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nLab > Latest Changes: structural ZFC

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  1. all the ZFC axioms have structural counterparts, which means that one could come up with a structural set theory which is equivalent in strength to ZFC.

    Anonymous

    v1, current

    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 13th 2022
    • (edited Nov 13th 2022)

    Are Axioms 3 and 4 a bit too weak? I could imagine the fibre axiom permitting something too small, and the product axiom something too big. Is this remedied by the other axioms?

    Edit: oh, I see why 3 should be ok. Still thinking about 4, though.

  3. fixed the axiom of fibers as it seemed to be missing the universal property for fibers

    Anonymous

    diff, v4, current

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 14th 2022

    Fixed typo in Axiom of fibers. The function gg needs codomain AA.

    I wonder, though whether one could posit the existence of an injection i:f −1(b)↪Ai:f^{-1}(b) \hookrightarrow A, not just a function, and then the universal property involving gg could be replaced by a weak universal property (i.e. existence but not uniqueness).

    diff, v7, current

  5. added singleton subset function operator

    Anonymous

    diff, v11, current

  6. I think the old version is not correct. If f:A→Bf : A \rightarrow B is not surjective, then there is more than one relation ϕ:A↬B\phi : A \looparrowright B such that ∀a∈A.ϕ(a,f(a))\forall a \in A. \phi(a,f(a)), because we are free to choose whether ϕ(a,b)\phi(a,b) holds for all b∉imfb \notin \im f.

    Robin Adams

    diff, v14, current

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