The continuum is one of the most important open problems in set theory. It is also a problem that has been the subject of some of the most intense debates in mathematics.
In many ways, it is the most basic question that mathematicians ask–how many points are on a line?
It is a simple problem, but it has some very interesting and complex connections with other questions. And it is a problem that has lingered with mathematicians throughout the centuries.
In the nineteenth century, the famous set theorist Georg Cantor was trying to resolve this problem. He was able to make some progress. But eventually he found that it was not possible to find a way to solve this problem on the basis of his set theory axioms. This meant that the continuum hypothesis was not solvable on its own.
Later, the great mathematical philosopher and set theorist David Hilbert began to struggle with this problem. In fact, the problem was placed first on his list of “open problems” that mathematicians should attempt to solve during the 20th century.
Although he did not succeed in solving the continuum hypothesis on his own, Hilbert continued to believe that it could be solved in some future. He even said that nothing less than “the glory of human existence” would depend on his ability to find a solution to this problem!
But then, in the 1930s, a new and different mathematician, Kurt Godel, took up the issue. He was a relatively late arrival to the field, but his contributions have been visible in virtually every aspect of the continuum problem since then.
He used a technique called the “universe of constructible sets.” In this model, there are infinitely many nontrivial universes, and each of these universes contains a number of sets that is as small as is conceivable by throwing out all of the other elements in the universe that are not absolutely essential.
This model sounded very limiting to many mathematicians, but it was a breakthrough in understanding how it might be possible to build a model for the failure of the continuum hypothesis. But it was still a long and difficult process.
What we are seeing is that, after a hundred years of trying to solve this problem, mathematicians have finally started looking for new methods that might be able to do this. And, if they succeed in these new methods, we will probably find that the continuum hypothesis is actually solvable.
Another important mathematician, Saharon Shelah, has done some very spectacular work in the area of cardinal arithmetic. Shelah has shown that for a large number of different sets, it is possible to compute the size of the continuum for all k by computing the size of k-subsets of the set.
Shelah’s work is a very significant development in the history of mathematics, as it is the first time that a new type of method has been used to solve an old and important problem. The new method is called “pcf-theory” and it has had a major impact on the way that mathematicians approach the problems of their day.