= Countability of finite subsets of a countable set
{title2=$\left|\{A\subseteq\mathbb N:|A|<\infty\}\right|=\aleph_0$}
A finite <subset> $A$ of the <natural numbers>, numbered from one, has the binary code $\sum_{j\in A}2^{j-1}$. Distinct finite <subsets> give distinct codes, because the largest differing exponent exceeds the sum of all smaller powers of two. Thus the finite <subsets> form a <countable set>, although the entire <power set> is an <uncountable set> by the <Cantor diagonal argument>. An enumeration transfers this coding to any <countable set>.
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