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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?

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    $\begingroup$ The short answer is that you use the same set theory you would use when you do other kinds of mathematics. Model theory of set theory sometimes has special requirements, and reverse mathematics has other requirements, but model theory is not only about foundations. $\endgroup$
    – Zhen Lin
    Jan 27 at 9:02
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    $\begingroup$ All model theorists I'm aware of are also pretty good set theorists, so generally speaking the answer is 'whatever kind they need to discuss the phenomenon at hand'. $\endgroup$
    – Alec Rhea
    Jan 27 at 9:10
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    $\begingroup$ ZFC and sometimes its popular extensions. If you mix too much set theory like in your third paragraph, it will be called, “set theoretic model theory,” or maybe, “Shelah-stuff.” $\endgroup$ Jan 27 at 9:44
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    $\begingroup$ Chocolate and walnuts with salted caramel and choice. $\endgroup$
    – Asaf Karagila
    Jan 27 at 10:06
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    $\begingroup$ The vote to close is puzzling. This is not a matter of opinion. $\endgroup$ Jan 27 at 21:39

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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.

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Certainly this is not true for most work in model theory, but there is some work in model theory that makes use of sophisticated set theoretical tools, ones which require a ZFC-style approach (or something very akin to that) and so aren't portable to just any old set theory.

For example, Keisler [1] used $\diamondsuit$ in some constructions. Shelah in later work [2] did some rather general work about absoluteness with these sorts of constructions, establishing that $\diamondsuit$ is not necessary for Keisler's results, so they are theorems of ZFC.

Roman Kossak has told me that it's still an open question to find a truly $\diamondsuit$-free proof of Keisler's theorem, i.e. one which doesn't go through $\diamondsuit$ and absoluteness. So, at least until someone figures an alternate, direct proof, we only have those theorems when formulated in a set theory that admits stuff like the construction of $L$, Shoenfield's absoluteness lemma, etc. This definitely puts them outside the purview of naive set theory. I think this should also put them outside of NF (though a NF expert might correct me on what can be done there).

  1. H. Jerome Keisler. “Models with tree structures”. In: Proceedings of the Tarski Symposium (Proc. Sympos. Pure Math., Vol. XXV, Univ. California, Berkeley, Calif., 1971) . Amer. Math. Soc., Providence, R.I., 1974, pp. 331–348.

  2. Saharon Shelah. “Models with second order properties II. Trees with no undefined branches”. Annals of Mathematical Logic 14.1 (1978), pp. 73–87.

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In mainstream model theory, like in most mainstream mathematics (as far as I know, anyway), whenever we actually care about set-theoretic foundations (which is, I think, a bit more often than usual in mainstream mathematics), we usually assume ZFC.

Beyond that, many authors (usually tacitly) assume the existence of an unbounded class of strongly inaccessible cardinals (to have arbitrarily large saturated models of all first order theories), but I don't know any result that actually relies on this hypothesis (it is just a bit more convenient in exposition).

Regarding the continuum hypothesis in particular, it is sometimes used, but in that case, the result is either qualified (a typical example is when you consider a countable ultraproduct of strutures of cardinality $\leq \mathfrak c$ - assuming CH, it is unique up to isomorphism, because it is a saturated model of cardinality $\aleph_1$), or (in one instance that I vaguely recall), its use is justified by the absoluteness of the statement that it is used to prove.

I do not know of any significant, modern results in pure model theory that rely on any non-ZFC set theory. There are probably a lot of results that can be made agnostic (especially those which talk about specific models or theories), but without some flavour of choice (more precisely, the compactness theorem, which is a bit weaker than AC under ZF), most general results will quickly fall apart.

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    $\begingroup$ Jonathan Kirby has an agenda that more results in model theory should use more large cardinal axioms (e.g. Ramsey cardinals) to simplify the argument, understanding that these axioms are generally redundant for the arguments. This will help people feel more comfortable with large cardinal axioms and also provide additional opportunities for grad students to "work out the LC-free details" if desired. $\endgroup$
    – Asaf Karagila
    Jan 28 at 12:58
  • $\begingroup$ Does he have an example of where this simplifies an argument? $\endgroup$ Jan 28 at 13:23
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    $\begingroup$ @KamerynWilliams: Yes (he literally passed by my office when I read your comment). Ramsey cardinals provide you with significantly easier proofs of existence of nice indiscernible sequences. Sure, you can probably get those with Erdős cardinals, and actually from ZFC if all you need is "enough of them" for your proof to go through. But the point is convenience, rather than exact consistency strength. The recent preprint of Halevy–Kaplan about removing the need for inaccessibles by forcing is similar, in that it "removes" the need by appealing to technically more difficult routes. $\endgroup$
    – Asaf Karagila
    Jan 28 at 14:14
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As many have pointed out by now, model theory, as usually practiced nowadays, doesn't require a lot of attention to foundations; the answers to questions in tame model theory like neostability theory are usually absolute. But I wanted to mention a few examples where moving beyond ZFC was important to bring core model-theoretic phenomena to light.

