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:
Now, we are able to define exactly what it means for a set to be finite or to be infinite.
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}
Chapters: 1
Corollaries: 2 3 4
Definitions: 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39
Explanations: 40 41 42 43 44
Lemmas: 45 46
Motivations: 47
Parts: 48 49
Proofs: 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89
Propositions: 90 91 92 93 94 95 96 97 98 99 100 101 102
Theorems: 103 104 105 106 107 108
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 $X\in |n|.$ ↩
Please note that the existence of infinite sets is ensured by the axiom of infinity and therefore, there are also infinite cardinal numbers. ↩