Is any elementary extension of a weakly small structure again weakly small?

Question

Recently, I've been reading a paper in which the author makes use of the following definition apparently first introduced by Oleg V. Belegradek:

A structure $\mathcal{M}$ in a language $\mathcal{L}$ is called weakly small, if it has only countably many 1-types over every finite set $A$ of parameters.

My question is whether there is an example for a weakly small structure $\mathcal{M}$ which posesses an elementary extension $\mathcal{M} \prec \mathcal{N}$ that is not weakly small.

Answer 1

Here's a combinatorial example.

The language is $L = \{P_n\mid n\in \omega\}\cup \{R_n\mid n\in \omega\}$, where each $P_n$ is a unary relation symbol and each $R_n$ is a binary relation symbol.

Consider the universal theory $T$ asserting for all $n$:

• $P_{n+1}\subseteq P_n$
• $R_n\subseteq P_n\times P_n$ (so if $R_n(x,y)$, then $P_n(x)$ and $P_n(y)$).
• $R_n$ is a graph relation (symmetric and anti-reflexive).

Take the model companion $T^*$ of $T$. For any $k\in \omega$, writing $L_k = \{P_n\mid n<k\}\cup \{R_n\mid n<k\}$, the class of finite models of $T|_{L_k}$ is a Fraïssé class $\mathcal{C}_k$. If $M_k$ is the Fraïssé limit of $\mathcal{C}_k$, then $T^*$ is the union $\bigcup_{k\in \omega}\mathrm{Th}(M_k)$. It follows that $T^*$ has quantifier elimination, since any formula is an $L_k$-formula for some $k$, and $\mathrm{Th}(M_k)$ has quantifier elimination.

By QE, $T^*$ has countably many $1$-types over $\varnothing$: $\{p_n\mid n\in \omega\}\cup \{p_\infty\}$. Each type $p_n$ is axiomatized by $\{P_k(x)\mid k < n\}\cup \{\lnot P_k(x)\mid k\geq n\}$. So $p_\infty$ contains $P_n(x)$ for all $n$. The type $p_\infty(x)$ containing $P_n(x)$ for all $n$ is non-isolated, so there is a countable model $M\models T^*$ omitting $p_\infty(x)$. I claim that $M$ is weakly small.

Let $A\subseteq M$ be finite, say $A = \{a_1,\dots,a_k\}$. Since none of the $a_i$ realize $p_\infty$, we can choose $N$ least such that $\lnot P_N(a_i)$ for all $i$. By QE, a $1$-type $q(x)$ over $A$ is determined by:

• Which of the countably many $1$-types over $\varnothing$ $q$ extends.
• Which of the formulas $R_n(x,a_i)$ are contained in $q$.

But no formula $R_n(x,a_i)$ is contained in $q$ for $n\geq N$. Hence there are only finitely many (at most $(2^k)^N$) $1$-types extending each of the countably many $1$-types over $\varnothing$. So $M$ is weakly small.

Now let $M\preceq M'$ be an elementary extension such that $M'$ contains $a$ realizing $p_\infty$. Then there are continuum-many $1$-types over $\{a\}$, since for all $X\subseteq \omega$, the type $$q_X(x) = \{P_n(x)\mid n\in \omega\}\cup \{R_n(x,a)\mid n\in X\}\cup \{\lnot R_n(x,a)\mid n\notin X\}$$ is consistent, and if $X\neq X'$, $q_X$ and $q_{X'}$ are contradictory. So $M'$ is not weakly small.

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By the way, recall that a theory $T$ is small if for all $n\in \omega$, there are only countably many $n$-types over $\varnothing$ consistent with $T$.

For a weakly small structure $M$ with $T = \mathrm{Th}(M)$, the following are equivalent:

1. Every elementary extension of $M$ is weakly small.
2. Every model of $T$ is weakly small.
3. $T$ is a small theory.

Proof: $2\implies 1$ is trivial.

$3\implies 2$: Suppose $N\models T$ and $A\subseteq N$ is finite. Enumerate $A$ as $\{a_1,\dots,a_k\}$. Now there is an injective map $S_1(A)\to S_{k+1}(\varnothing)$ given by $p(x)\mapsto \mathrm{tp}(a_1,\dots,a_k,b)$, where $b$ is any realization of $p(x)$. This map is injective, so if $T$ is small, then $|S_1(A)|\leq |S_{k+1}(\varnothing)|\leq \aleph_0$. Thus $N$ is weakly small.

$1 \implies 3$: Let's show by induction on $n\geq 1$ that $|S_n(\varnothing)|\leq \aleph_0$. In the base case, $|S_1(\varnothing)|\leq \aleph_0$ since $M$ is weakly small. Now assume $|S_n(\varnothing)|\leq \aleph_0$. There is a surjective map $S_{n+1}(\varnothing)\to S_n(\varnothing)$ which restricts each $(n+1)$-type to its first $n$ variables. Since $|S_n(\varnothing)|\leq \aleph_0$, it suffices to show that each fiber of this map is countable, i.e., for each $p\in S_n(\varnothing)$, $p$ has only countably many extensions to an $(n+1)$-type. Let $M\preceq M'$ be an elementary extension of $M$ containing a tuple $(a_1,\dots,a_n)$ realizing $p$. Let $A = \{a_1,\dots,a_n\}$. Then there is a bijection between the $(n+1)$-types over $\varnothing$ extending $p$ and the $1$-types over $A$. But since $M'$ is weakly small, $|S_1(A)|\leq \aleph_0$. $\square$

Thus we see that the notion of weakly small fits nicely into a common pattern in model theory: it's a condition on formulas/types in $1$ variable in a structure $M$, and it has a stronger version (smallness) that means that it is true of all structures elementarily equivalent to $M$, i.e. it is really a property of $\mathrm{Th}(M)$.

For example:

• A strongly minimal theory is one whose models are all minimal.
• A strongly o-minimal theory is one whose models are all o-minimal (though it's a theorem that strongly o-minimal and o-minimal are equivalent).
• A strongly C-minimal theory is one whose models are all C-minimal. etc.

See some discussion here. The weakly small condition doesn't exactly have the same flavor as these notions of minimality, but it's still an instance of the pattern that if we have a strong condition in $1$ variable (few $1$-types over finite sets), then we can lift it to a conclusion in any number of variables (in a small theory, we have few $n$-types over $\varnothing$ for all $n$).