Continuing the hopefully productive discussion of the Axiom of Choice, this time with maximality considerations brought in.

I have been discussing elemental ways to express the Axiom of Choice in a logical framework.

**THEOREM 1**. *AxC, over ZF, can be expressed in the form such and such (purely universal) sentence of predicate calculus with equality has a model on any given infinite domain, and also in the form such and such (purely universal) sentence of predicate calculus with equality has a model on any given nonempty domain.*

NOTE: I brought in the true sentences into the previous discussion, to formulate the priveleged position of AxC over ZF in this logical realm. However, I should not have brought the sentences in until I stated the above.

**THEOREM 2.** *AxC cannot be expressed over ZF in the form such and such purely universal sentence of predicate calculus with equality in only relation symbols has a model on any given infinite domain, nor in the form such and such purely universal sentence of predicate calculus with equality in only relation symbols has a model on any given nonempty domain. There is a decision procedure for the true ones of these forms, and they all follow from ZF + “every set can be linearly ordered”.*

We now want to discuss how we can express the Axiom of Choice as “maximality”, also in a logical framework.

**QUESTION:** Can AxC be expressed over ZF in the form *such and such a purely universal sentence in relation symbols only, has a maximal model on every nonempty domain*? Clearly, every such sentences are provable in ZFC.

In the past, I would never throw out a possible jewel like the above without putting a fair amount of effort into answering it. But I am 66 and trying to go into new fields like piano, math physics, mechanics, statistics, etc., to make good on a teenage ambition. So if by good fortune, the above Question turns out to be a jewel (maybe, maybe not), have fun.

**THEOREM 3.** *AxC, over ZF, can be expressed in the following form. Every model of such and such purely universal sentence with relations only, has a maximal extension with the same underlying domain, satisfying the same purely universal sentence with relations only. Clearly, every such sentence is provable in ZFC.*

Proof: Use R,S,P, where R,S are binary and P is unary. Assert that

i. iff not . ii. and implies ).

QED

**PROJECTS**. We would like to get a general understanding of what sentences over ZF can be expressed in these various ways. Various test problems can be formulated, including decision procedure issues.

Consider countable choice, dependent choice, and “the reals are well ordered”. Can these be stated over ZF in any of the forms we have discussed? Or if we restrict the forms above by restricting the domains?

Harvey