= Solution
A <set> is a <countable set> if it admits an <injective function> into the <natural numbers>; this includes <finite sets> and <countably infinite sets>. A <countably infinite set> can be listed by a <bijection> from $\mathbb N$.
Let $Z$ be the <set> of infinite <binary sequences>. Suppose it were countable and list all its members as $s^{(1)},s^{(2)},\ldots$, writing $s^{(j)}_n$ for the $n$th entry of row $j$. A finite listing could be repeated to make such a list as well. Define the <binary sequence> $y$ by $y_n=1-s^{(n)}_n$. It differs from the $n$th row in coordinate $n$, for every $n$. Thus it belongs to $Z$ but is missing from the list, a contradiction. \b[The <set> $Z$ is uncountable], by <Cantor's diagonal argument>.
Back to article page
