I don't understand Dedekind cuts

[Warp]

I have now tried for an hour to make ChatGPT make me understand how this works, without success, so I humbly ask for a clearer explanation.

The succinct way to express what I don't understand is this:

A "Dedekind cut" simply splits the set of numbers into two, without dropping any values and retaining ordering, so that every value on the left of the cut is less than every value on the right of the cut. Let's apply this to the set of rational numbers:

1. Every rational number splits the set of rationals into two subsets, and this split is unique.
2. Every irrational number splits the set of rationals into two subsets, and this split is unique.

(In this context "unique" means that two different numbers used to do the split never produce the same subsets of the rationals. Every unique number produces a unique split, no duplicates.)

This makes it sound like the two sets are of the same size. Yet, somehow, there are (uncountably many) "more" of the second type of cuts than the first type. How can a countably infinite set have uncountably many unique splits of this kind? It defies intuition and logic.