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    • CommentRowNumber1.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 8th 2012
    • (edited Nov 8th 2012)

    I’d like to announce the arrival of the pre-preprint Con(ZF+¬WISC)Con(ZF+\neg WISC), available here. Abstract:

    By considering a variant on forcing using a symmetric model for a proper class-sized group, we show that the very weak choice principle WISC—the statement that there is at most a set of incomparable surjections onto every set—is independent of the rest of the axioms of set theory, in particular those of ZF. Our result applies to any set theory which gives rise to a well-pointed boolean topos with nno. The proof does not rely on the axiom of choice, nor does it make any large cardinal assumptions.

    Comments and criticisms, please.

    • CommentRowNumber2.
    • CommentAuthorTobyBartels
    • CommentTimeNov 13th 2012

    Maybe this is clear if I read the paper, but this doesn't sound right:

    Our result applies to any set theory which gives rise to a well-pointed boolean topos with nno.

    Surely ZF + WISC is a set theory that gives rise to a well-pointed boolean topos with NNO, but WISC is not independent of ZF + WISC.

    • CommentRowNumber3.
    • CommentAuthorTobyBartels
    • CommentTimeNov 13th 2012

    Well, looking at the first few pages, you seem to mean that given any well-pointed boolean topos SS with NNO, you construct a boolean SS-topos EE in which WISC fails (and there's something about local connectedness in there too); but that doesn't mean that given a set theory TT and a (or the free?) model SS of TT that this EE is a model of TT. Whereas, presumably you do want that EE is a model of ZF.

    So rather than

    Our result applies to any set theory which gives rise to a well-pointed boolean topos with nno.

    it should be

    Our result applies to any well-pointed boolean topos with nno.

    (Or so it appears based on reading three pages!)

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 13th 2012

    Ok, really it should mean something like ’relatively consistent with’. My aim is to show that Con(ZF)Con(ZF+¬WISC)Con(ZF) \Rightarrow Con(ZF + \neg WISC), or possibly with Con(ZFC)Con(ZFC) instead of Con(ZF)Con(ZF), but I think the LHS can be replaced with Con(SEAR[C])Con(SEAR[-C]) with the result being Con(SEAR+¬WISC)Con(SEAR + \neg WISC), and ditto for ETCSETCS (by the way, what do people call ETCSETCS without AC?)

    In fact (thanks to comments from Mike) I know I haven’t quite got the material set theory version yet, but I think the structural version is complete (I haven’t got the structural\tomaterial translation done).

    So your last comment is correct.

    • CommentRowNumber5.
    • CommentAuthorTobyBartels
    • CommentTimeNov 13th 2012
    • (edited Nov 13th 2012)

    [never mind]

    • CommentRowNumber6.
    • CommentAuthorTobyBartels
    • CommentTimeNov 13th 2012
    • (edited Nov 13th 2012)

    by the way, what do people call ETCSETCS without AC?

    There's CETCSCETCS (constructive ETCS), but that's missing more. Maybe ‘ETSETS’ (^_^)?

    • CommentRowNumber7.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 13th 2012

    There’s CETCS (constructive ETCS), but that’s missing more.

    I definitely need classical logic for the current proof to work…

    • CommentRowNumber8.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 16th 2012
    • (edited Nov 16th 2012)

    After Mike’s comments alluded to above, I have resurrected the following material set theory version:

    If ZF is consistent, then so is ZFA+( a proper class of atoms)+¬WISC + (\exists \text{ a proper class of atoms}) + \neg WISC

    I’m still working on the independence from ZFZF.

    So as it stands, I have only given a pre-Cohen-style independence result, but as a corollary, I now understand the so-called Fraenkel model: it is the material set theory arising from the topos of sets with an action of an open subgroup of (/2) (\mathbb{Z}/2)^\mathbb{N} for a certain natural topology on this group. Open subgroups are the finite-index subgroups iIH i×(/2) I\prod_{i\in I} H_i\times (\mathbb{Z}/2)^{\mathbb{N} - I} for finite II\subset \mathbb{N} and H i/2H_i \le \mathbb{Z}/2. Arrows in this topos are allowed to be equivariant for an open (possibly proper) subgroup of the groups acting on the domain and codomain.

    I have put the above description at Fraenkel model of ZFA.