The continuum is a set of real numbers with the cardinality of infinity. Its existence is not only theoretically important, but also a problem for mathematicians and philosophers of mathematics.
The Continuum Hypothesis
A number of mathematicians, including Weierstrass and Dedekind, have made a mathematical argument for a continuum of real numbers, and the notion has gained a place in contemporary mathematics. However, it is not a universally accepted hypothesis and is considered to be philosophically inadmissible.
According to the intuitionist philosophy, the mathematical continuum cannot be constructed in the mind; therefore, it does not exist. This approach is largely indebted to Kant and Aristotle.
In contrast, the German mathematicians of the Berlin school, especially Kronecker, have dedicated themselves to constructing the continuum on an entirely different basis. They have developed a theory of the natural numbers based on the concept of “set.”
They have defined a set of incommensurables that does not include any actual infinity; thus, they reject an actual infinity as well.
There is another theorem which would entail a further bridge between the countable assemblage and the continuum: It follows that every system of infinitely many real numbers (that is, points) is either equivalent to the assemblage of natural integers, 1, 2, 3, etc. or to the assemblage of all real numbers and therefore to the continuum; this is called the “definition of incommensurables.”
This theorem is independent of ZFC. Its consequences have been extensively studied, but it is still not known whether or not the continuum hypothesis is true.
The continuum is a complex idea that encompasses a variety of different meanings and is used in both physics and metaphysics. The most common usage is that of a manifold whole. This includes both extensive and nonextensive manifolds. It is the basis of a variety of physical and mathematical concepts, such as analytic continuum, compositive continuous, extensive continuous, and so forth.