Cardinals allow us to distinguish between finite and infinite sets very easy. But to really understand how this is accomplished, we have to make a connection between two facts:

- Classic mathematics allows some
*atomic objects*which are not sets themselves, like numbers, points, etc. In modern mathematics, almost*all*mathematical objects are sets. We will see later in number systems, how numbers, including natural numbers, can be defined from a purely set-theoretical perspective. For the time being, just please keep in mind that not only the*set of natural numbers*$\mathbb N$ is a set, but also*every natural number*$n\in\mathbb N$ is a set. - For this reason, each natural number $n$ is a representative of its own cardinal equivalence class $|n|.$

Now, we are able to define exactly what it means for a set to be finite or to be infinite.

**Definition**: Finite Set, Infinite Set

A set $X$ is called **finite**, if both, $X$ and a natural number $n\in\mathbb N$ belong to the same cardinal, formally $|X|=|n|$ (we could even write $X\in |n|$ to express this^{1}). If such a natural number $n$ does not exist, then the set $X$ is called **infinite**.^{2}

^{1} Please note that the cardinal $|n|$ is an equivalence class, by definition, and must not to be mixed up with the absolute value of $|n|$, with the same notation! This fact justifies the notation $X\in |n|.$

^{2} Please note that the existence of infinite sets is ensured by the axiom of infinity and therefore, there are also infinite cardinal numbers.

| | | | | created: 2014-10-10 20:41:12 | modified: 2016-12-22 23:04:03 | by: *bookofproofs* | references: [656], [983]

[983] **Ebbinghaus, H.-D.**: “Einführung in die Mengenlehre”, BI Wisschenschaftsverlag, 1994, 3

[656] **Hoffmann, Dirk W.**: “Grenzen der Mathematik – Eine Reise durch die Kerngebiete der mathematischen Logik”, Spektrum Akademischer Verlag, 2011

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