This article has been reviewed according to Science X's editorial process and policies. Editors have highlighted the following attributes while ensuring the content's credibility: It is one of the great questions of mathematics: What does "infinity" actually mean? Which infinities are larger than others?
Can infinities be arranged in a sequence, like the rungs of an endless ladder, so that for any two of them we can always say which one lies above the other? In the 20th century, mathematicians discovered that some of these questions are, in principle, impossible to answer. William Chan of TU Wien has now taken a somewhat more modest approach: He abandons the requirement that there must be a single, clear hierarchy of infinities.
And precisely by doing so, he arrives at something remarkable. If infinities are not arranged along a single line but placed in a two-dimensional system, apparent difficulties disappear and a surprisingly simple order emerges—a kind of two-dimensional map of infinities. The research is published in the journal Proceedings of the National Academy of Sciences.
How can two numbers be compared in the first place? Quite simply: We match every element of one set with an element of another. If there are five cats and five food bowls, we can place one cat in front of each bowl.
We then know that the number of cats and the number of bowls are the same. If some bowls are left over, there are more bowls than cats. "The same idea can be applied to infinite sets," explains Chan from the Institute of Discrete Mathematics and Geometry at TU Wien.
"For example, the set of natural numbers is the same size as the set of even numbers." That may sound strange, because only half of all natural numbers are even. But the even numbers can all be numbered: 2 gets number 1, 4 gets number 2, 6 gets number 3, and so on. No even number is ever left out.
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