I need to find the cardinality of the set $S$ of all infinite subsets of $\mathbb{Q}$.
It's easy to prove $\operatorname{card} S=\mathfrak c$ if you first prove that the cardinality of the set of all finite subsets of $\mathbb{Q}$ is $\aleph_0$.
But is there some other proof for the statement without using the second one (about finite subsets)? Probably there is =)
Best Answer
To each real number $r$, you can associate the set $\{q\in\mathbb Q:q<r\}$. Distinct real numbers produce in this way distinct infinite sets of rational numbers. So there are at least $\mathfrak c$ infinite sets of rational numbers. Since you seem to already know tha there are at most $\mathfrak c$ such sets, the Schröder-Bernstein Theorem finishes the proof.