Mathematics

Georg Cantor

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08 October 2020                

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Introduction

Georg Ferdinand Ludwig Philipp Cantor, (March 3, 1845-January 6, 1918)
Illustration of Georg Cantor by Maney Imagination

Georg Cantor, also known as Georg Ferdinand Ludwig Philipp Cantor, (March 3, 1845-January 6, 1918) was a German mathematician who is famous as the founder of set theory and is also responsible for introducing the mathematically meaningful concept of transfinite numbers, indefinitely large but distinct from one another.

In fact, until the end of the nineteenth century, no mathematician had managed to describe the infinite, beyond the idea that it is an unattainable value. Georg Cantor was the first to comprehensively address such an abstract concept. He achieved this by developing set theory, which led him to the surprising conclusion that there are infinities of different sizes. Faced with the rejection of his counterintuitive ideas, Cantor doubted himself and suffered successive nervous breakdowns, until dying interned in a sanatorium. Nowadays, it is difficult to understand mathematics without his revolutionary insights.


Early Life And Training

Cantor’s parents were Danish. His artistic mother, a Roman Catholic, came from a family of musicians, and his father, a Protestant, was a prosperous merchant. When his father became ill in 1856, the family moved to Frankfurt. Cantor’s mathematical talents emerged before his 15th birthday while he was studying in private schools and at gymnasien at Darmstadt first and then at Wiesbaden; eventually, he overcame the objections of his father, who wanted him to become an engineer.

Georg Ferdinand Ludwig Philipp Cantor, (March 3, 1845-January 6, 1918) After briefly attending the University of Zürich, Cantor in 1863, transferred to the University of Berlin to specialize in physics, philosophy, and mathematics. He was taught by the mathematicians Karl Weierstrass, whose specialization of analysis probably had the most significant influence on him; Ernst Eduard Kummer, in higher arithmetic; and Leopold Kronecker, a specialist on the theory of numbers who later opposed him. Following one semester at the University of Göttingen in 1866, Cantor wrote his doctoral thesis in 1867, De Aequationibus Secundi Gradus Indeterminatis (“On the Indeterminate Equations of the Second Degree”), on a question that Carl Friedrich Gauss had left unsettled in his Disquisitiones Arithmeticae (1801). After a brief teaching assignment in a Berlin girls’ school, Cantor joined the faculty at the University of Halle, where he remained for the rest of his life, first as a lecturer (paid by fees only) in 1869, then as an assistant professor in 1872, and full professor in 1879.

In a series of 10 papers from 1869 to 1873, Cantor dealt first with number theory; this article reflected his fascination with the studies of Gauss and the influence of Kronecker. 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 on trigonometric series and on the function of a complex variable done by the German mathematician Bernhard Riemann in 1854, Cantor in 1870 showed that such a function can be represented in only one way by a trigonometric series. Consideration of the collection of numbers (points) that would not conflict with such a representation led him, first, in 1872, to define irrational numbers in terms of convergent sequences of rational numbers (quotients of integers) and then to begin his major lifework, the theory of sets and the concept of transfinite numbers.


Set Theory

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.

In 1873 Cantor demonstrated that the rational numbers, though infinite, are countable (or denumerable) because they may be placed in a one-to-one correspondence with the natural numbers (i.e., the integers, as 1, 2, 3,…). He showed that the set (or aggregate) of real numbers (composed of irrational and rational numbers) was infinite and uncountable. Even more, paradoxically, he proved that the set of all algebraic numbers contains as many components as the set of all integers and that transcendental numbers (those that are not algebraic, as π), which are a subset of the irrationals, are uncountable and are therefore more numerous than integers, which must be conceived as infinite.

But Cantor’s paper, in which he first put forward these results, was refused for publication in Crelle’s Journal by one of its referees, Kronecker, who henceforth vehemently opposed his work. On Dedekind’s intervention, however, it was published in 1874 as “Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen” (“On a Characteristic Property of All Real Algebraic Numbers”).

While honeymooning the same year with his bride, Vally Guttman, at Interlaken, Switzerland, Cantor met Dedekind, who gave a sympathetic hearing to his new theory. Cantor’s salary was low, but the estate of his father, who died in 1863, enabled him to build a house for his wife and five children. Many of his papers were published in Sweden in the new journal Acta Mathematica, edited and founded by Gösta Mittag-Leffler, one of the first persons to recognize his ability.

Cantor’s theory became a whole new subject of research concerning the mathematics of the infinite (e.g., an endless series, as 1, 2, 3,…, and even more complicated sets), and his theory was heavily dependent on the device of the one-to-one correspondence. In thus developing new ways of asking questions concerning continuity and infinity, Cantor quickly became controversial. When he argued that infinite numbers had an actual existence, he drew on ancient and medieval philosophy concerning the “actual” and “potential” infinite and also on the early religious training given him by his parents. In his book on sets, Grundlagen einer allgemeinen Mannigfaltigkeitslehre (“Foundations of a General Theory of Aggregates”), Cantor in 1883 allied his theory with Platonic metaphysics. By contrast, Kronecker, who held that only the integers “exist” (“God made the integers, and all the rest is the work of man”), for many years heatedly rejected his reasoning and blocked his appointment to the faculty at the University of Berlin.


