> Two Dedekind reals (A, B) and (A', B') are equal if and only if they have identical representations. […] Can you elaborate on how you're thinking about decidability?

Direct:

Make one of the sets uncomputable, at which point the equality of the sets cannot be decided. This happens when the real defined by the Dedekind cut is itself uncomputable. BB(764) is an integer (!) that I know is uncomputable off the top of my head. The same idea (defining an object in terms of some halting property) is used in the next proof.

Via undecidability of Cauchy reals:

Equality of Cauchy reals is also undecidable. The proof is by negation: consider a procedure that decides whether a real is equal to zero; consider a sequence (a_n) with a_n = 1 if Turing machine A halts within n steps on all inputs, 0 otherwise; this is clearly Cauchy, but if we can decide whether it’s equal to 0, then we can decide HALT.

Cauchy reals and Dedekind reals are isomorphic, so equality of Dedekinds must also be undecidable.

Hopefully those two sketches show what I mean by decidable; caveat that I’m not infallible and haven’t been in academia for a while, so some/all of this may be wrong!