Infinity lives in smaller infinities

Georg Cantor: The master of infinity

Saturday is a public holiday! At least in Austria - in Germany and Switzerland January 6th is only a public holiday in some regions. The “Three Wise Men”, the “Wise Men from the Orient”, are celebrated. But I'm not really interested in them. But if you should have extra free time on that day, you can use it to think about a real “sage from the east”. Georg Cantor namely, who died on January 6, 1918 in Halle an der Saale and is thus celebrating his 100th anniversary of death this year (or not, because he is dead).

For me, Georg Cantor is one of the most fascinating people in the history of mathematics. He was manic-depressive, repeatedly in psychiatric treatment; has participated in conspiracy theories about the “true” authorship of Shakespeare's works; was convinced that he had received some of his mathematical knowledge directly from God and died impoverished in a sanatorium in Halle an der Saale. And yet he revolutionized mathematics with his work on infinity.

The infinite always fascinates people; probably because we can't really imagine it. And if we try, sooner or later we always end up with contradictions, paradoxes and other problems. Our brains are not made for infinity. But the math! This is exactly one of Cantor's great achievements: he mathematically captured infinity. And not only that: He was even able to show that “Infinity” does not exist, but different degrees. “Infinite” is not always “infinite”; there are infinities that are greater than other infinities.

That sounds absurd, but it is a mathematical reality. And one that can be understood without a great deal of mathematical expertise. All that is needed is the famous “Cantor's second diagonal argument”. And this is how it works:

Let's face the real numbers in front. That is the totality of all rational and irrational numbers. The rational numbers are those that can be represented as fractions. The irrational numbers are the ones that do Not represented by a fraction; which have an infinite number of decimal places without a pattern. 1, 8 and 76423 are rational numbers; as well as 17/23, 3/4 or 1/2. The number pi or the square root of two, on the other hand, are irrational numbers. So far, so clear. The question now is: How many of these real numbers are there?

It seems like the answer can only be “infinitely many”. But appearances don't count in mathematics, you need an exact proof. And that's where Cantor comes in. Let's imagine all real numbers between 0 and 1. We write them down in a long list and use the decimal fraction expansion. Put simply, we get a list that looks like this:


Z1 = 0, a11, a12, a13, ...
Z2 = 0, a21, a22, a23, ...
Z3 = 0, a31, a32, a33, ...
.
.
.

Z1 is the first real number in our list. It starts with “0,” and then there are lots of places after the decimal point, which I have described with a11, a12, a13,…. Strictly speaking, there is always an infinite number of decimal places (for example I can also write the number “0.1” as “0.1000000… ..”). Next on the list is number 2, then number 3, and so on. And here, too, it is clear: there must be an infinite number of numbers in the list. But - and that is the big question - are also available all real numbers between 0 and 1 in the list?

To find out, we'll construct one Diagonal number. It works like this: The new number - let's call it X - has to look like this:


X = 0, x1, x2, x3, ...

X is also a real number with an infinite number of decimal places x1, x2, x3, etc. But not just any decimal places! We choose them according to a simple rule: x1 is equal to 3. Unless a11 is equal to 3, then we choose x1 equal to 4. With this we achieve that x1 always has a different value than a11. For x2 we play the same game, only now with a22: We choose 3, unless a22 = 3, then we choose 4. And so on, for all of the infinite number of decimal places.

Now it comes: In the end we made a number for which made sure is that it differs from the number Z1 (because the first decimal place is different). It is also ensured that it differs from Z2 (because the second decimal place is different). And so on: We have constructed a number that differs from each of the infinite numbers Z1, Z2, Z3, ... in our list.

In other words: We have constructed a real number that Not is included in our infinitely long list! But at the beginning we assumed that our list all contains real numbers. Obviously, this is not the case. The “mistake” lies in the assumption that one can write down the real numbers in an infinitely long list. In mathematics the property is called “countability”: We can count all numbers in our list. We will never come to an end because there are infinitely many. But if we go through the natural numbers (1, 2, 3, etc), we will be able to assign exactly one number to our list and none will be left over.

If we had limited our list to the rational numbers only, we would have no problem. The can to count; here you can assign exactly one natural number to each entry in the list (Cantor proved this in his first diagonal argument). The rational numbers are “countably infinite” or what we normally understand by “infinite”. As Cantor's second diagonal argument impressively demonstrates, this is not possible for real numbers. You can do not count: Even if you assign a number from the list to every natural number, numbers remain that have not been given a number (for example our diagonal number).

There are “more than infinitely many” real numbers. Or, as Cantor formalized it mathematically: There are different ones Powers infinitely large quantities. And the power of the set of real numbers is greater than the power of rational or natural numbers. And it doesn't stop there: You can also construct sets that are “more infinite” than the set of real numbers. And infinities that are even “bigger”. Cantor has shown that there are infinitely many infinities, each of which is greater than the previous one ...

That fascinates me every time. Still, you can't intuitively understand all this stuff. But despite all the paradoxes, inconsistencies and the limitation of our brain, one can grasp the infinities mathematically! And that's kind of very impressive!

P.S. If you want to know more about all the infinities, I strongly recommend the book discussed here.

Saturday is a public holiday! At least in Austria - in Germany and Switzerland, January 6th is only a public holiday in some regions. The “Three Wise Men” are celebrated, the ...