Some compact logics --- results in ZFC

by Mekler and Shelah. [MkSh:375]
Annals Math, 1993
We show that if we enrich first order logic by allowing quantification over isomorphisms between definable ordered fields the resulting logic, L(Q_Of), is fully compact. In this logic, we can give standard compactness proofs of various results. Next, we attempt to get compactness results for some other logics without recourse to diamondsuit, i.e., all our results are in ZFC. We get the full result for the language where we quantify over automorphisms (isomorphisms) of ordered fields in Theorem 6.4. Unfortunately we are not able to show that the language with quantification over automorphisms of Boolean algebras is compact, but will have to settle for a close relative of that logic. This is theorem 5.1. In section 4 we prove we can construct models in which all relevant automorphism are somewhat definable: 4.1, 4.8 for BA, 4.13 for ordered fields. We also give a new proof of the compactness of another logic -- the one which is obtained when a quantifier Q_{Brch} is added to first order logic which says that a level tree (definitions will be given later) has an infinite branch. This logic was previously shown to be compact, but our proof yields a somewhat stronger result and provides a nice illustration of one of our methods.

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