The continuum is a series or range of things that gradually change, with no clear dividing lines or points. It is usually used to describe things that are very different from one another, such as the colors of a rainbow or the range of experiences experienced by people of different ages.
It is also used to describe a set of things that are all on the same scale, like the way that behavior can range from effective to severely abnormal. For example, children may be considered part of a continuum of growth and development, with developmental delays and other conditions falling on one end of the spectrum and a complete lack of function or even physical damage on the other.
Continuum models are used to study a huge variety of phenomena, from the movement of air and water to snow avalanches and blood flow. These types of models are called fluid continuum mechanics, and they involve a simplification that allows us to model the motion of large numbers of individual particles without their clumping or collisions.
Many of these phenomena can be modeled mathematically, using the principles of differential calculus. For example, when studying the behavior of air or water, it is important to ignore the fact that each particle can move individually, and instead focus on the average motion of a large number of atoms.
Mathematicians have been struggling with the problem of the continuum hypothesis (CH) since it was first proposed by Georg Cantor in the late nineteenth century, but they could not solve it. Nevertheless, it was listed as one of the open problems for the 20th century by Hilbert, and a number of important developments have taken place in the field.
In the early 1930s, Kurt Godel began thinking about CH. He was a relatively newcomer to the field, but he became a crucial figure in its history.
His work helped solve one of the most important open problems in set theory: he proved that the continuum hypothesis is inconsistent, meaning that it can be proven false using current mathematical methods. But his work on the subject did not stop with CH: he continued to study other aspects of the problem, as well.
For instance, he developed a method to solve an even more general form of the problem, in which the question is not “How many points are there on a line,” but rather “How many smaller subsets of a given line you need to cover all the points by a few of them.” In this method, Shelah showed that the answer to the question can be much larger than what Hilbert was looking for, and so that it might prove true.
The fact that Shelah has been able to do this is particularly noteworthy because it indicates that mathematicians are now in the process of developing new methods that will eventually allow them to solve the continuum hypothesis, and so to resolve it.
Mathematics is a very dynamic discipline, and it is not confined to any particular time or place. It is always evolving and expanding, with each generation bringing in new ideas and new techniques. This is true especially of the problem of the continuum hypothesis, which is now in its fifth generation and still proving to be a challenge for the mathematics community.