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nLab > Latest Changes: Cantor-Schroeder-Bernstein theorem

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    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeApr 22nd 2019

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

    diff, v30, current

    • 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 CFinSet^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.

    diff, v31, current

    • 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.

    diff, v32, current

    • CommentRowNumber6.
    • CommentAuthorDavidRoberts
    • CommentTimeJan 27th 2020

    Added mention of and link to

    • {#Escardo20} Martín Hötzel Escardó, The Cantor-Schröder-Bernstein Theorem for ∞-groupoids, blog post (2020), Agda proof

    diff, v33, current

  7. Added arxiv preprint of Escardó’s contribution.

    diff, v34, current

    • 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 GFinSet^G is Boolean for any finite group GG, 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 CFinSet^C).

    diff, v36, current

    • 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 CC refers to the possibility that Fin CFin^C may be be non-Boolean – perhaps it would be better just to cite the specific example C=2C = \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 𝒞≠0,1\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.

    diff, v39, current

    • CommentRowNumber17.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 21st 2024

    Added clarification that the counterexample to the result that CSB => LEM is a non-Grothendieck topos, and that the assumption of an NNO (for the theorem CSB+NNO=>LEM) holds in a Grothendieck topos.

    diff, v40, current

    • CommentRowNumber18.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 21st 2024

    Added the detail that Freyd’s 1994 proof of ’CSB’+NNO=>LEM relies on a version of CSB starting not just from monomorphisms but monos with retractions.

    diff, v40, current

  19. Added example that the CSB property holds for Archimedean ordered fields and ring homomorphisms.

    Anonymouse

    diff, v41, current

    • CommentRowNumber20.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 12th 2024

    Added reference to Dedekind’s proof

    and prose mentioning the specific date and context of the proof. The previous text was the much more tentative “Wikipedia reports that”.

    diff, v43, current

    • CommentRowNumber21.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 12th 2024

    Regarding Cantor’s proof, I’m fairly sure he only originally proved the case of sets with cardinality ≤ω 1\leq \omega_1 (in his 1883 Grundlagen). For finite sets its obvious, for infinite countable sets this is not too bad, and Cantor used sets equipped with a well-ordering iso to ω 1\omega_1 itself for the proof of the last case. In 1887 Cantor published the statement of the theorem for arbitrary sets but didn’t give a proof at all. In November 1882 Cantor wrote to Dedekind saying he didn’t know how to prove the result. Putting the general claim in his 1887 paper without finding a proof is a very gutsy move.

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