Find the number of ways to choose a countable subset of R I need to find how many ways there are to choose countable subset from $\mathbb R$ the set of real numbers.
This confuse me because $\mathbb R$ is uncountable, although he has countable subsets like Netural Numbers set.
The cardinality is $\mathfrak{c}$. To show this it suffices to find an injective map from the set of infinite sequences of subsets of $\mathbb N$ to subsets of $\mathbb N$. Consider the map
$$ (A_1, A_2, \cdots ) \mapsto \bigcup_{n = 1}^\infty \{p_n^{a} : a \in A_n \}$$
where $p_n$ is the $n$th prime.
The cardinality is $\mathfrak{c}$. To show this it suffices to find an injective map from the set of infinite sequences of subsets of $\mathbb N$ to subsets of $\mathbb N$. Consider the map
$$ (A_1, A_2, \cdots ) \mapsto \bigcup_{n = 1}^\infty \{p_n^{a} : a \in A_n \}$$
where $p_n$ is the $n$th prime.