Cantor went mad before he could could answer whether there is a size of infinity between \(|\mathbb{N}|\) and \(|\mathbb{R}|\). Now we know that there is no answer.

This surprising and controversial result, known as Cantor’s Theorem, was proven by Georg Cantor, in 1981. Before Cantor infinity had been treated as a vague and intuitive notion.

Cantor mathematized infinity, and with this result about the power set created/discovered an elaborate infinite hierarchy of different sizes of infinity.

Since that time, set theory has continued to develop, with many bizarre and counterintuitive surprises along the way.

Here we will tell the story of one surprise -- the Continuum Hypothesis -- which was the focus of Cantor's mathematical work in his later life.

In the second and third modules, we saw that \(|\mathbb{R}| > |\mathbb{N}|\). And in this module we showed that \(| P(\mathbb{N})| > |\mathbb{N}|\). How does \(|\mathbb{R}|\) compare to \(| P(\mathbb{N})|\)? It turns out they are the same size (Cantor also proved this).

But the question is: is there a size of infinity between \(|\mathbb{N}|\) and \(|\mathbb{R}|\)?

In his transfinite arithmetic (discussed in the first module), Cantor defined \(|\mathbb{N}| = \aleph_0\), and defined \(\aleph_1\) to be the next larger size of infinity. So we can rephrase the above question as: is \(\aleph_1 = |\mathbb{R}|\)? The Continuum Hypothesis, one of the most famous mysteries in the history of mathematics, is the hypothesis that the answer is yes.

In the 1880s, Cantor worked obsessively on this question. He wrote to colleagues claiming he had a proof that it was true, but wrote again having found a mistake. Switching tacks, he then attempted to prove the Continuum Hypothesis was false, but failed to find a proof. The evidence and his intuition were inconclusive, and the question teased him back and forth.

"In mathematics the art of asking questions is more valuable than solving problems." -- Georg Cantor

In spring 1884, Cantor had a nervous breakdown. He recovered, but in following years suffered more breakdowns. Soon it got worse, and he was hospitalized in the mental asylum in Halle.

When he wasn't trying to prove or disprove the Continuum Hypothesis, he worked on establishing evidence for his theory that Francis Bacon was the real author of Shakespeare's work. He spent the rest of his life, until he died in 1919, in and out of the mental asylum, obsessed with these two questions.

Legend has it that he was driven mad by either the Continuum Hypothesis, or by the hostility and ostracization he experienced from the mathematics community following his controversial results of the 1870s. Of course, there is no answer to the question of why someone loses touch with reality in the way Cantor did.

It turns out that there is also no answer to the Continuum Hypothesis, at least in the framework of mathematics within which most mathematicians work.

In 1940, Kurt Gödel (1906-1978) showed that the Continuum Hypothesis is consistent with the collection of mathematical axioms known as ZFC. The ZFC axioms are considered the standard, safe, uncontroversial assumptions that mathematicians are allowed to make, and upon which we try to build all our mathematical results.

Gödel's result showed that the Continuum Hypothesis can't be disproved.

"The essence of mathematics lies entirely in its freedom." -- Georg Cantor

But in 1963, Paul Cohen (1934-2007) showed that the opposite of the Continuum Hypothesis (that is, assuming that \(\aleph_1\) is not the same size as \(|\mathbb{R}|\)) is also consistent with the ZFC axioms.

In other words, using the ZFC axioms, it is impossible to prove the Continuum Hypothesis and it is impossible to disprove the Continuum Hypothesis.

One can essentially choose whether to assume the Continuum Hypothesis is true or is false -- whether there is or is not a size of infinity between that of \(\{1,2,3…\}\) and that of a continuous line -- and build on that assumption without ever arriving at a contradiction.