I'm studying real analysis and I know from previous courses that there are countably infinite rationals but uncountably infinite irrationals. However, I haven't done a formal proof about uncountability of irrationals.
I've been thinking about which irrationals are "unique" (in terms of being able to be expressed in terms of other irrationals / rationals), and for this informal thought, I've come up with the following situation which I'll describe formally:
Suppose we define equivalence classes on irrationals, such that
$$[r] = \{ p + r : \forall p \in \mathbb{Q} \}$$
Or stated in other words,
$$x \in [y] \lor y \in [x] \rightarrow x - y \in \mathbb{Q}$$
How many such distinct equivalent classes would exist? Countably infinite? uncountably infinite?