The continuum is one of the most important open problems in set theory. It is also one of the most difficult. It has been the subject of many attempts by mathematicians throughout the history of the field, but no one has been able to resolve it.
A continuum is an infinite series of points that is not distinguishable from other parts of the series, except by arbitrary division. The term is used to describe a vast array of phenomena, from the movement of air and water in everyday life to the study of rock slides, snow avalanches, blood flow, and galaxy evolution.
Continuum Mechanics, a subdiscipline of applied mathematics, focuses on the behaviour of fluids such as liquids and gases. It is also the basis of the study of the movement of materials such as rocks, ice, and snow, and the movement of molecules in the atmosphere.
It postulates that the properties of a fluid are resolved at an upper scale, called a representative elementary volume (REV), which is defined by a small sampling volume as large as needed to resolve spatial variations in a fluid’s properties, but not so large that molecular action prevents the resolution of such a small sample. Once a REV is defined, any activity below that level is suppressed and the fluid has a perfectly homogeneous property.
The continuum hypothesis was first presented by Cantor, who believed it to be solvable and placed it on his list of open problems for the 20th century. However, Cantor was not able to prove that the continuum hypothesis was true.
This was a major blow to Cantor, because it meant that he had failed to solve a problem that he was attempting to resolve. Later, both Godel and Hilbert attempted to solve the continuum hypothesis, but without success.
Both Cantor and Godel believed that it was solvable, but both men could not prove it to be true. Even so, both men were adamant that it would be solved.
In order to solve the continuum hypothesis, the mathematician must carefully add real numbers to Godel’s model, a task that is incredibly difficult. It is like trying to add a new card to a house of cards, which already contains a lot of other cards.
Fortunately, it is possible to make a surprisingly powerful tool for solving the continuum hypothesis: a model for how it can be made to fail. This model involves an axiom of choice that Godel introduced, and which he had to show was consistent with his continuum hypothesis.
If the axiom of choice is valid, then a continuum function will be able to resolve the same set of points as the corresponding regular function for any countable cofinality. This is a surprising result, and shows that the continuum function can be constrained in some way by the axiom of choice.
The axiom of choice can also be used to prove that the continuum hypothesis is not solvable. It is true that some provably undecidable statements do exist, but these are not relevant to the solvability of the continuum hypothesis. This is because the continuum hypothesis is not a theory about the smallest possible universe, but rather a way of modeling the consistency of the continuum hypothesis.