FOM: Simpson on AFA and Categories

Allen Hazen a.hazen at
Sun Jun 11 04:32:11 EDT 2000

   Stephen Simpson sketched an "easy" theorem in his 8 June post to the
effect that the category of sets (and mappings) corresponding to Aczel's
"anti-founded" universe is the same as (is isomorphic to) that of the usual
well-founded universe.
    Nothing with categories is easy for me!
    If I understand it correctly, the idea is this:
    (a) On the categorical approach, sets of the same cardinality are
indistinguishable.  (So the objects in the "skeletalization" of the
category of sets are essentially the cardinal numbers.)
    (b) The well-founded universe and Aczel's are alike in containing
absolutely infinitely many sets of every cardinality.
    (c)  So, by global choice, there is a cardinality-preserving (class-)
bijection between the two universes of sets.
   The relative consistency of global choice (in NBG; if you want it in 1st
Order ZF, add a selection OPERATOR to the language) over ZFC is one of my
favorite results pedagogically, since it seems to require only part of the
machinery of an ordinary forcing proof: conditions, but no generic.  Start
with your favorite model of ZFC (class models like "all the sets" **are**
allowed), let a condition be a choice function (=SET of ordered pairs)
defined over part of the model, let one condition extend another if it is a
superset of it, note that
     For every set of nonempty sets x, "the" choice function is
        defined on x
is forced, and there you are!

Allen Hazen
Lecturer in Philosophy
University of Melbourne

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