The continuum is the name given to the set of real numbers. It is a mathematical concept used in a variety of different fields, such as fluid dynamics and rock-slides. The idea is that instead of looking at the behavior of individual particles, such as atoms, we look at the behaviour of whole areas of matter, which are divided and divided infinitely; this approach has applications in many different disciplines, including the study of the motion of air, water, blood flow and even galaxy evolution.
Continuum theories explain variation as involving gradual quantitative transitions without abrupt changes or discontinuities; in contrast, categorical models explain variations as involving qualitatively different states. These theories have been a subject of intense research throughout the centuries, with mathematicians and philosophers seeking to resolve a series of open problems that involve them.
One of the most famous open problems in set theory is the continuum hypothesis (CH). It is a problem that Cantor and Hilbert struggled to solve, and it is still a central problem. Ultimately, however, both mathematicians and philosophers came to the conclusion that CH cannot be resolved using the axioms they employ.
This is a serious problem for a number of reasons. First, it shows that we cannot solve the continuum hypothesis using our current methods. If we can’t, then there is no way to build a model of the mathematical universe in which the continuum hypothesis holds.
Second, if we can’t solve the continuum hypothesis using our current methods, then it shows that there is no information about it lurking in the standard machinery of mathematics. This would be a very bad thing for mathematics.
Third, it shows that the continuum hypothesis is a very difficult problem to solve, and that it’s very likely impossible to find a model that solves it. This is an important point for the future of mathematics, because it suggests that there might be some kind of mathematical universe in which the continuum hypothesis does not hold.
The continuum is also a problem in real analysis, a field that seeks to understand the properties of complex numbers. In fact, the continuum is so hard to resolve that it was considered the hardest problem in real analysis until Kurt Godel figured out how to do it.
In the 1990s, researchers were able to extend Godel’s work by adding more real numbers. This required very careful planning, because it was possible that a few new real numbers could make the difference between the continuum hypothesis and its failure.
This was a very difficult task because it meant that you had to make sure to add only a tiny amount of real numbers at a time, a process that is very similar to adding points to a line. This was an extremely hair-raising task, and it’s one that remains a problem to this day.
Fourth, it shows that the continuum hypothesis isn’t completely consistent with Zermelo-Fraenkel set theory extended with the Axiom of Choice (ZFC). This is a result that sheds some light on the underlying question of whether the continuum hypothesis can be solved using the axioms in ZFC.