A continuum is a series of things that change gradually and have no clear dividing lines. It can be used to describe a variety of different things, including the colors in a rainbow or a range of things within one line or category.
Continuum mechanics is a mathematical discipline that involves studying the behaviour of fluids by assuming they exist as continua. It is useful in understanding a huge number of physical processes, from rock slides and snow avalanches to blood flow and even galaxy evolution.
The word “continuum” comes from the Greek words (kynas) and (skenas), which both mean “infinite” or “continuous.” The term has been used in many scientific fields, most notably for fluid physics and quantum mechanics. It is the basis of a wide variety of practical applications in science and engineering, such as the behavior of air and water.
It also serves as a theoretical tool in the study of the motion of matter on large scales. This is because it ignores the particulate nature of matter, focusing on the average properties of large numbers of particles instead of individual ones.
Several seminal figures have worked on the continuum hypothesis, including Cantor and Hilbert. Both of them attempted to solve it, but did not succeed.
Cantor was particularly interested in the problem because it had an important connection to the Zermelo-Fraenkel set theory extended with the Axiom of Choice. In the late 19th century he tried to find a way to resolve it, but was unable to do so.
Then in 1930, Godel started working on the question. He was not able to solve it until 1937, but did prove that the continuum hypothesis is consistent with current mathematics methods.
However, he did not consider this to be evidence that the hypothesis was true. He argued that it was not important whether the hypothesis was true or false, only that it was consistent.
This led to the development of a theory called the universe of constructible sets, which is based on the idea that all sets can be reduced to an infinite set, called the continuum. This model was not a physical universe, but it still provided a powerful and consistent proof of the consistency of the continuum hypothesis.
A similar result was made by Saharon Shelah in 1978. Shelah used a similar idea to make some remarkable results on cardinal arithmetic, but the problem was much more general than that of a single Borel set.
Shelah showed that the continuum function on regular cardinals is relatively unconstrained in ZFC, but that it can jump at singular cardinals of countable cofinality. This is provable in ZFC by a deep result of Magidor.
Another important result in this area was made by Silver and Woodin, who figured out that the continuum function can first fail at o, as long as o is a supercompact cardinal. The fact that this result is a good example of the generalized form of the continuum hypothesis is not really relevant for this discussion.