What things are infinite

The myth of infinity

The starry night sky has always touched people deeply and made people think. How many stars are there? Is the universe infinitely large? These questions come to mind when you look into the seemingly immeasurable black of the sky speckled with twinkling stars. But we can neither really imagine an infinite universe nor a spatially limited universe.

"Two things are infinite, the universe and human stupidity," Albert Einstein once joked, "but I'm not quite sure about the universe yet." And to this day, scientists are not entirely sure of this. If the big bang model is accepted, then the universe can only have inflated to a certain size in the 13.7 billion years of its existence to date. The radius of the visible universe is therefore a maximum of 13.7 billion light years, and it would be finite. Beyond this volume would be a fundamentally inconceivable terra incognita. Yet the human mind continues to ask: what lies beyond the edge of our universe if it should be finite?

Most astrophysicists are currently convinced that the universe is expanding and that it is infinite, at least in perspective. Just a few years ago, another theory was being discussed: If the mass in space were to exceed a certain value, then the currently observable expansion of the universe would come to a standstill due to gravitational forces. Then it could even collapse again. The discovery of so-called dark matter and the also quite fresh knowledge that neutrinos - tiny elementary particles that are produced everywhere in stars - are not free of mass, spoke in favor of this theory of a "back to the big bang" and a finiteness of space. Scholars had already represented this view of things in antiquity - without any telescopes or research satellites. Aristotle believed in a finite universe, which, however, should be surrounded by an infinite void.

But at the moment the astrophysicists are again more on the line of the philosopher Immanuel Kant, who came to the conclusion in his Koenigsberg study that the universe is infinite. Only a few years ago researchers discovered the extremely mysterious "dark energy" that is uniformly distributed in space and acts like an antigravity force. This overcompensates for the attraction of visible and invisible matter: The universe expands infinitely. "The eternal silence of these infinite spaces terrifies me," said the French philosopher Blaise Pascal as early as the 17th century. And Johann Wolfgang von Goethe shuddered: "Where can I get hold of you, infinite nature."

Mathematicians, who are not even afraid to use a symbol for "infinite" in their formulas, undoubtedly have the most informal relationship to the infinite:. Exactly 350 years ago, in 1656, the British mathematician John Wallis invented this symbol. Possibly he was inspired by a mythological image that was known as early as 1600 BC: a snake coiled into an eight and biting its own tail.

In the world of mathematics, one inevitably comes across the phenomenon of infinity. One does not have to be a scholar to recognize that the sequence of the natural numbers 1, 2, 3, 4 ... obviously does not end and that for every number n one can find a number n + 1 that is even greater than n. In short, there are an infinite number of natural numbers and most people do not worry about accepting them. It is more difficult to see that the set of natural numbers is not greater than the set of natural numbers, divisible by two, 2, 4, 6, 8, ... One could intuitively assume that one infinity is twice as powerful as the other. But two times remains, just as two times zero is unchanged zero. The zero and the infinite are closely related anyway. An Indian mathematician recognized as early as 678 that if you divide any number by zero, you would reach infinity. Therefore it is forbidden to divide numbers by zero. Try it out with your calculator. It will report "Error" to you or, at best, display ""

Many areas of mathematics are surrounded by an aura of infinity. Vibrations of all kinds are represented as the sum of an infinite number of sinusoids or limit values ​​are calculated from infinitely long sequences of numbers. Another reason why the famous number attracts so much attention is that it contains a never-ending, infinitely long sequence of decimal places. In addition to the infinitely large, mathematicians also master the infinitely small, which they deal with just as unabashedly in differential calculus, for example. At the point of contact between infinitely small and infinitely large, things get particularly exciting. For example, with the simple series 1 + 1/2 + 1/3 + 1/4 + 1/5 ... the question arises whether the result of this infinite arithmetic problem is a certain number, a so-called limit value, or whether it is always ever-smaller summands in their infinite succession finally manage to jump into infinity. The French bishop Nicholas of Oresme was the first to find the answer in 1350: This sum is infinitely large. However, if one squares the denominators, the infinitely many summands 1 + 1/4 + 1/9 + 1/16 + ... no longer succeed in overcoming the hurdle to infinity. The total is then astonishingly 1/6 2 as Leonhard Euler (1707-1783) was able to show.

The mathematician Georg Cantor (1845-1918) is considered to be the grand master of infinity. He succeeded in proving that there are different, differently sized infinities. On the one hand there are "small" infinite sets, which are referred to as "countable infinite". The prototype for this are the natural numbers. Cantor showed, however, that there are also infinite sets that are "larger" in a predictable sense, that is, "uncountably infinite". And he went one better: There are even an infinite number of "sizes" of infinity. For every infinity, however great, there is still a greater infinity. One can no longer really imagine this, but we can believe the genius Cantor that he is right about it.

"No other question," the famous mathematician David Hilbert (1862-1943) was convinced, "has never moved the human mind as deeply as the infinite." When dealing with the infinite, however, a distinction was not always made between the fact that this term has at least three meanings: the infinite in physics, the infinite in mathematics and finally absolute infinity as it is understood theologically. It is by no means certain that the infinite in physics can be described at all with the infinity concepts of mathematics.

In fact, physicists try to avoid "singularities" in their theories, in which a physical quantity takes on the value. At school we learned that electrons are point-like elementary particles with a certain mass and electrical charge. If that is true, then the matter density and the charge density in the electron would be infinitely large. Mathematicians have no problem with that. But this gives physicists a stomachache. In the past few decades they have therefore developed theories that describe electrons and other elementary particles as spatially extended objects - so-called strings.

Many associate the term infinity not so much with physics or mathematics, but with the religious and the mythical. "Man has a deep need to recognize meaning in his life. In this respect, the longing for the transcendent and the infinite in the spiritual sense is clearly anchored in the human brain," explains Bremen-based brain researcher Professor Gerhard Roth.