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 this1). 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: , 
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Bibliography (further reading)
 Ebbinghaus, H.-D.: “Einführung in die Mengenlehre”, BI Wisschenschaftsverlag, 1994, 3
 Hoffmann, Dirk W.: “Grenzen der Mathematik – Eine Reise durch die Kerngebiete der mathematischen Logik”, Spektrum Akademischer Verlag, 2011