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.

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