# Did I find a few (small) errors in the Pradic and Brown 2021 paper that Schroeder-Bernstein implies excluded middle?

I'm looking at the https://arxiv.org/abs/1904.09193 paper (version 2, from 2021) and think it has a few errors. I think I found three small places where the paper needs to be corrected (in the sense that the corrected version is valid - none of the key conclusions are undermined by this).

1. On page 5 it says "Lemma 3.5. Let Q ∈ $$2^{N_∞}$$ be given. If Q(ω) = 1 and ∀n ∈ N. Q(n) = 1, then ∀p ∈ $$2^{N_∞}$$ . Q(p) = 1." I believe the second $$2^{N_∞}$$ should be $$N_∞$$ (because Q ∈ $$2^{N_∞}$$, you'd want to evaluate Q at an element of $$N_∞$$)

2. A few lines down is the statement "Theorem 3.6 ([Esc13, Theorem 3.15]). There is a function ε : $$2^{N_∞}$$$$N_∞$$ such that for every Q ∈ $$2^{N_∞}$$ , if Q(ε(Q)) = 1, then ∀p ∈ $$2^{N_∞}$$ . Q(p) = 1.". The last $$2^{N_∞}$$ should be $$N_∞$$ for the same reason.

3. At the end of the proof of that theorem, in "Hence Q(p) = 1 for every p ∈ $$2^N$$", $$2^N$$ should be $$N_∞$$ again for the same reason.

Perhaps this is a social question - how do I contact an author? - but I also wanted to post here to see whether what I am proposing is right and in case other people are hitting this when trying to read this paper.

P.S. I formalized these proofs at https://us.metamath.org/ileuni/nninfall.html and https://us.metamath.org/ileuni/nninfsel.html in case that helps.

• The preprint lists the authors' institutional affiliations (but not their emails, unfortunately). The institution website will have homepage for their faculty where you can find their contact info - googling authorname institutionname will usually work. If what you found are indeed typos (even if only one out of the three is a genuine typo), the authors will probably appreciate your feedback. Aug 13, 2022 at 20:53
• Assuming you have a valid arXiv account (or if you know someone who does), on the arXiv page you can click "view email" of the corresponding author who uploaded the pre-print. That e-mail will most likely be current (as every time you upload/update a pre-print arXiv asks you to update it if necessary). The uploader for the pre-print you mentioned is swansea.ac.uk/staff/p.r.a.pradic Aug 13, 2022 at 21:13

It may interest you that Martín Escardó also formalized the proofs, see Cantor-Schröder-Bernstein-gives-EM in his TypeTopology library.