A continuum is a range of things that are always present and change slowly over time. It can be a physical range, like the seasons; it can be a social range, such as gender expectations or the severity of crimes; and it can be a mathematical range, like the students in a high school who move on to different math courses.
Continuum Hypothesis
The continuum hypothesis is one of the most important open problems in set theory. It is considered a foundational problem of the field and, at its peak, was listed as the first on Hilbert’s list of open problems for the 20th century.
Although the problem has not been solved, it is still an important subject of study for both mathematicians and philosophers alike. In particular, it has a deep connection with the history of mathematics and is one of the central questions that sets theory attempts to answer.
In the twentieth century, there were many seminal results that supported or challenged the continuum hypothesis. In addition, a new way of thinking about the problem was introduced in the 1970s, when Saharon Shelah posed an interesting question about the size of sets.
If a set is large enough, its members are more similar to each other than they are to those of smaller sets. This makes the continuum hypothesis easier to understand than previously.
Moreover, a large set can be broken up into many small sets. In this sense, the continuum hypothesis is a model of how sets in general can be broken up into many small and more distinct parts.
These small and more distinct parts can then be analyzed and understood using mathematical methods, such as differential calculus or spectral theory. This is what enables us to learn more about the nature of the universe, and it has also been used to make discoveries that are difficult to explain without mathematics.
For example, the idea that all matter is made up of individual particles (atoms) was discarded in favor of a simpler model of matter, called continuum mechanics. In this model, atoms are replaced by infinitesimally small volumes of material (called fluid particles).
Each volume is a sample of the whole medium and, since each particle contains the same amount of material at all times, there is a direct correspondence between each space point and a fluid particle in that volume. The volume is then called the sampling volume or REV.
When this REV gets so small that the fluid properties are no longer able to change, it degenerates into a point. This point can then be interpreted as an equivalent to a mathematical quantity, such as the coordinates of the flow domain or the average value of the fluid property.
Despite the success in proving that the continuum hypothesis holds for a special class of sets, mathematicians continued to speculate that it was not solvable in general. This was particularly true for Hilbert, who had considered it an open problem until 1900.