Countability of finite subsets of a countable set ($\left|\{A\subseteq \mathbb{N}:|A|<\infty \}\right|={\aleph }_0$)

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.