We all intuitively have a more or less certain idea about what infinity is. ChatGPT proposes a definition that is quite close to the approximation with which most of us probably feel comfortable: “Infinity is an idea that describes something that has no end or limit. It’s like imagining something that goes on and on forever, never stopping (…) Although we can’t see infinity, we use it to think about things that are very big or endless.”

If we stick to the domain of mathematics, it is interesting that we do not overlook that in the field of numbers we can always add one more, so we will never reach a maximum value above which there is no other number. We will always have a larger number if we continue adding one. However, and here comes a surprising twist, mathematicians realized more than a century ago that there is more than one type of infinity.

It is possible to construct sets of increasingly larger infinities.

At the end of the 19th century, in 1878, the German mathematician, although of Russian origin, Georg Cantor demonstrated for the first time that the infinite set that incorporates real numbers, and which, therefore, includes negative numbers and decimals, is larger than the infinite set of natural numbers or integers. This idea is quite intuitive, but, nevertheless, since it is not possible to count them all, Cantor had to prepare a very meticulous comparison of both sets.

If we think for a moment about the idea we have just explored, we will realize that we have just come to the conclusion that there is more than one infinity. As we have just seen, infinity linked to real numbers is greater than infinity associated with natural numbers. This discovery allowed mathematicians to realize that they could construct sets of increasingly larger infinities, thus creating an infinite hierarchical scale of sets.

For mathematicians, the concept of infinity is very interesting, which has invited some of them to continue researching to understand its properties a little better. In fact, a group of researchers from the University of Vienna (Austria) proposes two new sizes of infinity, which they have called exact and ultra-exact cardinals. The most surprising thing is that these “new” infinities do not obey the rules that describe the infinities that were known until now.

These mathematicians have collected their work in a very interesting article available in the open access repository arXiv. Juan P. Aguilera, one of the researchers who participated in this study, has explained that “these two new sizes of infinity they don’t fit quite well into the linear hierarchy. They interact in a very strange way with other notions of infinity.” We are flirting with very abstract ideas, it is true, but there is no doubt that the discovery these mathematicians have made is exciting.

In fact, they have defined these two sets by making them so large that they contain mathematically exact copies of their entire structure. In some way we can consider that the two newly discovered ones rule in the kingdom of the infinite. However, this is not all. According to Philipp Lückeanother of the mathematicians who participated in this research, if the mathematical community finally accepts the exact cardinals, this discovery “would strongly suggest that chaos reigns.” And, as a consequence, it will support the existence of a new type of astonishing infinity.

Image | Frank Cone

More information | arXiv

In techopiniones | This 17-year-old Chinese girl is a mathematics prodigy. He has defeated students from MIT, Cambridge and Stanford