A continuum is a series or range of things that gradually change and have no clear dividing points, though the extremes may be quite different. This is similar to the way that the color of a rainbow changes gradually without any clear line or divider between one color and another.

#### Continuum and fluid mechanics

A common principle in classical hydrodynamics is that fluids exist as continua, meaning that they fill their entire volume. This idea can be used to model a wide variety of physical phenomena, including the flow of air and water, the formation and evolution of snow avalanches, blood flow, and galaxy formation and evolution.

Continuum theory is distinguished from categorical theories of fluid behavior, which explain variations in the properties of a fluid as involving gradual quantitative transitions without abrupt changes or discontinuities. This concept allows the application of differential calculus to fluid dynamics, which is a branch of applied mathematics that is useful in a variety of disciplines.

It is not a hard-and-fast rule, however, that fluids must not contain any particles smaller than an infinitesimally small volume called the representative elementary volume (REV). The REV is a geometric volume of infinity, as well as a mathematical point with unique coordinates in the flow domain, and contains the same amount of material at all times. The fluid’s average property value tends to a maximum within the REV, and the properties of the fluid below this point are suppressed by a sharp cut-off filter.

There are some exceptions, however, and if the size of the REV approaches zero, fluid behavior is governed by a physics that can be described only with the aid of a linear model. This is the case with the fluid behavior of rock slides, and it also applies to the evaporation and condensation of water and other liquids.

The continuum hypothesis, which was first proposed by Georg Cantor in the nineteenth century, asserts that the set of all real numbers is a continuous one and that the size of this set is 2. It’s important to note that the cardinal 20 is part of the continuum, so we can say that 2a=2+ for all a.

#### Godel and the continuum hypothesis

The problem with the continuum hypothesis is that it’s provably unsolvable using current methods, even if you have a complete picture of the universe in which it holds. But this does not mean that mathematicians haven’t found new ways of solving it. In fact, they are still in the process of doing so!

When Godel began working on the problem in 1930, he did not know how to make it solvable. It took him years to get the right answer, but he was eventually able to prove that the continuum hypothesis is consistent.

It’s interesting to note that this discovery is only one step in a much longer history of work on the continuum hypothesis, which was first introduced by Cantor and worked through several iterations. The problem is a prime example of the fact that mathematics is not a static thing–it’s a constantly evolving enterprise that requires a lot of time and effort to develop new theories.