Power Set of Set of Finite Strings is Uncountable

Theorem

Let $\Sigma$ be a finite alphabet.

Let $\Sigma^*$ be the set of finite strings of $\Sigma$.

Let $\powerset {\Sigma^*}$ be the power set of $\Sigma^*$

Then $\powerset {\Sigma^*}$ is an uncountable set.

Proof

From Set of Finite Strings is Countably Infinite, $\Sigma^*$ is a countably infinite set.

The result follows from Power Set of Countably Infinite Set is Uncountable.

$\blacksquare$

Sources

• 1979: John E. Hopcroft and Jeffrey D. Ullman: Introduction to Automata Theory, Languages, and Computation ... (previous) ... (next): Chapter $1$: Preliminaries: $1.4$ Set Notation: Infinite sets