The continuum is a range of things that changes slowly over time. It is a natural range in human experience, and in mathematics. It can be used to describe the range of mathematical objects that are known, from simple numbers to complex algebraic structures, and the various methods used to study them.

It is important to understand that in mathematical science, it is not necessarily true that the same thing exists everywhere, or even that there is one single universal method to solve all problems, but that many different methods can be used to find a model for any given problem. It is also a fact that in mathematics, each generation builds on the work of its predecessors.

In the early twentieth century, Kurt Godel began working on the continuum hypothesis. He was a relative newcomer to the problem, but he made a significant contribution to its development.

He was also a key figure in a series of developments that helped to make the continuum hypothesis successful. He was one of the first to explore its implications, and in the 1930s he developed a series of methods that he believed would help to make the hypothesis consistent.

But he also discovered that the continuum hypothesis had a number of problems. He first found that it was possible to show that it was not a valid model for the real numbers, but this was not enough because it still left some important questions unanswered.

Another important issue was whether the continuum hypothesis had any effect on the way that large cardinals were computed. It is a common assumption that for infinite cardinals the hypothesis can be used to find k+ or cf(k)+, so that 20 is equal to 1.

However, this is not always the case; sometimes it is possible for the statement to fail at o (assuming the consistency of a supercompact cardinal). In 1977 Magidor showed that this is the case.

This led to a theorem in ZFC that showed that the jump from o to d is constrained by a very precise set of rules. In 1978 Shelah showed that it was even more constrained, by a result that is reminiscent of what Woodin found.

The problem is that if the statement is true in this model, it is provably false using current methods of computation. This makes the question of how to build a universe in which the continuum hypothesis fails a central problem for mathematicians.

In the context of set theory, it is particularly interesting to ask this question, because most objects in set theory are infinite. So it is not surprising that the continuum hypothesis is linked to most of the open problems in set theory, as well as with the very first problem on the list, which is about how many points are on a line.

The fact that this problem is so closely linked to the question of how many points are on a line indicates that it is very difficult for the continuum hypothesis to be solved with current mathematical methods. But it is not impossible, and mathematicians are in the process of developing new methods that will allow them to resolve this question.the continuum