Stationary logic and its friends. I (Q1820149)

This is the first of two papers that deal with L(aa) and related logics. Here we establish: every consistent \(L_{\omega_ 1\omega}(Q)\) sentence has an \b{F}-determinate model (\b{F} is a countable fragment of \(L_{\omega_ 1\omega}(aa))\); and it is consistent that L(Q) has the weak Beth property. A logic has the weak Beth property if it satisfies Beth's theorem where the hypothesis has been strengthened to require that implicit definitions guarantee existence as well as uniqueness.