[ot]

> like finding larger and larger natural numbers is not the same as having a single number larger than all natural numbers

Very nice analogy!

> König's lemma implies that for tilings both statements are in fact equivalent.

Exactly, and instead of working by shifting the tiling (which would produce a sequence of incompatible tilings) it works by finding a sequence of finite tilings where each is a subset of the next, so it makes sense to take the union of the sequence.