebesgue.
• Logic: the diagonal argument reappears in Gödel’s incompleteness theorems, in Alan Turing’s proof that the halting problem is undecidable, and throughout computability theory.
• Open questions: the continuum hypothesis, independent of ZFC by the work of Gödel and Cohen, remains at the centre of research on new axioms for set theory.
• Critics: the intuitionism of Brouwer and the finitism of Kronecker continue as constructive mathematics, which accepts Cantor’s countability results and rejects his completed infinities.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Georg Cantor, MacTutor History of Mathematics.
2. Georg Cantor, Wikipedia.
3. Georg Cantor, Mathematics Genealogy Project.
4. J. W. Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite, Harvard University Press, 1979; Princeton, 1990.
5. H. Meschkowski, Probleme des Unendlichen: Werk und Leben Georg Cantors, Vieweg, 1967.
6. I. Grattan-Guinness, Towards a biography of Georg Cantor, Annals of Science 27 (1971).

Mathematics

1. G. Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, ed. E. Zermelo, Springer, 1932.
2. G. Cantor, Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen, J. reine angew. Math. 77 (1874). See Cantor’s first set theory article.
3. G. Cantor, Über eine elementare Frage der Mannigfaltigkeitslehre, Jahresbericht der DMV 1 (1891). See Cantor’s diagonal argument.
4. G. Cantor, Grundlagen einer allgemeinen Mannigfaltigkeitslehre, Teubner, 1883.
5. G. Cantor, Contributions to the Founding of the Theory of Transfinite Numbers, trans. P. E. B. Jourdain, Open Court, 1915; Dover reprint.
6. Cantor set and Controversy over Cantor’s theory, Wikipedia.
7. J. Ferreirós, Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics, 2nd ed., Birkhäuser, 2007.
8. A. Kanamori, The mathematical development of set theory from Cantor to Cohen, Bull. Symbolic Logic 2 (1996).