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    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 4th 2015

    I have started an article countable chain condition. So far most of the discussion is about topological spaces. (Additional small recent edits at separable space and metric space.)

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeAug 4th 2015

    That’s a lot of info!

    I wonder what the “right” constructive version of this condition is. The notion of antichain doesn’t seem very constructive as-is.

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 4th 2015

    These and other nLab things I’ve started are notes made on the side as I dip occasionally in Kunen’s set theory book, which I took with me on a recent vacation, thinking that I might try once and for all to do a sympathetic reading of consistency results as written by ZF-ists (more or less). Their style is somewhat alien to me.

    I didn’t try to think too hard about constructive versions, but I agree this would be nice to know. Maybe Toby has thought some about this.

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 4th 2015
    • (edited Aug 4th 2015)

    The more general case of the κ\kappa-cc for regular κ\kappa is what is mostly used, and to preserve power sets on taking sheaves. Since the notion of regular cardinal is a bit difficult without AC, I expect for the constructive definition of ccc one really wants a statement that would usually be a theorem in classical maths.

    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 4th 2015
    • (edited Aug 4th 2015)

    More generally, I believe there should be a relative form of this. Given a fibration of sites a la Moerdijk, there should be a relative κ\kappa-cc, corresponding to the case when the (inverse image part of the) induced geometric morphism preserves the appropriate power objects.

    • CommentRowNumber6.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 4th 2015
    • (edited Aug 4th 2015)

    Sorry, I’m being a bit slow. Here’s a nice perspective on ccc (and κ\kappa-cc) that I came up with a little while ago. The exact statement may have to be fine tuned, but it was something like: a poset \mathbb{P} satisfies the ccc (resp. κ\kappa-cc) if (and only if?) sieves (on \top? possibly for all objects) for the double-negation topology can be refined by sieves generated by no more that 0\aleph_0-many maps (resp. κ\kappa-many). Equivalently, I think, there is an equivalent site equipped with merely a coverage such that all covering families have size bounded by 0\aleph_0 (κ\kappa). A set theorist put it to me once as: there’s not much difference between an maximal antichain and a dense subset, the latter obviously

    I’m not sure if this is best phrased (in the general case) as: the (indexing set of the) family of maps in such a covering family is a subquotient of κ\kappa. This plays well with the situation with sheaves as I mentioned above, and has clear extensions to arbitrary sites, not just (,¬¬)(\mathbb{P},\neg\neg)

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeAug 5th 2015

    Wow, neat! I can sort of vaguely see how that might work, but it would be nice to have it written out. Does that mean that the ¬¬\neg\neg-topology is an “ 1\aleph_1-ary site” as here?

    • CommentRowNumber8.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 5th 2015

    Maybe just a weakly κ\kappa-ary site, for a poset satisfying κ\kappa-cc, and κ\kappa here is the minimum possible. The prelimit stuff might be related to what is called being <κ\lt \kappa-directed (or poss the related κ\kappa-closed). Note one can get off-by-one errors here (as in, these are ordinals, so there’s a κ\leq\kappa version)

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeAug 5th 2015

    Ah, right. Anyway, cool! If you ever feel like adding that proof to the page, it would be great…

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