How to Solve the Continuum Hypothesis
A continuum is a range of things that tend to stay the same, changing very slowly over time. It can be the range of temperatures in a season, or it can be the range of skills that students learn in school. It can also be a term used to describe a line that runs through something, like the lines of a map or a chart.
The Continuum Hypothesis (CH) is one of the most prominent open problems in set theory. Mathematicians such as Cantor and Hilbert tried to resolve it, but they never did successfully.
This is a problem that has plagued mathematicians for a long time. Even though Godel proved that the continuum hypothesis is consistent, he still did not prove that it is solvable with current methods.
Despite this, many mathematicians continue to believe that the continuum hypothesis is solvable. In fact, both Godel and Hilbert thought that it was solvable.
It was not until Godel introduced a model of the universe that the continuum hypothesis became a problem. In this model, which he called the universe of constructible sets, a set of real numbers was made as small as possible by deleting all but the most essential elements.
In order to solve the problem, mathematicians had to figure out where to add in the new real numbers that they needed. It was a difficult task, because the model was so tiny that it was almost impossible to find space for the new set of real numbers.
When you do find space, it is important to make sure that the new set of real numbers does not violate any other axioms in Zermelo-Fraenkel set theory extended with the Axiom of Choice. In particular, it is important to ensure that the new set does not violate the Axiom of Choice, which says that all axioms must be true for all possible logical statements.
This is a problem because if you do not ensure that the new set of real numbers does not break any other axioms, then you are just making a big mistake. In other words, you are adding a lot of real numbers to the model without really understanding how it works.
Another way to approach this problem is to build a model that shows that the continuum hypothesis is not true, and then see if it can be shown that the new set of real numbers breaks any other axioms in Zermelo-Fraenkel. This is a very interesting technique that Saharon Shelah has used to find a solution to this problem.
This method has resulted in many remarkable results, but the main problem has been that it has shown that there is a provable limit to how large a jump a function can make when it is applied to a continuum. In particular, Shelah has found that the jump cannot be much bigger than 2o = 20 + o4 if all of the cardinals are countable cofinal.