A Jonsson $\omega_1$-like model of set theory

This is a talk at the CUNY Set Theory Seminar, November 15, 2013.

In 1962, Bjarni Jónsson asked whether there is a cardinal $\kappa$ such that every first-order structure (in a countable language) of size $\kappa$ has an elementary substructure of size $\kappa$. In his honor, a first-order structure is called Jónsson if it has no proper elementary substructures of its cardinality, and a cardinal $\kappa$ is called Jónsson if there are no Jónsson structures of size $\kappa$. It turns out that Jónsson cardinals are large! For instance, if there is a Jónsson cardinal, then $0^\sharp$ exists. It is known that $\omega_1$ cannot Jónsson, meaning that there are Jónsson first-order structures of size $\omega_1$. In this talk, I will prove a theorem of Julia Knight that there is a Jónsson $\omega_1$-like model of set theory [1].

Keisler and Morley [2] proved that every countable model has a top extension using an omitting types argument. Suppose that $M\models{\rm ZFC}$ is countable. Extend the language of set theory by adding constants $c$ for all elements of $M$ and an additional constant $e$. Let $T_M$ be the theory ${\rm Th}(M,c)_{c\in M}$ together with the sentence "$e$ is an ordinal" and the collection of sentences $\{e>\alpha\mid\alpha\in{\rm ORD}^M\}$. For every $c\in M$, we define the corresponding type $p_c(x)$ consisting of the formula $x\in^M c$ and the formulas $\{x\neq b\mid b\in^M c\}$. Using the Omitting Types Theorem, it is not difficult to see that $T_M$ has a model omitting all types $p_c(x)$ and this is our desired top extension.

Another folklore proof uses a variation on the Skolem ultrapower construction. Skolem ultrapowers were introduced by Skolem who used them to show that there is a countable nonstandard model of the Peano Axioms. The ultrapower is built out of definable functions $F:{\rm ORD}^M\to M$ and an ultrafilter on the definable classes of ordinals of $M$. If $M$ has a definable global well-order, the ultrapower satisfies the Łoś Theorem. If it doesn't, we can use forcing (see here) to extend it to a model of ${\rm GBC}$ (Gödel-Bernays set theory) and build the ultrapower out of this larger collection of classes. By choosing an ultrafilter with the property that every class function $F:M\to V_\alpha^M$ is constant on some set in it, we ensure that the ultrapower is a top extension.

Julia Knight's construction of a Jónsson $\omega_1$-like model hinges on the following observation. Let $p(x,y)$ be the type expressing that $x$ and $y$ are not definable from each other. It is easy to see that any model of size $\omega_1$ which omits $p(x,y)$ must be Jónsson. Knight shows that that if $M$ itself omits $p(x,y)$, then the theory $T_M$ has a model which omits $p(x,y)$. Since we can omit countably many types at once, we can omit all $p_c(x)$ together with $p(x,y)$ to obtain a top extension.

Ali Enayat's [3] construction of a Jónsson $\omega_1$-like model uses superminimal extensions obtained via Skolem ultrapowers. Suppose that $M\models{\rm ZFC}+V={\rm OD}$ (every element is ordinal definable). An elementary extension $N$ of $M$ is called superminimal if for every $a\in N\setminus M$, we have $N={\rm Scl}(a)$, the Skolem hull of $\{a\}$. It is easy to see that if $M$ is an elementary chain of length $\omega_1$ of countable models, where the successor stage extensions are superminimal, then it is Jónsson. Enayat shows that every countable model $M\models{\rm ZFC}+V={\rm OD}$ has a Skolem ultrapower that is a superminimal top extension by carefully choosing the ultrafilter.