The set of real numbers $\mathbb{R}$ is uncountable. Proof: Assume $\mathbb{R}$ is countable. List all reals in $(0,1)$ as a sequence $r_1, r_2, r_3, \dots$. Each $r_i$ has decimal expansion $0.a_{i1}a_{i2}a_{i3}\dots$. Construct $b = 0.b_1b_2b_3\dots$ where $b_i = 5$ if $a_{ii} \neq 5$, else $b_i = 6$. Then $b$ differs from every $r_i$ at the $i$-th decimal place, so $b$ is not in the list. Contradiction. $\square$

For any set $A$, $|A| < |\mathcal{P}(A)|$. Proof: The map $f: A \to \mathcal{P}(A)$ defined by $f(a) = \{a\}$ is injective, so $|A| \leq |\mathcal{P}(A)|$. To show strict inequality, suppose a bijection $g: A \to \mathcal{P}(A)$ exists. Define $B = \{a \in A : a \notin g(a)\}$. Then $B \in \mathcal{P}(A)$, so $B = g(b)$ for some $b \in A$. But $b \in B \iff b \notin g(b) = B$, contradiction. $\square$ Join Discord PreviousCardinality and CountabilityNextCombinatorics & Probability