When I read statements like ‘first order theories can’t control cardinalities of their models’ I wonder, what flavor of set theory is used in a (meta)model theory? (I hope not a naïve set theory, lol).
Can I use, say, New Foundations theory with a universal set of all sets, talking about ZFC models, or vice versa?
Can I ask if any first order theory can have a model with a cardinality in between countable and continuum, thus asking how in a meta-set theory, used in model theory, the continuum hypothesis is settled?
Best Answer
The answer to the question is similar to the answer to the question "what flavor of set theory is used in topology or in algebra?".
A "typical" topologist or algebraist can function quite comfortably with the toolbox afforded by naive set theory (in the sense of Halmos' canonical textbook), whereas, in contrast, there are a good number of topologists and algebraists whose work is highly sensitive to principles of set theory that go well beyond those of ZFC.
Similarly, in model theory, in one extreme we have the work of Shelah and his school, intimately intertwined with extensions of ZFC set theory (including large cardinals), and at the other extreme, ZFC is more than enough for what is nowadays dubbed "tame model theory". It goes without saying that the area of "model theory of set theory" is inseperable from higher set theory.
It is also noteworthy that Appendix A to Chang and Keisler's Model theory (arguably the most influential textbook in model theory) includes three flavors of set theory relavant to model theory: Zermelo set theory , Bernays set theory (otherwise known both as Gödel-Bernays set theory, and as von Neumann-Gödel-Bernays set theory), and Bernays-Morse set theory (otherwise known as Kelley-Morse set theory). The idea being: one needs stronger set theories for certain kinds of model theoretic constructions.
Finally, regarding the interaction between NF (Quine's New Foundation) and model theory: the variant NFU of NF (due to Ronald Jensen) has proved to be an interesting platform for category theory. This was first explored by Solomon Feferman, for an overview and refinement of his work, see this paper by Gorbow, McKenzie and myself (a later version appeared in this volume).
John Baldwin's recent paper Exploring the Generous Domain gives an up-to-date overview of set-theoretical foundations of model theory and category theory.