Meaning of Infinity

Cantor’s first ten papers were on number theory, after which he turned his attention to calculus (or analysis as it had become known by this time), solving a difficult open problem on the uniqueness of the representation of a function by trigonometric series. His main legacy, though, is as perhaps the first mathematician to really understand the meaning of infinity and to give it mathematical precision.

Back in the 17th Century, Galileo had tried to confront the idea of infinity and the apparent contradictions thrown up by comparisons of different infinities, but in the end, shied away from the problem.

Cantor’s procedure of bijection or one-to-one correspondence to compare infinite sets
Figure 1: Cantor’s procedure of bijection or one-to-one correspondence to compare infinite sets

He had shown that a one-to-one correspondence could be drawn between all the natural numbers and the squares of all the natural numbers to infinity, suggesting that there were just as many square numbers as integers, even though it was intuitively obvious there were many integers that were not squares, a concept which came to be known as Galileo’s Paradox.

He had also pointed out that two concentric circles must both be comprised of an infinite number of points, even though the larger circle would appear to contain more points. However, Galileo had essentially dodged the issue and reluctantly concluded that concepts like less, equals, and greater could only be applied to finite sets of numbers, and not to infinite sets. Cantor, however, was not content with this compromise.

Cantor’s starting point was to say that, if it was possible to add 1 and 1, or 25 and 25, etc, then

it ought to be possible to add infinity and infinity. He realized that it was actually possible to add and subtract infinities and that beyond what was normally thought of as infinity existed another, larger infinity, and then other infinities beyond that. In fact, he showed that there may be infinitely many sets of infinite numbers – an infinity of infinities – some bigger than others, a concept which clearly has philosophical, as well as just mathematical, significance. The sheer audacity of Cantor’s theory set off a quiet revolution in the mathematical community and changed forever the way mathematics is approached.


The Beginning of Nervous Breakdowns

Both Cantor and Dedekind were exceptional mathematicians in their time, but neither ever obtained a prominent professional position. While Cantor remained at the University of Halle, a small institution, for most of his life, Dedekind never went beyond being a high school teacher in his hometown of Brunswick. Cantor’s main detractor was Leopold Kronecker, who prevented him from entering the prestigious University of Berlin every time he tried. Kronecker argued that mathematics should be based on whole numbers, and systematically rejected that incipient new branch of mathematics. Kronecker’s attacks ended up provoking in 1884 the first of Cantor’s nervous breakdowns, which he suffered periodically for the rest of his life.

Cantor spent the end of his days in a psychiatric hospital in Halle, where he died. However, his set theory ended up becoming the common language used in the various branches of modern mathematics. A few years after his death, the renowned mathematician David Hilbert said that transfinite arithmetic is “the finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity.”


Appendix A: Bijective Functions

It would seem that the set of natural numbers (N = {0, 1, 2, 3, 4, 5, 6…}) comprises more elements than the subset containing prime numbers (P = {2, 3, 5, 7, 11, 13…}). However, Cantor showed that both have the same cardinal, and therefore the same infinite number of elements. To demonstrate this, he paired each of the elements that form a set with the elements of the other, which is known as establishing a bijective function (or one-to-one correspondence) between both sets. In the case of natural numbers and primes, the pairing could be 0-2, 1-3, 2-5, 3-7, 4-11, 5-13…

Bijective Functions
Figure 2: Illustration of bijection. Source: Wikimedia

On the contrary, in one of his most famous demonstrations, Cantor proved that it was impossible to establish a bijective function between the set of natural numbers and the set of points that form a number line. He thus concluded that the cardinal of the set of real numbers was greater than that of natural numbers: they were infinities of different sizes. He demonstrated that the infinite cardinal of the real numbers is followed by another greater one, which is followed by an even greater one, and so on. To the smallest of all these infinite cardinals, the cardinal of natural numbers, he called it aleph-zero (aleph is the first letter of the Hebrew alphabet). The next ones he called aleph-one, aleph-two, aleph-three, etc. All these cardinals of infinite sets are known by the name of transfinite cardinals.


Frequently Asked Questions (FAQs)

What is the Cantor Set?

In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties.

Who is Georg Cantor?

Georg Cantor was a German mathematician who founded set theory and introduced the mathematically meaningful concept of transfinite numbers, indefinitely large but distinct.

What is Georg Cantor most famous for?

Georg Cantor is famously credited for introducing the Set theory in Mathematics. One cannot imagine understanding mathematics today without the concept of set theory. This also led him towards the discovery of different quantities of Infinity.


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