Shelah's 'dividing lines' philosophy, which has had a huge impact on the development of model theory, suggests that it is interesting to find properties that divide the space of first order theories into tame and wild. Good dividing lines have the property that you can prove interesting structure theorems on the tame side and interesting nonstructure theorems on the wild side. But they also have the property that they can be characterized by both 'inside' (i.e. syntactic) and 'outside' (i.e. semantic or set-theoretic) criteria. The outside characterization usually has a set-theoretic flavor and this sometimes requires working outside of ZFC to obtain the equivalence. Here are some examples:

1. NIP - A theory $T$ is NIP if no formula has the independence property, i.e. there is no $\varphi(x;y)$ and sequence $(a_{i})_{i \in \mathbb{N}}$ in a model of $T$ such that $\{\varphi(x;a_{i}) : i \in X\} \cup \{\neg \varphi(x;a_{i}) : i \not\in X\}$ is consistent for all $X \subseteq \mathbb{N}$. This is equivalent to an outside condition: define $\mathrm{ded}(\kappa)$ to be the supremum of cardinals $\lambda$ such that there is a linear order of size $\lambda$ with a dense subset of size $\kappa$. Shelah proved that the number of types over a set of size $\kappa$ in a countable NIP theory can be bounded above by the cardinal $\mathrm{ded}(\kappa)^{\aleph_{0}}$, while the number of types over a set of size $\kappa$ in any theory with the independence property will be $2^{\kappa}$. So one can characterize the (countable) NIP theories by their type counting function, provided $\mathrm{ded}(\kappa )^{\aleph_{0}} < 2^{\kappa}$ for some $\kappa$. This can be arranged by forcing results of Mitchell.

2. Simple theories - A theory $T$ is simple if no formula has the tree property, i.e. there is no $\varphi(x;y)$, tree of tuples $(a_{\eta})_{\eta \in \omega^{<\omega}}$ in a model of $T$, and $k < \omega$ such that

(a) For all $\eta \in \omega^{\omega}$, $\{\varphi(x;a_{\eta | i} : i < \omega\}$ is consistent.

(b) For all $\eta \in \omega^{<\omega}$, $\{\varphi(x;a_{\eta^{\frown}\langle i \rangle}) : i < \omega\}$ is $k$-inconsistent.

This is equivalent to an outside condition. Define the saturation spectrum $\mathrm{SP}(T)$ to be the set of pairs of cardinals $\kappa \leq \lambda$ such that every model of $T$ of size at most $\lambda$ has a $\kappa$-saturated elementary extension of size $\lambda$. It follows from a standard argument that if $\lambda = \lambda^{<\kappa}$, then $(\kappa,\lambda)$ is in the saturation spectrum of any $T$ (in a countable language, say) and Shelah proves that, modulo a forcing axiom he calls $\mathrm{Ax}_{0}\mu$, a theory $T$ is simple if and only if there is some pair $(\lambda, \kappa)$ in the saturation spectrum of $T$ which does not satisfy $\lambda = \lambda^{<\kappa}$. Shelah proves the consistency of this forcing axiom by a class forcing argument.

3. NSOP$_{2}$ A theory $T$ is said to be NSOP$_{2}$ if there is no formula $\varphi(x;y)$ and tree of tuples $(a_{\eta})_{\eta \in \omega^{<\omega}}$ in a model of $T$ such that

(a) For all $\eta \in \omega^{\omega}$, $\{\varphi(x;a_{\eta | i}) : i < \omega\}$ is consistent.

(b) For all $\eta \perp \nu$ in $\omega^{<\omega}$, $\{\varphi(x;a_{\eta}), \varphi(x;a_{\nu})\}$ is inconsistent.

This has an outside characterization as well, via something called the interpretability order (sometimes also called the `triangle star order'). Suppose $T_{1},T_{2}$ are countable theories and $\lambda$ is a cardinal. Say $T_{1} \unlhd^{*}_{\lambda} T_{2}$ if there is a theory $\tilde{T}$ in a language of size $< \lambda$ that interprets both $T_{1}$ and $T_{2}$ and, moreover, has the property that, in any model of $\tilde{T}$, if the interpreted model of $T_{2}$ is $\lambda$-saturated, then the interpreted model of $T_{1}$ is $\lambda$-saturated. Then say $T_{1} \unlhd^{*} T_{2}$ if $T_{1} \unlhd^{*}_{\lambda} T_{2}$ for all sufficiently large $\lambda$.

Assuming GCH, a countable theory is maximal in this pre-order $\unlhd^{*}$ on theories if and only if that theory has SOP$_{2}$. This is really three theorems: Džamonja-Shelah proved that maximality implies a property called SOP$_{2}''$ assuming GCH. Shelah-Usvyatsov proved that SOP$_{2}''$ and SOP$_{2}$ are equivalent for theories, and then Malliaris-Shelah proved recently that SOP$_{2}$ implies maximality.

The point of these examples is that set-theoretic characterizations have been established as part of the method for recognizing dividing lines in model theory and often these characterizations require making assumptions that go beyond what can be established in ZFC alone. It is no doubt true that most questions about simple or NIP theories are absolute, but the fact that they can be given this sort of abstract set-theoretic description is part of what contributes to the sense that they mark meaningful notions of complexity.

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