The continuum is a range, series or spectrum that changes gradually and has no clear dividing points or lines. Think of the rainbow, which is a continuous whole of different colors, or a continuum of care in nursing homes, which ranges from crisis response to stabilization to safe return to the community.
Continuum is also used to describe something that keeps on going, such as the seasons. It may be a range of colors or a range of math skills.
It can also refer to a broad range of opinions on an issue, such as left-wing or right-wing politics. It can also refer to a range of things, such as gender expectations for women and men.
In classical hydrodynamics, the fluid continuum hypothesis postulates that the fluid has a homogeneous mass distribution and fills completely the space it occupies. It does this by resolving the properties of the fluid in a sphere defined by a representative elementary volume (REV). Once the REV is defined, all activity below it is suppressed. This means that the average value of any fluid property tends to a limit as the size of the REV approaches zero. This is because fluid properties are all constant within the REV, and their linear dimensions and moments of inertia about any axis are equal to zero.
Mathematics is a very dynamic field, where each generation develops new methods to solve old problems. Sometimes these new methods take some time before they are accepted, as was the case with set theory in the nineteenth century.
One of the most important open problems in set theory is the continuum hypothesis, which was first introduced by Georg Cantor. It was so significant that it was placed on Hilbert’s list of open problems in 1900. It has persisted to this day and has been a challenge to mathematicians, despite their efforts to resolve it.
This problem was a tour de force for Godel, who was able to show that the universe he had constructed was consistent with the continuum hypothesis. But it is not the real universe, so there is no way to prove that this is true.
However, this achievement does provide an important example of how the continuum hypothesis can be resolved using current methods. It is very difficult to build a model of the mathematical universe in which the continuum hypothesis fails, and it is even more difficult to build a model in which it holds.
To resolve the continuum hypothesis, mathematicians have had to find a model in which it fails, and then prove that it is inconsistent with the standard axioms of set theory. This is not an easy task, since it requires careful addition of real numbers to the already large number of axioms that are in place for setting the world.
In the case of regular cardinals, the continuity function at k is relatively unconstrained in ZFC, but at singular cardinals of countable cofinality, it is constrained. This is because the jump of k at o can only occur in the case of supercompact cardinals, and the axioms of ZFC do not suffice. It is therefore necessary to use more sophisticated axioms, which are in turn provably in ZFC.