We defined absoluteness of formulas with respect to two classes ; for example, every formula is absolute with respect to and , if is transitive. Once we establish a sufficiently long list of properties that are absolute with respect to a transitive model of (enough of) and , we will be able to prove a few relative consistency results. The main application will be in the proof that Gödel’s constructible universe is a model of (and of choice and ), but a few other examples will be presented as well.
We proved the reflection theorem and some of its consequences, in particular, that no consistent extension of is finitely axiomatizable.
An important application of these techniques is the use of basic model-theoretic tools to establish combinatorial facts. Some examples will be explored in the next homework set.