(1)Given Omega-completeness, and assuming compactness implies upward Lowenheim-Skolem, can one get the failure of compactness by showing failure of upward l.s.? I ask because it would seem to take care of "two birds with one stone". (2) if one can proceed in this way, is my proof sketch below correct?
My proof sketch would be as follows:
Claim: Compactness and upward l.s. fails in Omega logic.
Proof: Suppose Omega-logic is complete. We show that upward l.s. fails by constructing a sentence which characterizes Omega logic and which has only countable models. Then for a given T characterizing Omega logic, T has a model. So by the Omega rule we can construct a countably infinite conjunction PHI s.t. T|- PHI. By completeness PHI must have a model, so |= PHI. Now since there are only Omega-many constants in the model, PHI can only have a countable model. Now compactness implies upward l.s.; upward fails, so compactness fails as well.