Since model theory must be founded in set theory, do you get different model theoretic results depending on your choice of set theory?

Question

As I understand it, model theory tells you a lot about set theories (plural) but is also founded upon set theory, which means you need to pick one set theory to do model theory within. But different choices of set theory may yield different versions of model theory. So, does that mean it’s impossible to be agnostic about your set theoretic universe if you want to consider model-theoretic results? How much of a difference does your choice of set theory actually make when it comes to model theory?

Answer 1

It ultimately depends on what question(s) you want to ask.

Some natural questions in model theory are highly sensitive to the ambient set theory. For example:

• The existence of saturated models of various theories and cardinalities is very set-theoretic: both basic combinatorial principles like CH and more complicated principles like inaccessible cardinals quickly become relevant.

• Certainly once you move past first-order logic, set theory has a tendency to creep in to even very basic questions. For example:

  • Whether an infinitary sentence has a model is in general independent of set theory, and can be changed by forcing.

  • There is a second-order sentence which is a tautology iff the continuum hypothesis holds, so even the validity question for second-order logic involves nontrivial set theory.

More basic questions, however, are set-theoretically "robust." For example:

• Suppose $M$ and $N$ are transitive models of (a large enough, but still very very small, fragment of) ZFC. Then if $\mathcal{A}$ is a structure in both models and $\varphi$ is a first-order sentence, we have $$M\models\mathcal{A}\models\varphi\quad\iff\quad N\models\mathcal{A}\models\varphi.$$ In particular, we can't change satisfaction by forcing over or taking inner models of a transitive model of ZFC.

On the other hand, there can be surprising failures of robustness:

• If we try to generalize the previous bulletpoint too far, we run into trouble; so there is some very weak set theory underlying even basic satisfaction claims.