Mathematicians Accidentally Discover Two New Types of Infinity

The concept of infinity may seem simple at first glance, but it becomes increasingly complex upon closer examination. Infinity refers to an endless sequence of numbers stretching into eternity. However, it also implies the existence of an infinite number of different infinities, forming a complex hierarchy.

Mathematicians and scientists have long studied the nature of infinity, and for over a century it has been known that there are several types. For example, one familiar type of infinity is the infinite set of natural numbers: 1, 2, 3, and so on. But there is also the set of real numbers, which includes negative values and fractions. Delving deeper, one can find an infinite set of infinities.

Recently, researchers from the Vienna University of Technology and the University of Barcelona described two new types of infinity: precise cardinals and ultra-precise cardinals. These infinities stand out due to their unusual properties and do not fit into the familiar linear hierarchy.

Precise cardinals are so large that they contain copies of themselves—like a house that contains smaller replicas of itself. Ultra-precise cardinals include additional mathematical rules that define their creation, similar to a house decorated with blueprints of itself. These new infinities display unexpected properties when compared with the axiom of choice—one of the key principles in set theory, which states that it is possible to form a new set from any collection of sets.

Theorists distinguish three categories of infinities: those that fit standard set theory, those that belong to the realm of chaotic mathematics, and those that are intermediate. It was assumed that the new cardinals occupy a place in this intermediate region, but their exact position is still difficult to determine. Moreover, their properties may contradict a concept known as a hereditary partially ordered set, which suggests that the axiom of choice imposes order even on the largest infinities.

Although the mathematical community has not yet confirmed these discoveries, the study of new types of infinity continues to expand our understanding of this fundamental concept. Work in this area shows that the exploration of infinity is far from complete.