## Order in Infinity

Researchers have known for a long time that infinity is sometimes not just infinity, but actually much more. For decades they puzzled over the size of certain infinite sets. As they have now found out, they are all different - and you can classify them.

Simple math concepts like counting seem firmly embedded in the natural mindset. Studies show that even very young children and animals seem to have limited such abilities. This is not surprising, since counting is extremely useful from an evolutionary point of view - for example, it helps to estimate whether an enemy group is larger than one's own and whether it is worth attacking or retreating.

Over the past few millennia, mankind has advanced these concepts in remarkable ways: starting with dealing with a handful of objects, one found that the methodology can easily be applied to completely different scales. A mathematical framework was soon created that can be used to describe both huge quantities such as the distance between galaxies or the number of elementary particles in the universe, as well as the almost incomprehensible distances in the microcosm, from atoms to quarks.

We can even handle numbers that exceed anything that is currently known to be relevant to the description of the universe: Thus, \(10^{10^{100}}}) (a one followed by 10^{100} zeros, where 10^{100} has one hundred zeros) and do all sorts of calculations with it. However, if you were to represent them in the usual decimal notation, you would need more elementary particles than probably exist in the universe, even if you only use one particle per digit. Because physicists estimate that our cosmos contains fewer than 10^{100 particles.}

But even such unimaginably large numbers are vanishingly small compared to infinite sets, which have been an integral part of mathematics for more than 100 years. Simply counting objects creates the set of natural numbers that many encounter in school. But even this supposedly simple concept seems problematic: There is no largest natural number, you can always count on and find a larger element.

Can infinite sets exist at all? …