If you open a textbook, in high school or college, in the first chapter you will be introduced to set theory and the theories of finite numbers, infinite numbers, and irrational numbers. The development of many theories of math took years upon years and the input of many mathematicians, as in the example of non-Euclidean geometry. This was the case with most math theories, however set theory was primarily the result of the work of one man, Georg Cantor. In his time, these hypotheses were considered greatly controversial by other mathematicians. However, now they are an integral part of the study of mathematics. Georg Cantor received more criticism than complement in his time and it eventually led him to mental illness. However, one must remember that many other things, once thought to be controversial are now considered to be fact. Georg Cantor should be considered one of the pioneers of modern mathematical theory.
Mathematics can be considered a language in its own right. It is the language that we use to describe our world. Math tells us vital information such as how big, how fast, and describes the relationship between two things. A set is a group of things that can be treated as a single unit. There are two ways to describe a set. The first method is to simply list the elements of a set. The second way is to describe the members of a set and define what characteristics determine which elements will be included or not be included in a particular set.
Family and Early Life
Georg Ferdinand Ludwig Phillip Cantor was born on March 3, 1845 in St. Petersburg, Russia. The family lived in Russia for eleven years until his father's failing health forced the family to move to the milder climate of Frankfort, Germany in 1856. It was here that Cantor would spend the rest of his life. Georg was the eldest of the three children. His father was a wealthy merchant, Georg Walsematr Cantor and his mother was a famous artist, Maria Boehm. The other children had exceptional artistic talents like their mother. Georg's brother, Constantine, was an army officer and also a fine pianist. His sister, Sophie Nobiling, was an accomplished designer. However, Georg excelled in Math (Johnson, 1997). Georg Cantor came from a family with a wealth of talent in math, physics, and philosophy. His brothers and sisters also displayed talent in math.
Cantor had a strict religious upbringing, and he carried a strong religious sense all through his life. His father was Jewish, but later converted to Protestantism around the time of Georg's birth. His mother was a devout Catholic. This difference of religious opinions did not sway Georg's own beliefs and he became a knowledgeable theologian as well as mathematician (Johnson, 1997). It was no doubt that this diverse religious background made him the type to question his surroundings and stand by his ideas, even when everyone else said he was wrong.
Georg attended several private schools in Frankfurt, and in 1859, entered the distinguished Grossherzoglich Hessiche Provinzialrealschule in Darmstadt. He left this institution in 1860 with high recommendations in mathematics. His father discouraged the study of math due to the fact that he wished him to become an engineer, a job that paid considerably more than mathematics. He originally attended Grossherzogliche Hoehere Gewerbeschule (Grand-Ducal Higher Polytechnic, later changed to Technische Hochschule) at Darmstadt following his father's wishes and studying Engineering. Later, when Georg convinced his father that his heart was truly in math, his father relented and he began the study of Mathematics in 1862 (Johnson, 1997).
Cantor began his higher studies in Zurich, the fall of 1862. He left in Spring of 1963 due to the death of his father. In Fall of 1863, he entered the University of Berlin to study mathematics, physics, and philosophy. The University of Berlin was home to three famous mathematicians Ernst Eduard Kummer, Karl W.T. Weierstrass and Leopold Kronecker. These three men made the University of Berlin one of the top schools for the study of mathematics in the entire world. The student population was small and therefore the students were in close contact with these three great minds. Cantor was heavily influenced by the works of Weierstrass. Kronecker was also a great influence, but would later become one of his greatest critics. It was customary in Germany, at the time, to study at another University for a period time. He studied at the Cantor attended the University of Gottingen during the summer term of 1866 (Johnson, 1997). Cantor received the degree of doctor on December 14, 1867. His dissertation was based on a study of the Disquisitiones Arithmeticae of Carl Friedrich Gauss, another contemporary mathematician of his time, and on the number theory of Adrien-Marie Legendre (Johnson, 1997).
Cantor's thesis centered around one of the ideas that Gauss had left aside concerning the solutions in integers x, y and z of the indeterminate equation ax2 + by2 + cz2 = 0, where a, b and c are any given integers. The full title of the thesis was "De aequationibus secundi gradus indeterminatis" ("On indeterminate equations of the second degree") and as was customary, dedicated to his guardians, Eduard Flersheim and Bernhard Horkheimer. As was also the custom for Doctoral candidates, Cantor also defended three theses against opposing doctors. All fo which were translated from the original Latin, the theses were "In arithmetic merely arithmetic methods far surpass analytic methods," "Since it is disputed, the question of the absoluteness of space and time is more important than its solution" and "In mathematics the art of proposing a question must be held of higher value than solving it." (Johnson, 1997). Cantor's early works were considered to be excellent among his peers, but no on ever suspected the genius that would emerge in his later writings.
In Spring of 1869, Cantor began his career as a Privatdozent at the University of Halle on the basis of his paper "De transformatione formarum ternariarum quadraticarum" ("On the transformation of ternary quadratic forms") (Johnson,1997). Cantor specialized in Number theory. Cantor's became Extraordinarius at Halle in 1872 and Ordinarius in 1879. He was released from his official duties in 1905 and resigned his post altogether in 1913.
In 1874,Cantor published his first paper on the Theory of Sets. In that same year, he married Vally Guttman in the summer. They had two sons and four daughters, none of whom, it might be noted were gifted in mathematics. One of Cantor's daughters, Frau Gertrud Vahlen, was an important source of information for the biography of Cantor by A.A. Fraenkel, which was written in the latter part of his life. When Cantor received his professorship on the Theory of Sets in 1879 (Breen, 2000), not everyone agreed with and readily accepted Cantor's ideas. His lectures on the theory were not well attended. They criticized his set theory and believed that only numbers were integers and that negatives, fractions, and imaginary numbers did not belong in the field of mathematics proper, but rather as a type of metaphysics. Cantor's career at Halle would not be considered to be a success by many standards. He produced few researchers and doctorate candidates, unlike those who had influenced him. Cantor stood by his theories and now is considered to be one of the greatest mathematicians in history (Breen, 2000).
In 1874, Cantor wrote a paper which appeared in Crelle's Journal. The paper proposed that there are two different orders of infinity. Cantor showed that the set of real numbers can be put in one-to-one correspondence with the set of natural numbers. However the real numbers cannot be put into one-to-one correspondence with the set of natural numbers. This theory in itself raised no eyebrows and stirred no controversy. The fact was established in this paper that the set of real numbers is larger than the set of natural numbers. Cantor was the first to use nested intervals to proved that the set of real numbers is not countable rather than use his diagonal processes produced in his later work.
Cantor commented on his theory as such,
The fear of infinity is a form of myopia that destroys the possibility of seeing the actual infinite, even though it in its highest form has created and sustains us, and in its secondary transfinite forms occurs all around us and even inhabits our minds (George Cantor in Rucker, 1995).
Cantor's next paper appeared in 1878 with the central idea of one-to-one correspondence and a number of theorems concerning such correspondences given along with suggestions for classifying sets based on these assumptions. This paper contains the proof that the set of rational numbers is countable. Cantor used the word "power" ("Machtigkeit") for the first time to establish that two sets, which can be put in one-to-one correspondence with each other, have the same power. Cantor discusses in some length sets as having the smallest infinite power and demonstrates that…