How do we prove the existence of uncountably many transcendental numbers? I know how to prove the countability of sets using equivalence relations to other sets, but I'm not sure how to go about proving the uncountability of the transcendental numbers (i.e., numbers that are not algebraic).

If you accept the premise that $|\mathbb R| = |P(\mathbb N)|$ then you know it is uncountable, due to Cantor's theorem.
Take all polynomials in rational numbers, each have only finitely many roots in $\mathbb R$.
Since the set of polynomials is equivalent to $\bigcup_{n\in\mathbb N} \mathbb Q^n$ (the union of polynomials from all degrees), which is countable you have only countably many possibly roots for all rational polynomials and thus only countably many algebraic numbers.
Now consider $\mathbb R\setminus A$ where $A$ is the set of all algebraic numbers, if it was not uncountable then $\mathbb R = \mathbb R\setminus A\cup A$ which was a union of two countable sets, which then would be countable.
