# Start a new discussion

## Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

## Site Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorMike Shulman
• CommentTimeApr 22nd 2019

Cited today’s preprint by Pradic-Brown showing that CSB is equivalent to LEM.

• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeApr 22nd 2019

I don’t have much time to look into this now, but this equivalence should be reconciled with the statement made just before this section, that LEM isn’t implied by CSB if we consider topos models. Perhaps I was missing some subtlety when I first wrote that (I think it was me).

• CommentRowNumber3.
• CommentAuthorMike Shulman
• CommentTimeApr 22nd 2019

Interesting, I didn’t see that remark! I think the point is that Pradic-Brown’s proof uses the natural numbers, while $FinSet^C$ doesn’t have a NNO. They remark on this very example in their footnote 5. This should be clarified on the page – but I don’t have time to do it right now.

• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeApr 24th 2019

At the cost of slight redundancy, I fixed the slight inconsistency on the page about whether or not LEM follows from CSB.

• CommentRowNumber5.
• CommentAuthorMike Shulman
• CommentTimeApr 24th 2019

Thanks! I added some more redundancy by mentioning the need for the axiom of infinity in a couple places.

• CommentRowNumber6.
• CommentAuthorDavidRoberts
• CommentTimeJan 27th 2020

• {#Escardo20} Martín Hötzel Escardó, The Cantor-Schröder-Bernstein Theorem for ∞-groupoids, blog post (2020), Agda proof
1. Added arxiv preprint of Escardó’s contribution.

• CommentRowNumber8.
• CommentAuthorGuest
• CommentTimeFeb 3rd 2021
hi. i was looking for arie hinkis book about cantor-bernstein, and wikipedia points to this page. your statement that every mono is iso in "nontrivial" powers of FinSet does not seem true - for *any* topos. an exercise-level proof that cantor-bernstein in a topos implies boolean was spelled out on CATEGORIES mailing list a long time ago, in a conversation with peter freyd:
https://www.mta.ca/~cat-dist/archive/1994/94-2

:)
-- dusko
• CommentRowNumber9.
• CommentAuthorGuest
• CommentTimeFeb 3rd 2021
PS sorry, it is late here. corrections: "for any topos" --> for any nondegenerate topos (different from 0 and 1). the claim is that CB<=>boolean, and the relevant part if boolean => CB. CB => bool is due to dana scott.
• CommentRowNumber10.
• CommentAuthorThomas Holder
• CommentTimeFeb 3rd 2021

The proof on the mailing list uses a NNO, doesn’t it? Actually Freyd mentions there more or less the same counterexample otherwise. See also remark #3 above.

Nevertheless, since $FinSet^G$ is Boolean for any finite group $G$, one should replace ’any” with ’some’ in the nLab entry or use finite monoids that are not groups as counterexample in the footsteps of Freyd instead of mere finite cats.

• CommentRowNumber11.
• CommentAuthorTodd_Trimble
• CommentTimeFeb 17th 2021

Fixed the mistake “every mono is an iso” to what was obviously the intended statement: every endo-mono is an iso (in $FinSet^C$).

• CommentRowNumber12.
• CommentAuthorThomas Holder
• CommentTimeFeb 17th 2021

Thanks Todd! I obviously completely misunderstood what Dusko wanted to say here, sorry for that!

• CommentRowNumber13.
• CommentAuthorTodd_Trimble
• CommentTimeFeb 17th 2021

Actually, I don’t think you did – he was making several points. FWIW, if he’s reading: the “nontriviality” of the power $C$ refers to the possibility that $Fin^C$ may be be non-Boolean – perhaps it would be better just to cite the specific example $C = \mathbf{2}$, although I think any finite non-groupoid would do.

• CommentRowNumber14.
• CommentAuthorThomas Holder
• CommentTimeFeb 17th 2021
• (edited Feb 17th 2021)

I am still very embarrassed to have fed that semi-silly line to Dusko after his helpful comment. Concerning the second point I think it is ok to have the generality in the counterexample (which might be useful on other occasions) but as formulated it is a bit irritating and relies heavily on the reader’s collaboration (e.g. I was reading the ’nontriviality’ as $\mathcal{C}\neq 0,1$).

• CommentRowNumber15.
• CommentAuthorTodd_Trimble
• CommentTimeFeb 17th 2021

Huh, I thought I had announced the most recent edit – see if you think it reads better now. (You’re right – “nontrivial” was certainly suboptimal.)

• CommentRowNumber16.
• CommentAuthorThomas Holder
• CommentTimeFeb 18th 2021

Having posted my comments yesterday after already having called it a day, “as formulated” should be read as “as it was formulated a week ago”, meaning if you had already attended to the point no criticism of the altered version was intended. The current version sure looks ok to me!

I have also added a reference to Freyd’s proof.

• Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
• To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

• (Help)