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    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 31st 2018

    Work-up of an article that explains “Scott’s trick” for forming quotients of large classes as classes.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 31st 2018

    Perhaps we should have a page for the category of ZF-classes (if we don’t have one already, I can’t check atm). Aside from being a pretopos, it’s (small) cocomplete with NNO (and W-types more generally) and with subobject classifier. There is this paper which is relevant.

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 31st 2018

    Those are some good points, David.

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeJun 1st 2018

    The closest page we have might be algebraic set theory.

    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 2nd 2018

    I think the standard material set theory model for algebraic set theory is NBG, to that proper classes are first-class objects and not mere syntactic devices.

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeJun 2nd 2018

    But that’s basically equivalent.

    • CommentRowNumber7.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 3rd 2018

    I’m sure you know better than me, but aren’t the classes of ZF just the definable classes, as opposed to NBG classes which could be more arbitrary? I can certainly imagine that there could be classes in a model of NBG that don’t arise from propositions in the first-order language of sets.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeJun 4th 2018

    Yes, you’re right, that’s true. But NBG doesn’t allow you to construct any non-definable classes (in contrast to MK).

    I guess maybe my point is that I don’t think it would be worth having separate pages about the category of ZF-classes versus the category of NBG-classes, as the former is a special case of the latter.

    • CommentRowNumber9.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 4th 2018

    No, just perhaps it would be good for searchability to have something like category of classes, separate from ’algebraic set theory’, which isn’t a very enlightening title for a materially-trained mathematician.

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeJun 4th 2018

    Oh, I certainly agree; I was just pointing out AST as the closest thing we already had.

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