How many infinities are in one infinity

Infinite is not always infinite

In mathematics you come across infinity at practically every corner: There are infinitely many numbers, but also infinitely many even numbers or prime numbers. The circle number Pi in turn has an infinite number of decimal places. And all kinds of numbers together are also infinite.

Are these infinities all the same because nothing can be more than infinite? Or are there gradations of infinity after all?

In fact, there are different types of infinity - some infinities are even infinitely larger than others. Investigating what kinds of infinity there can be and how they are related to one another is one of the central research areas of logic and set theory.

Ten differently defined infinities

Cichoń's diagram, which the British mathematician David Fremlin created and which he named after his Polish colleague Jacek Cichoń, brings a little structure to the hierarchy of infinities. To put it simply, it contains ten differently defined infinities and indicates the relationship between them. So far it was unclear how many different infinities there could be in Cichoń's diagram.

Now the mathematicians Martin Goldstern and Jakob Kellner from the Vienna University of Technology, together with their colleague Saharon Shelah from the University of Jerusalem, succeeded in providing an important proof: all infinities in this diagram can be infinitely different. The variety of these infinities is maximal. This finding has now been published in the current issue of the "Annals of Mathematics", one of the most important specialist mathematics journals in the world.

How to count infinity

Which is greater - the set of natural numbers or the set of positive even numbers? Only every second natural number is a whole number, so one could believe that the set of even numbers has to be smaller - but set theory does not see it that way. This apparent contradiction is resolved if one takes a closer look at how one counts infinity in the first place.

Mathematicians use a trick to do this: if you want to compare the size of two sets, for example the set of all chickens and all hen eggs, you assign an egg to each chicken. In the end, if some elements of a set remain unpaired, then this set is larger than the other. If, on the other hand, you find exactly one egg for each chicken, then both quantities are the same.

In the specific numerical example, one would assign two to the even number one, the even number two to four, and so on. Each number gets its unique partner from the other set, so both are the same size.

"Uncountable infinite"

But what about the set of real numbers on the number line with an infinite number of decimal places? There is no way to number them consecutively. The infinity of all numbers is even greater than the infinity of whole numbers. It is said that there are "uncountably infinite" many.

The question now is: is there anything in between? Is there a set that has more elements than the set of natural numbers but fewer than the set of real numbers? The great mathematician Georg Cantor did not believe this and in 1878 formulated the continuum hypothesis: If a set is greater than that of natural numbers, then it must be at least as large as the set of real numbers. (In addition, of course, infinities can be constructed that are much larger.)

A hierarchy of infinities

Many people tried to prove the continuum hypothesis - but with little success: in the 1960s it was finally proven that the hypothesis could not be proven. "But regardless of whether the continuum hypothesis is correct or not, you can add some structure to the hierarchy of infinities," says Jakob Kellner, who conducts research together with Martin Goldstern at the Institute for Discrete Mathematics and Geometry at the Vienna University of Technology.

One can define a number of infinities and examine their relationships to one another. "If you pick out certain pairs from the possible definitions of infinity and prove that one must be greater than or equal to the other, a hierarchy of infinities emerges," says Martin Goldstern. These relationships create a network-like structure known as the Cichoń's diagram. A total of ten different definitions of infinitely large numbers are put in relation in this diagram.

Overview of Cichoń's diagram

But so far only individual connections in this network have been examined. Getting a global view of Cichoń's diagram is much more difficult. And so it was unclear for a long time how much freedom the diagram actually allows for different infinities: How many of them can be different? Is there perhaps a set of three entries in this diagram, of which two must be logically imperative?

"If one assumes that the continuum hypothesis is correct, then the matter is very simple: Then between the infinity of the natural numbers and the infinity of the real numbers, no further kind of infinity is possible, and all entries in the diagram are the same," says Jakob Kellner .

If one does not assume the validity of the continuum hypothesis, however, things look completely different: As Goldstern, Kellner and Shelah were able to show, ten different entries are actually possible. Cichoń's diagram allows the greatest variety of infinities that is conceivable.

This long-discussed riddle about Cichoń's diagram has now been solved. Nevertheless, the investigation and characterization of infinities remains an important research area in mathematics - a century and a half after Georg Cantor. Because the number of mathematical puzzles around infinity should also be infinitely large. (red, August 19, 2019)