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Georg Ferdinand Ludwig Philipp Cantor

/ CAN-tor /


German Mathematician

Mathematics Ranking 32nd of 46

Cantor set. Macao stamp from 2005.

Georg Ferdinand Ludwig Philipp Cantor, like many others before him, became interested in the concept of infinity, of things that go on forever without stopping. He introduced the mathematically meaningful concept of transfinite numbers, indefinitely large but distinct from one another. Because the word infinity had a long history with much baggage attached, Cantor introduced the term transfinite numbers to denote all of his infinite numbers, both cardinal numbers (numbers denoting quantity - one, two, three, etc.) and the ordinal numbers (a number defining a thing's position in a series, such as "first," "second," or "third").


From his investigations he also developed the foundation of modern set theory called in German “Mengenlehre,” or theory of aggregates. Set theory is accepted today as the logical basis of all mathematics.(1) Set theory has become an essential part of any mathematics course, because it gives a convenient and versatile language in which to describe the subject. Today, every area of mathematics, pure and applied, is firmly based in the formalism of set theory.(2) Without the language of sets, mathematicians now find it impossible even to specify what they are talking about. 



Cantor published his first paper on set theory before he was thirty. His radical paper on infinite sets was not warmly received by Germany’s leading mathematicians, most of whom were intellectually conservative and still laboring in Gauss’s long shadow. Because Cantor’s ideas seemed to call into question the validity of much of modern mathematics, he was not welcomed by the mathematicians who had invested so much of their careers and their reputations in shoring up conventional mathematics. It was this animosity that kept Cantor from obtaining the position he coveted at the University of Berlin and prompted such bitter attacks by his one-time mentor Kronecker. This controversy seems to have been instrumental in causing Cantor’s career to languish at the University of Halle—an institution that ranked low on the academic pecking order—where he passed his entire professional life as a professor of mathematics.


Even though Cantor eventually had the last word and mathematicians everywhere came to accept his work in set theory as one of the most important contributions to mathematics in the modern era, Cantor gained little satisfaction from this change in his fortunes.(3) A series of nervous breakdowns had effectively ended his mathematical career before he celebrated his fiftieth birthday even though he had stopped doing important mathematical work nearly a decade before. But Cantor did live to see mathematicians take up the gauntlet and approach their subject with the rigor and thoroughness that he had brought to the study of set theory. He died in a Halle asylum in 1918 at age seventy-three.


Theory of Sets and Transfinite Numbers

In a series of ten papers from 1869 to 1873, Cantor dealt first with the theory of numbers. On the suggestion of Heinrich Eduard Heine, a colleague at Halle who recognized his ability, Cantor then turned to the theory of trigonometric series, in which he extended the concept of real numbers. Starting from the work done by the German mathematician Bernhard Riemann in 1854, Cantor in 1870 showed that the function of a complex variable could be represented in only one way by a trigonometric series. This led him to begin his major lifework, the theory of sets and the concept of transfinite numbers.


An important exchange of letters with Richard Dedekind, a mathematician at the Brunswick Technical Institute, who was his lifelong friend and colleague, marked the beginning of Cantor’s ideas on the theory of sets. Both agreed that a set, whether finite or infinite, is a collection of objects (e.g. the integers {0, +1, +2, . . .)} that share a particular property, while each object retains its own individuality. But when Cantor applied the device of the one-to-one correspondence (e.g. {a, b, c} to {1, 2, 3}) to study the characteristics of sets, he quickly saw that they differed in the extent of their membership, even among infinite sets. (A set is infinite if one of its parts, or subsets, has as many objects as itself.) His method soon produced surprising results. Dedekind took the natural numbers (positive integers including 0 i.e. - 0, 1, 2, 3, 4, . . .) as the paradigm example of an infinite set and defined a set as infinite if the natural numbers could be put into a one-to-one correspondence with that set, or a subset of it. Thus, the natural numbers are infinite by definition and so are the integers (all whole numbers and their opposites: -2, -1, 0, 1, 2 . . .), the rational numbers (any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q), and the real numbers (a value of a continuous quantity that can represent a distance along a line) because every natural number is also an integer, a rational number, and a real number. (see illustration below of all the categories that are part of the overall category of "real numbers.")


With Dedekind having accomplished that, Cantor asked two interesting questions: First, can infinity be recognized without making reference to the natural numbers? Second, are there different degrees of infinity?

Cantor answered his first question by defining a set as being infinite if it could be put into a one-to-one correspondence with the set of natural numbers ({0, 2, . . .}). The set of natural numbers trivially satisfy this condition. The set of natural numbers can be put into a one-to-one mapping with a subset of itself (consider, for example, the mapping that

takes 0→1, 1→2, 2→3, 2→3, 3→4, . . .). Therefore, any set that satisfies Cantor’s definition of the infinite automatically satisfied Dedekind’s definition.


By making the infinite more general than the natural numbers, Cantor opened the possibility of there being different degrees of the infinite. The list of natural numbers runs on for ever: 1, 2, 3, 4, 5., . . There are also infinitely many real numbers (decimal numbers such as .5 or π or 0.1234101981. . .).  These two types of infinity are known as “countable infinity” and the “continuum,” respectively. To the dismay of his contemporaries, Cantor proved that these are actually different sizes. In a very real sense, the collection of decimal numbers is a bigger infinity than that of the whole numbers. This was not the end of it: Cantor identified more levels of infinity than these two (infinitely many in fact). But for most ordinary mathematics, these are the two most important types of infinity. Cantor had shown that the continuum is a bigger infinity than the countable level. What he didn’t know was whether there were any intermediate levels between them. He believed that there were not, and this conjecture became known as the “continuum hypothesis.” In 1940, Kurt Gödel showed that the Continuum Hypothesis cannot be disproved using the standard axioms of set theory; then in 1963, Paul Cohen proved that it cannot be proved using those axioms either!




(1) Lloyd Motz and Jefferson Hane Weaver, The Story of Mathematics (New York, 1993), p. 267; Ian Stewart, Significant Figures – The Lives and Work of Great Mathematicians (New York, 2017) p. 165.

(2) Stewart, p. 175.

(3) Motz and Weaver, p. 266.

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