Tag Archives: Logical maximality

Re: Paper and slides on indefiniteness of CH

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 T_\text{extend}, whose language is that of T extended with a new unary predicate symbol P. The axioms of T_\text{extend} are

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

Note that the models of T_\text{extend} are the (M,P), where M is a model of T and P carves out a submodel of T, where in both cases the schemes are taken over all formulas in the extended language.

We say that T is logically maximal if and only if T_\text{extend} proves \forall x\ P(x). I.e., every such (M,P) above has P = dom(M).

THEOREM 1. \textsf{PA} (Peano arithmetic) is logically maximal. \textsf{Z}_2 (formulated as a single sorted theory in the obvious way) is logically maximal. More generally, for n \geq 2, \textsf{Z}_n (formulated as a single sorted theory in the obvious way) is logically maximal. However, no consistent extension of \textsf{Z} 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 T_\text{elex}, for elementary extension. Of course it is stronger than elementary extension. T_\text{elex} has language that of T extended with a new unary predicate symbol P as before. The axioms of T_\text{elex} are

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

Note that the models of T_\text{elex} are the (M,P), where M is a model of T and P carves out an elementary submodel of M for the language of T, where for both M and the submodel, the schemes hold when taken over formulas in the extended language.

We say that T is elementarily maximal if and only if T_\text{elex} proves \forall x\ P(x). I.e., every such (M,P) above has P = \text{dom}(M).

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 \omega + n” 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