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    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 27th 2021


    For a treatment in homotopy type theory see

    • Dan Frumin, Herman Geuvers, Léon Gondelman, Niels van der Weide, Finite Sets in Homotopy Type Theory, (pdf)

    diff, v46, current

  1. Re-organised slightly. Added a brief introductory section, moved a couple of paragraphs of existing content into it. Removed the two context menus ’foundations’ and ’mathematics’ which I don’t think really fit here (the latter is arguably too general to be useful on any page).

    Intend to add a new section in a subsequent edit.

    diff, v47, current

  2. Added a section explaining how to view a finite set as a scheme (over any base).

    diff, v48, current

    • CommentRowNumber4.
    • CommentAuthorUlrik
    • CommentTimeFeb 22nd 2021

    Richard, maybe we can mention (in the “Viewing as schemes” section) that a finite coproduct of affine schemes SpecR iSpec R_i, i=1,,ni=1,\ldots,n, is again affine, Spec(R 1××R n)Spec (R_1 \times \cdots \times R_n). Taking R i=R_i=\mathbb{Z}, we can view the finite set XX as the (affine) scheme Spec( X)Spec (\mathbb{Z}^X). This agrees with what you wrote, but seems more canonical.

    • CommentRowNumber5.
    • CommentAuthorRichard Williamson
    • CommentTimeFeb 22nd 2021
    • (edited Feb 22nd 2021)

    Nice! Great if you can go ahead and make an edit if you have time, as I’ll be tied up until the evening European time!

    Maybe keep the explicit description, but add the nice canonical one in addition?

    • CommentRowNumber6.
    • CommentAuthorUlrik
    • CommentTimeFeb 22nd 2021

    Alternative description of finite sets as affine schemes.

    diff, v51, current

  3. Thanks for the edit!

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeDec 5th 2021

    Added link to MO question mentioning stronger forms of Dedekind-finiteness.

    diff, v55, current

  4. added a predicative definition of finite sets: the important thing to note is that finite subsets are decidable subsets, so one could use the collection of decidable subsets 2 S2^S instead of the power set P(S)P(S) in the definition of finite set.

    Owen Coyle

    diff, v60, current

  5. Correction: not all finite subsets are decidable subsets. Any singleton in the real numbers is finite but generally not decidable unless the real numbers themselves have decidable equality. But finite subsets of sets with decidable equality are decidable subsets, and every finite set has decidable equality.

    Owen Coyle

    diff, v60, current

  6. Reference “Constructively Finite?” paper by A. Spiwack and T. Coquand.

    Stéphane Desarzens

    diff, v62, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeAug 21st 2023
    • (edited Aug 21st 2023)

    Thanks. I have touched the formatting and moved the item into chronological order (now here).

    Also copied it to the author’s entries (at Arnaud Spiwack and Thierry Coquand).

    diff, v63, current

    • CommentRowNumber13.
    • CommentAuthorncfavier
    • CommentTimeNov 8th 2023

    Mentioned that sets with split surjections from [n][n] are finite.

    diff, v64, current

    • CommentRowNumber14.
    • CommentAuthormartinescardo
    • CommentTimeDec 21st 2023
    The first bullet point of "Finiteness predicatively without infinity" is impredicative: it says "let ... be the smallest ... such that ...".
  7. Is B(S)B(S) a decidable subset of 2 S2^S if SS is finite? If so I think one could just replace “smallest subset” with “smallest decidable subset” in that definition.

    • CommentRowNumber16.
    • CommentAuthormartinescardo
    • CommentTimeDec 22nd 2023
    • (edited Dec 22nd 2023)
    How do you know that this "smallest" subset exists if you are working predicatively and without a type of natural numbers?
  8. Merged definition involving decidable subsets into the section “finiteness without infinity”.

    Whoever added the section “finiteness predicatively without infinity” was probably thinking about the definition in Agda, because somebody was talking about this topic in the Univalent Agda discord server. And while Agda is predicative in that it doesn’t have propositional resizing, Agda isn’t predicative in that one still has universes U iU_i, and thus a U iU_i-large subobject classifier Prop iProp_i, for which every impredicative definition still makes sense and is constructible but does not lie in U iU_i.

    Also, the fact that the universes exist means that Agda does have an axiom of infinity, just not the one that we are usually used to via the second-order Peano axioms. Given a universe U iU_i, define that a type AA is finite if it satisfies the constructive definition without infinity but with Prop iProp_i. Then you could collect all the types in U iU_i which are finite using the dependent sum type A:U iisFinite(A)\sum_{A:U_i} isFinite(A). Then since one has Prop iProp_i, one also has the impredicative definition of quotient sets through Prop iProp_i, and if you quotient A:U iisFinite(A)\sum_{A:U_i} isFinite(A) by the mere existence of an equivalence, then you gets the natural numbers.

    In set theory, imagine if you did not have an axiom of infinity or power sets, but you did have function sets and an internal model VV of finitist CZF. Then you could take the set of all subsingletons in VV and quotient it by existence of an isomorphism to get a set Ω V\Omega_V of truth values that you could use in the definition of finite set. Then take the set of all finite sets in VV and quotient it by existence of an isomorphism to get a set N VN_V of natural numbers.

    Edwin Michaelson

    diff, v67, current