The Problem of the Continuum

A continuum is a range or series of points, in which no part can be distinguished from the adjacent parts except by arbitrary division. It is also a type of set.

The continuum hypothesis is one of the most important open problems in set theory, but it is not yet solvable using current methods. Mathematicians have attempted to resolve it twice before.

In the 19th century, the problem was first encountered by Georg Cantor. He was a leading figure in the development of set theory and he struggled to resolve the problem without success. Afterwards, it was taken up by David Hilbert, who also thought that the problem was not solvable but believed that it would be solved in the future.

Kurt Godel, on the other hand, took a more critical view of the problem and began to think about it in 1930, while he was still in his 20s. His work on the problem lasted until his death in 1976.

His contribution to the problem was based on his belief that there are models in which the continuum hypothesis is true, and there are models in which it is not. This led him to develop a model of the universe that he called the “universe of constructible sets” where he removed everything from it that was not essential for mathematics and a consistent continuum hypothesis.

This model was used to demonstrate that the continuum hypothesis was consistent, as it made sense to eliminate anything from a universe that was not necessary to support mathematics. It was a tour de force in its own right, but it did not solve the problem.

Since the 1970s, a number of new developments have occurred in the field of set theory. These developments have radically changed the way that we understand the continuum hypothesis.

They have made it much harder to prove that the continuum hypothesis is provably unsolvable with current methods. This has raised the question of whether a new model is needed to resolve the problem.

There are several ways that this may be done. The most important method involves the use of a new set-theory model called the Zermelo-Fraenkel-Axiom of Choice (ZFC) extended with some additional axioms. This method is now being used to solve other open problems in set theory.

In a ZFC-extended model, it is possible to solve the continuum hypothesis using large cardinals. In particular, it is possible to produce outer models in which d13 > 2 and even ThL(R) 3; but this is not known how to do this with smaller cardinals.

Another method is to use some additional axioms of ZFC, like the so-called scale axioms. These axioms are not present in the standard set-theory machinery, and they show that the continuum hypothesis is independent of the axioms used by modern mathematicians.

In this article, I will describe some of these new developments and their implications for the continuum hypothesis. I will also explain some of the history of the continuum hypothesis, from the point of view of Kurt Godel, who helped make some of these developments possible.

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