ok here's a more concrete + human understandable proof of uncountability
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RT @pawnofcthulhu
cute
one quote "the tile admits uncountably many tilings": is it even possible to have a set of tiles that aperiodically tile the plane without having uncountably many tilings? https://twitter.com/cs_kaplan/status/1637996332475359232
https://twitter.com/pawnofcthulhu/status/1638004102352498688

for any tilings of the plane T,S we write T->S if every finite patch of tiles in S occurs infinitely often in T.

(1) we first show that if T->T either T is periodic or there are uncountably many tilings

lets imagine building up a tiling S step by step (for concreteness in say, a square grid we can enumerate the squares in a spiral around the origin and say at each stage we're adding a tile to cover the uncovered square with the smallest number, so we reach everywhere eventually)