## 116c- Lecture 14

We defined absoluteness of formulas with respect to two classes $M\subset N$; for example, every $\Delta_0$ formula is absolute with respect to $M$ and $V$, if $M$ is transitive. Once we establish a sufficiently long list of properties that are absolute with respect to a transitive model of (enough of) $\mathsf{ZFC}$ and $V$, we will be able to prove a few relative consistency results. The main application will be in the proof that Gödel’s constructible universe $L$ is a model of $\mathsf{ZF}$ (and of choice and $\mathsf{GCH}$), 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 $\mathsf{ZF}$ 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.