There is infinity

Is there order among the infinities?

Over the next few decades, a number of mathematicians tried to confirm or disprove Cantor, but all failed. In 1900, in his famous speech of the century at the International Congress of Mathematicians in Paris, David Hilbert named the continuum hypothesis as the first of ten of the most important mathematical problems to be addressed in the coming century.

Kurt Gödel presented the first sobering result in 1940: The negation of the continuum hypothesis - that is, the statement that there are sets whose size is between that of the natural and that of the real numbers - cannot be proven with the means of the ZFC. The last spark of hope of still being able to prove the continuum hypothesis died out in 1963 with a publication by the American mathematician Paul Cohen. In this he brought out a new mathematical method called forcing, from which it follows that the statement ℵ1 = 20 can not be proven with the means of the ZFC.

The results of the two researchers show: If the usual ZFC set theory does not lead to contradictions (which is also unprovable with the means of the ZFC), then ZFC is also consistent with the continuum hypothesis; and also ZFC with the negation of the continuum hypothesis. So you can build a mathematical framework that allows there to be infinities between ℵ1 and 20 there, or one that forbids - neither case will cause problems.

Tidying up in infinity

Assuming that the continuum hypothesis is wrong, mathematicians actually managed to define ten infinities that were at least ℵ in the years to come1 and at most 20 are. The definitions of these infinities are rather complicated. For example, one of them, cov (N), is the smallest required number of so-called zero sets that cover the real numbers. Zero sets are not to be confused with the empty set, for example all countable subsets of the real numbers are zero sets, that is, among other things, the natural or the rational numbers.

However, the scientists could not find out whether the ten infinities really differ from one another or whether, for example, six of them are necessarily the same size. One only knew that they are greater than or equal to or less than or equal to some other infinity. Mathematicians summarized the relationships between the infinite quantities in a so-called Cichoń diagram.

Martin Goldstern and Jakob Kellner from the Vienna University of Technology, together with their colleague Saharon Shelah from the University of Jerusalem, have now proven that eight of the ten infinities can actually be of different sizes - the other two sizes each correspond to two others, which was already known.