This will be my last message about general maximality before I turn – in the next message after this – to “intrinsic maximality of the set theoretic universe”, starting with a discussion of “there exists a non constructible set of integers”. (Recall that I reported receiving enough encouragement for me to continue trying to turn this thread into something more productive).

Here we discuss some kinds of “logical maximality”.

Let T be a theory in the first order predicate calculus with equality, with finitely many constant, relation, and function symbols. We assume that T is given by individual axioms and axiom schemes. A particularly important case is where there are finitely many axioms and finitely many axiom schemes.

We define the associated theory , whose language is that of T extended with a new unary predicate symbol P. The axioms of are

- The extension of P contains the constants and is closed under the functions.
- The axioms, and the schemes of T, the latter being treated as schemes in the extended language.
- The axioms of T, with quantifiers relativized to P.
- The schemes of T, with quantifiers relativized to P. These schemes are treated as schemes in the extended language.

Note that the models of are the , where is a model of and carves out a submodel of , where in both cases the schemes are taken over all formulas in the extended language.

We say that is logically maximal if and only if proves . I.e., every such above has .

**THEOREM 1**. * (Peano arithmetic) is logically maximal. (formulated as a single sorted theory in the obvious way) is logically maximal. More generally, for , (formulated as a single sorted theory in the obvious way) is logically maximal. However, no consistent extension of by axioms is logically maximal.*

The last negative claim is rather cheap. Consideration of it suggests a stronger notion.

Let T be as above. We define the associated theory , for elementary extension. Of course it is stronger than elementary extension. has language that of T extended with a new unary predicate symbol P as before. The axioms of are

- The extension of P carves out an elementary substructure with respect to the language of T.
- The axioms, and the schemes of T, the latter being treated as schemes in the extended language.
- The axioms of T, with quantifiers relativized to P.
- The schemes of T, with quantifiers relativized to P. These schemes are treated as schemes in the extended language.

Note that the models of are the , where is a model of and carves out an elementary submodel of for the language of , where for both and the submodel, the schemes hold when taken over formulas in the extended language.

We say that is elementarily maximal if and only if proves . I.e., every such above has .

Note that elementarily maximal is a nice logical notion. Let’s see what happens when we apply it to ZFC and its extensions.

**THEOREM 2.** *Logically maximal implies elementarily maximal (trivial). ZC + “every set has rank some ” is elementarily maximal. No consistent extension of ZF by axioms is elementarily maximal.*

Another example is the real closed fields. There are two particularly well known axiomatizations. One is ordered field, every positive element has a square root, every moonic polynomial of odd degree has a root. The second is ordered field, plus the scheme of least upper bound.

**THEOREM 3.** *The first axiomatization of real closed fields is not elementarily maximal. The second axiomatization of real closed fields is logically maximal.*

Next time I want to get into real set theoretic issues. Particularly, the relationship between “intrinsic maximality of the set theoretic universe” and the existence(?) of a nonconstructible subset of omega.

Harvey