Controversy and resistance from the mathematical community.

While this result is now famous among mathematicians, the technique to prove it -- called Cantor's diagonal argument -- is equally well-known. Variations of "diagonal arguments" have become powerful tools in proving a wide range of other results.

In fact, Cantor's first proof that \(|(0,1)|\) is larger than \(|\mathbb{N}|\), published in 1874, used a different method, less elegant or illuminating. It wasn't until seventeen years later in 1891 that he reproved the result using the diagonal argument.

Recognizing that the result would be seen as controversial, it was quietly included in his 1874 paper, amidst other more conventional results, nominally about a "property of the algebraic numbers."

Yet this paper began the controversy and resistance from the mathematical community, which never abated and caused Cantor much mental stress. The strongest resistance was from his old teacher, Kronecker, who was an influential professor at Berlin and had a large following.

Kronecker didn't even believe in the existence of the real numbers. He believed that, "God made the integers. All the rest is the work of man." He went out of his way to make life difficult for Cantor, including blocking his application for a professorship at Berlin, and seeing to it that some of his papers were rejected from publication.

"My theory stands as firm as a rock; every arrow directed against it will return quickly to its archer.

"How do I know this? Because I have studied it from all sides for many years; because I have examined all objections which have ever been made against the infinite numbers; and above all because I have followed its roots, so to speak, to the first infallible cause of all created things."

— Georg Cantor

Cantor was also approached about his ideas by members of the Catholic Church. In the late 1870s, Cantor had many exchanges with priests, and even wrote a letter to Pope Leo XIII. His work was quoted in Christian writings.

Cantor himself was deeply religious and saw no contradiction between his mathematical results and the absoluteness and vastness of his God. In fact, later in life, he insisted his theories had been communicated to him directly by God.

"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," Cantor said.

During this time, he also insisted -- and tried passionately to prove -- that Francis Bacon was the true author of Shakespeare's work. Sadly, he spent much of his last two decades in the mental asylum in Halle, Germany, dying there in the winter of 1919.