The Continuum and Applied Mathematics
The continuum is a line that contains infinite points on it. The problem is that there are only two things that can happen to a line with an infinite set of points marked on it: either the set is countable, or there is no third infinity between them.
In the world of mathematics, there are a number of interesting open problems that are related to the idea of the continuum hypothesis. The first of these is a simple question: how many points are there on a line?
Another important problem is the continuity of sets. This is a similar question, but it is much more difficult to solve. It is related to the idea that there are infinite sets of objects in the universe, and it turns out that this idea has a profound influence on many other areas of mathematics.
Continuum mechanics is an approach to studying the behaviour of matter that ignores its particulate nature and instead studies it on scales that are larger than the distance between the individual particles. It is used to study a huge range of phenomena, from the flow of air and water to rock slides and snow avalanches.
It is also a branch of Applied Mathematics that explores the movement of fluids, including liquids and gases. This applies to a wide variety of phenomena, such as rock slides, snow avalanches, blood flow and even galaxy evolution.
The concept of the continuum came into existence in the classical hydrodynamics field, where it was proposed that fluids exist as continuous spaces and are filled completely by them at all times. This hypothesis abrogated the heterogeneous atomic micro-structure of matter and allowed the approximation of physical properties at the infinitesimal level, which is reached by resolving fluid properties at a macroscopic level defined by a representative elementary volume (REV).
Once the REV is defined, it tends to have perfectly homogeneous properties, with all fluid properties being constant within the REV. As the REV approaches zero, it degenerates to a mathematical point that has unique coordinates in the flow domain and which is called a fluid particle.
In the early 1900s, German mathematician Hermann Cantor tried to prove that the continuum hypothesis was true. He thought he had done so. However, he later found out that his proof was wrong.
This discovery was a major blow to Cantor, who believed that his work was defective because it could not answer such an obvious question. He therefore gave up his work.
Since that time, mathematicians have sought a model in which the continuum hypothesis fails. This is like trying to add a new card to a big house of cards, but without knowing where to start.
Eventually, the idea of Godel’s universe of constructible sets was introduced into the discussion. This is a small universe, and it demonstrates that the continuum hypothesis is consistent.
But it is not the real universe, and so it is impossible to show that it is true. The only way to do that would be to build a model in which the continuum hypothesis holds, and a model in which it fails. It is a hair-raising task, but one that was necessary in order to demonstrate that the continuum hypothesis is consistent.