The continuum is a mathematical object that consists of many different points. This concept was first used in the 19th century by Georg Cantor. This was a major development in the field of set theory, and it became a very important part of mathematical practice.
There are a variety of ways to describe a continuum in mathematics, including as a compact connected metric space or as a set of points in a plane. The latter description is very common and has its own technical implications.
One of the most interesting things about the continuum is that it doesn’t seem to have a natural limit. It is an infinite set of points that doesn’t have a beginning or end (see image 12 below).
It also doesn’t have a singularity, which means that it’s not a point – and, in fact, isn’t even a region containing just one place. This makes it very different from the usual conception that a continuum is a collection of points arranged in a certain order.
In the physics and metaphysics community, this idea is often debated. It’s not just a simple problem of what a point is; it’s a question about whether there is a definite beginning and end to things and how they interact with each other.
The problem of how to define a continuum is an important one in the history of mathematics and has been a long time open to debate. The question was included as the first of David Hilbert’s 23 problems presented in 1900, and it still remains a prominent mathematical problem.
Some people think that the answer to this question is “no” because the concept of the continuum is undecidable, meaning that it can’t be proven or disproved from the axioms of set theory. Others believe that it is true because, as Paul Cohen pointed out in 1963, it’s independent of the Zermelo-Fraenkel axioms of set theory.
It’s not the first time that a mathematical question has been regarded as impossible to prove or disprove; it has also been raised in physics, and it is an important issue for some physicists. Today there are a lot of physicists that believe that the way to solve these problems isn’t by trying to reduce everything to a set of points, but instead by considering structures and relations as fundamental objects.
This is an interesting thought, because it’s similar to the idea of thinking of a physical continuum as a collection of atoms that are not necessarily connected with each other. It’s an interesting idea, but it seems that it isn’t necessarily the right approach.
But maybe, like some other ideas that seem to be ‘ultra-weird’, it turns out to have a really good reason.
The continuum hypothesis is a statement that asserts that the cardinal number of the real numbers is the same as aleph 1. It’s been around for a very long time, and it’s an open question as to whether it’s true or not. But it’s also very important for a mathematician, and it was one of the first things that Kurt Godel and Paul Cohen proved to be independent of the axioms of set theory.