A continuum is a continuous series or range of things that changes gradually without dividing lines or distinct points. For example, a rainbow consists of many different colors, each changing slowly but still forming one continuous range. A continuum can also be used to describe a series or range that is always present, such as the four seasons.
A Continuum of Mathematics
Mathematicians are continually expanding their work and methods. In fact, they are constantly finding new ways to solve old problems that they had previously found insoluble with their methods of the past. This is a normal part of any discipline, and it is a particularly important characteristic of mathematics as a whole.
When it comes to a problem like the continuum hypothesis, this is especially true. In 1900, when Hilbert listed it as the first open problem of his mathematical system, he was attempting to resolve it using methods that were then relatively new.
Throughout the century, there have been some seminal figures who have made significant contributions to solving this problem. Among them is Kurt Godel, who worked on the question on several occasions in the 1930s and on continuously from 1940 until 1976.
His contribution is perhaps most apparent in the way he formulated the idea that there are sets of real numbers in the universe of infinitely many objects, which have a natural structure in which we can describe them as being on a continuum. This is the foundation of set theory, a field that has had to deal with this problem.
He showed that this idea works not only for the usual sets that mathematicians use but also for a special class of sets called Borel sets.
There are many other interesting results, including a number of deep ones, that have been made possible by this technique, which is known as pcf-theory. It has been used to prove that there are certain kinds of sets with more than a billion real numbers in them, and it has also helped to reverse a long streak of independence results for cardinal arithmetic, an area of mathematics that had for the most part not seen much new progress in fifty years.
In the late 1980s, Saharon Shelah began to apply this strategy to a more general form of the continuum hypothesis, in which we are not so concerned with the number of points on a line but rather with how many “small” subsets of a given set you need to cover every smaller subset by only a few points.
Shelah adapted a technique from the early nineteenth century called “stratification” to make this approach more practical. By using this technique, she was able to make the problem more tractable by finding new sets of real numbers that were richer than the original ones, but also smaller than the larger ones.
This was a major accomplishment, and Shelah’s result has been a tremendous boost for the continuity hypothesis, which had been in the doldrums of mathematicians’ thoughts since the death of Hilbert. But, even more importantly, it has allowed mathematicians to explore a variety of other aspects of the mathematical universe and to try to understand how they might be related to each other. These discoveries are helping us to better understand how the problem was actually solved and why it took so long for it to be resolved.