Cardinality of Sets

Definition

The sets $A$ and $B$ have the same cardinality if and only if there is one-to-one correspondence from $A$ to $B$. When $A$ and $B$ have the same cardinality, we write $\left|A\right|=\left|B\right|$

If there is a one-to-one function from $A$ to $B$, the cardinality of $A$ is less than or the same as the cardinality of $B$ and we write $\left|A\right|\le \left|B\right|$.

If $A$ and $B$ are sets with $\left|A\right|\le \left|B\right|$ and $\left|B\right|\le \left|A\right|$, then $\left|A\right|=\left|B\right|$

Countable Sets

A set that is either finite or has the same cardinality as the set of positive integers is called countable. A set that is not countable is called uncountable. When an infinite set $S$ is countable, we denote the cardinality of $S$ by ${\aleph }_0$. We write $\left|S\right|={\aleph }_0$ and say that $S$ has cardinality "aleph null"

If $A$ and $B$ are countable sets, then $A\cup B$ is also countable.

The Continuum Hypothesis

Cantor states that the cardinality of a set is always less than the cardinality of its power set. Hence, $\left|{\mathbb{Z}}^{+}\right|<\left|\mathcal{P}({\mathbb{Z}}^{+})\right|$. We can rewrite this as ${\aleph }_0<2^{{\aleph }_0}$. We denote $2^{{\aleph }_0}={\aleph }_1$

We have $\left|\mathbb{R}\right|={\aleph }_1$

The continuum hypothesis is that there is no cardinal number $X$ between these numbers:

$${\aleph }_0<{\aleph }_1<{\aleph }_2<\cdots$$

where ${\aleph }_{n+1}=2^{{\aleph }_n}$