Definition 1. If we assume $\AC$, every set $X$ can be well-ordered (by Well-Ordering Theorem). Then there is at least one ordinal equinumerous to $X$. We define the cardinality of $X$ as the least such ordinal:
\[\lvert X \rvert = \min \{\alpha \in \Ord : X \equ \alpha\}.\]Another way to define the cardinality without relying on $\AC$ is to use Scott’s Trick (Wikipedia).
Proposition 1. For any sets $X$ and $Y$:
- \[X \equ Y \enspace\lrimp\enspace \lvert X \rvert = \lvert Y \rvert.\]
- \[X \lequ Y \enspace\lrimp\enspace \lvert X \rvert \leq \lvert Y \rvert.\]
- \[X \lnequ Y \enspace\lrimp\enspace \lvert X \rvert < \lvert Y \rvert.\]
References
- [1]
- Wikipedia contributors. Cardinality. Wikipedia, The Free Encyclopedia.
- [2]
- Thomas Jech. Set Theory. 3rd Millennium ed, rev. and expanded. Springer, 2002. p. 29.