The next key idea of Georg Cantor (1845 - 1918) was to use a given set $A$ as a representative for all sets, which are equipotent to it. In order to be able to do so, Cantor realized that any given set can be selected as a representative of a whole class of sets which are equipotent to it. This abstraction process allowed Cantor to ignore which elements a set has and to concentrate on how many elements it has.
Let $\mathcal X$ be a universal set. Being equipotent for any two sets \(A,B\subseteq \mathcal X\) is an equivalence relation \(\sim\) on \(\mathcal X\). Each equivalence class $[A]\in \mathcal X/\sim$^{1} is called the cardinal number or cardinality of $A$.
Instead of $[A]$, the notation $|A|$ is more commonly used for the cardinality of $A.$ Note that all cardinalities are disjoint.
Proofs: 1
Chapters: 1
Definitions: 2 3 4 5
Examples: 6
Explanations: 7
Lemmas: 8
Motivations: 9
Parts: 10
Proofs: 11 12 13 14 15
Propositions: 16 17 18 19 20 21
Theorems: 22 23
"$\mathcal X/\sim$" denotes the quotient set induced by the relation of being equipotent. ↩