The continuum is a range of things that is present at all times. This can be something like a school’s ninth graders and seniors, or it can be the four seasons. It can also be a person’s behavior, which is characterized by a variety of different elements.
The Continuum Hypothesis, or CH for short, is a difficult problem in set theory. It has been the focus of many efforts by mathematicians since it first appeared in the early twentieth century. It is often considered one of the most important open problems in the history of mathematics, and for good reason: if solved, it would prove that set theory was a complete theory, and that all objects in the mathematical universe are infinite.
In the late nineteenth century, Cantor attempted to resolve the continuum hypothesis. He was successful in proving that it holds for certain types of sets called Borel sets, but the problem persisted. Eventually, in the 1920s, Hilbert attempted to solve it, but he was unsuccessful.
It is now thought that this failure was due to the fact that it relied on axioms that were not fully established in modern set theory. In particular, the axiom of choice, which is the basis for Zermelo-Fraenkel set theory, was not fully developed in the early 20th century and thus was unable to resolve the continuum hypothesis.
Godel, who was very much against the solvability of the continuum hypothesis, had some success with it in the 1970s. He formulated three rules that, when applied to the continuum hypothesis, put a limit on the size of its bounds. These rules were not fully proven, but they still demonstrated that the continuum hypothesis is consistent and that it cannot be derived from any of the axioms employed by modern mathematicians.
He was also able to show that the continuum hypothesis is not consistent with the axiom of choice. This result, along with his other work on the same issue, helped make it clear that the axiom of choice does not solve the continuum hypothesis.
Saharon Shelah has been able to solve the continuum hypothesis in an entirely different way. Shelah shows that it is not the number of points on a line that determines the size of a continuum; rather, it is how much smaller you need to make the boundaries between sets.
This approach is called pcf-theory and it reverses the trend of fifty years of independence results in cardinal arithmetic. It also allows us to use the axiom of choice to prove that the continuum hypothesis is not consistent with Zermelo-Fraenkel sets.
Moreover, the pcf-theory is not only useful in the study of continuum hypotheses, it is also very useful for solving other kinds of complex systems, which were long regarded as unsolvable. In particular, it has been used to establish the solvability of many n-category systems, and to solve some questions about fractals that were previously viewed as solvable but which now seem impossible.