There is a nice solution to an even more attractive formulation. Look
at all sentences of set theory given by
*) Let be an infinite set. There exist constants, relations, and
functions obeying a given first order sentence in predicate calculus
EXAMPLE. Let be an infinite set. There exists one-to-one . An extremely simple sentence in predicate calculus.
CONJECTURE. The above Example is the simplest example under *) that is
provably equivalent to AxC over ZF.
Tarski proved that this example is provably equivalent to AzC over a
weak fragment of ZF.
THEOREM. It is provable in a weak fragment of ZFC that the set of true
instances of *) is complete co-r.e. Every instance of *) is either
refutable in a tiny fragment of ZF, or provable in ZFC together with
the true Pi01 sentences.
CONJECTURE. Every reasonably simple instance of *) is either refutable
in a weak fragment of ZF or provable in ZFC. For reasonably simple
instances of *), you can determine which implies which over ZF.
Coming back to “set theoretic maximality”, there is the general idea
that I have been playing with on this list. Namely, perhaps there is a
good robust notion of “imaginable property of the set theoretic
universe”, and we want to say that “any imaginable property of the set
theoretic universe is in some sense actualized”. I know this is
fraught with all kinds of non robustness, inconsistencies,
trivialities, and the like. More tractable might be “any imaginable
kind of set that can be added to the set theoretic universe is in some
sense already present in some form”.
But for statements with enough simplicity, my feeling is that there
may be some criteria whereby we can accept them or reject them as
I’m not ready to be able to put this all together into a legitimate
foundational program targeting “set theoretic maximality” — but
hopefully moving in that direction..