The missing piece of the puzzle
Is there a set that is larger than the natural numbers but smaller than the real ones? This fundamental question is one of the unprovable problems of mathematics. Experts are therefore looking for new laws that supplement the mathematical framework and eliminate this undecidability.
The concept of infinity has always caused difficulties: philosophers and theologians have puzzled over it for centuries - not to mention mathematicians, who only managed to work with the unimaginable magnitudes in the 19th century. In fact, they came across different types of infinities early on, but for a long time they didn't know how to describe or compare them.
In the 1870s, the German mathematician Georg Cantor finally made his breakthrough. By examining sets with an infinite number of elements, he was able to distinguish their sizes from one another and thereby founded modern set theory, on which all mathematics is now based.
But this step did not go smoothly. Scientists had to formulate a collection of so-called axioms - unprovable statements from which all mathematical relationships should follow without producing contradictions. This demanding task has now been largely solved. Since the beginning of the 20th century, a system of axioms called ZFC has been used, which has so far been consistent and includes an extensive theory of infinities.
Nevertheless, modern set theory has weaknesses. As Kurt Gödel showed at the beginning of the 20th century, there are fundamental questions that cannot be answered with it, you can neither prove nor disprove them. Logicians are therefore trying to extend the theory to solve at least some of the stubborn puzzles…