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

with equality.

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

exhibiting “maximality”.

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..

Harvey