Father of Set Theory (and Infinity Infinities) Part 3

Cantor’s work from 1874 to 1884 begins the real origin of set theory. Since set theory has become a fundamental part of modern mathematics and its basic concepts are utilized throughout multiple branches of mathematics, it is definitely worth noting this work. Sets had be used implicitly since the beginnings of mathematics (even back to Aristotle) however those ideas of sets had only covered finite sets. For distinction, the “infinite” was kept separate and was often a topic of philosophical and not mathematical discussion. Cantor showed that there could be infinite sets of different sizes some of which are countable and some of which are uncountable. Throughout the 1880s and 1890s his set theory was refined where he defined well-ordered sets and power sets. During this time he also introduced the concepts of ordinality and cardinality. Cantor was also attributed to the arithmetic of infinite sets. What is today known as Cantor’s theorem, states generally that, for any set A, the power set of A (the set of all subsets of A) has a strictly greater cardinality than A itself. Even more specific, the power set of a countably infinite set is uncountably infinite.

Cantor and his groundbreaking work on infinity and set theory is adopted as normal mathematics, today. However, during his time, the work was met with skepticism. It was often mistreated and misunderstood by mathematicians of his time. For example, Henri Poincaré claimed that,”Later generations will regard Mengenlehre (set theory) as a disease from which no one has recovered.” Whereas some mathematicians instantly saw the value of Cantor’s work and it potential to revolutionize mathematics. David Hilbert in 1926 stated that,” no one shall expel us from the Paradise that Cantor has created.” Cantor would discuss his work with few other mathematicians and most of them were unnerved by his concept of infinity. In the 1880s, he encountered some resistance, which at times would be fierce resistance. Some individuals that would challenge Cantor would be Leopold Kronecker (Cantor’s old professor), Henri Poincaré (as stated from his previous quote), Ludwig Wittgenstein (a philosopher) and Christian theologians. These individuals saw Cantor’s work as a challenge to their viewpoints and the nature of God. Cantor was a religious man himself, noted some annoying paradoxes thrown up by his work, but some went further and saw it as the complete destruction of the logical and comprehensible base of which mathematics was created from.

Since Cantor received a lot of pushback, it would not be a farfetched to say some of his mental illness could be linked to the constant disregard for his work. Even further, Cantor contemplated such complex, abstract and paradoxical concepts. As stated in his biography post, Cantor did spend some time off and on in Halle Sanitorium for his manic depression and paranoia. In the last few decades of his life, he did no mathematical work but honed in on his two newfound obsessions. His two obsessions were that Shakespeare’s plays were written by the English philosopher Sir Francis Bacon and that Christ was the natural son of Joseph of Arimathea. Overall, Cantor brought ideas that we take for granted in the world of mathematics today. Students just assume the idea of infinity was always present in math and that set theory was easily made. However, if it was not for Cantor, we might not have both of those mathematical ideas today as well as other ideas that stemmed from infinity or set theory. Or, we possibly could have those ideas and they just were founded in a different time. Case in point, because of Cantor and his groundbreaking work, we have an infinite list of possibilities to where mathematics can lead us.

Sources: Mastin, L. (2010) 19th Century Mathematics-Cantor. Retrieved from: https://www.storyofmathematics.com/19th_cantor.html