Saturated Models - Model Theory - Cambridge III notes

15 Saturated Models

Definition 15.1. Let $M$ be an infinite $L$ -structure, and $κ > | L | + ℵ_{0}$ . We say $M$ is $κ$ -saturated if for any $A \subseteq M$ with $| A | < κ$ , every type in $S_{n}^{M} (A)$ is realised in $M$ for all $n$ .

Remark 15.2.

(i) Restricting to complete types is not important, as every $n$ -type over $A$ with respect to $M$ can be extended to a complete type.
(ii) It suffices to assume $n = 1$ to prove $κ$ -saturation.
(iii) If $M$ is $κ$ -saturated then $| M | \geq κ$ . ${x \neq a : a \in M}$ is a consistent $1$ -type over $M$ in $M$ .

Definition 15.3 (Partial elementary / homogeneous). Let $M, N$ be $L$ -structures, $A \subseteq M$ , $B \subseteq N$ . A function $f : A \to B$ is partial elementary if for every $L$ -formulas $φ (x_{1}, \dots, x_{n})$ and $a_{1}, \dots, a_{n} \in A$ we have

M ⊨ φ (\bar{a}) ⟺ N ⊨ φ (f (\bar{a})) .

Given $κ \geq | L | + ℵ_{0}$ , $M$ is $κ$ -homogeneous if for any $A \subseteq M$ with $| A | < κ$ , any partial elementary map $f A \to M$ and any $c \in M$ there is some $d \in M$ with $f \cup {(c, a)}$ partial elementary. In other words, “partial elementary maps can be extended”.

For the rest of this section, assume $T$ to be a complete $L$ -theory with infinite models.

Definition 15.4. Define $S_{n} (T) = S_{n}^{M} (\emptyset)$ for any / some $M ⊨ T$ (because if $M, N ⊨ T$ , then $S_{n}^{N} (\emptyset) = S_{n}^{M} (\emptyset)$ as $T h (M) = T h (N) = T$ ).

Proposition 15.5. Assuming that:

$T$ a complete $L$ -theory with infinite models
$M ⊨ T$

Then

M

ℵ_{0}

-saturated if and only if

M

ℵ_{0}

-homogeneous and

M

realises all types in

S_{n} (T)

n \geq 1

Proof.

$\Rightarrow$ Assume $M$ is $ℵ_{0}$ -saturated. In particular, $M$ realises all types in $S_{n} (T)$ ( $\emptyset$ is finite).
Fix some finite $A \subseteq M$ , and a partial elementary map $f : A \to M$ . (Aim: find $d \in M$ such that $f \cup {(c, d)}$ is partial elementary). Given $c \in M$ , define $p \in S_{1}^{M} (f (A))$ to be such that
$φ (x, f (\bar{a})) ⟺ M ⊨ φ (c, \bar{a})$

(for all $φ (y, \bar{x})$ $L$ -formulas and $\bar{a} \in A$ ). Write $p = f ({t p}^{M} (c ∕ \bar{a}))$ .

To show $p \in S_{1}^{M} (f (A))$ , consider $φ (x, \bar{a}) \in p$ . Then $M ⊨ φ (c, \bar{a})$ , so $M ⊨ \exists x, φ (x, \bar{a})$ , os $M ⊨ \exists φ (x, f (\bar{a}))$ as $f$ is partial elementary.

So $p$ is finitely satisfiable, and completeness follows from ${t p}^{M} (c ∕ A)$ being complete. So as $M$ is $κ$ -saturated we have $d ⊨ p$ for some $d \in M$ . Then $f \cup {(c, d)}$ is a partial elementary map.
$\Leftarrow$ Fix $a_{1}, \dots, a_{n} \in M$ , $p \in S_{1}^{M} ({a_{1}, \dots, a_{n}})$ . We want to show $p$ is realised in $M$ .
Set
$q = {φ (y, x_{1}, \dots, x_{n}) : φ (y, \bar{a}) \in p} \in S_{n + 1}^{M} (T) .$

So by assumption, we have $d, b_{1}, \dots, b_{n}$ with $(d, \bar{b}) ⊨ q$ .

Consider $t p (\bar{a} ∕ \emptyset) = t p (\bar{b} ∕ \emptyset)$ . So $f : b_{i} \to a_{i}$ is partial elementary. Let $c \in M$ such that $f \cup {(d, c)}$ is partial elementary. Then ${t p}^{M} (c, \bar{a}) = {t p}^{M} (d, \bar{b})$ . So $(c, \bar{a}) ⊨ q$ and $t p (c ∕ \bar{a}) = p$ . □

Notation. Given $M$ , $\bar{a}, \bar{b} \in M^{n}$ , write $\bar{a} \equiv^{M} \bar{b}$ if ${t p}^{M} (\bar{a}) \equiv^{M} \bar{b}$ .

So $M$ is $ℵ_{0}$ -homogeneous if and only if whenever $\bar{a} \equiv^{M} \bar{b}$ and $c \in M$ , there exists $d \in M$ with $\bar{a} c \equiv^{M} \bar{b} d$ .

Lemma 15.6. Assuming that:

$T$ a complete $L$ -theory with infinite models
$M ⊨ T$

Then there is an

N ≽ M

with

| N | \leq | M | + | L |

and

N

ℵ_{0}

-homogeneous.

Proof. First claim: For any $M ⊨ T$ , there is $N ≽ M$ with $| N | \leq | M | + | L |$ and for any $\bar{a}, \bar{b}, c$ from $M$ such that $\bar{a} \equiv^{M} \bar{b}$ there is some $d \in N$ with $\bar{a} c \equiv^{N} \bar{b} d$ .

Proof of claim: Enumerate all $(\bar{a}, \bar{b}, \bar{c})$ as ${({\bar{a}}_{α}, {\bar{b}}_{α}, {\bar{c}}_{α})}_{α \leq | M |}$ . Now let $M_{0} = M$ , and use transfinite induction to form a chain ${(M_{α})}_{α < | M |}$ .

$α$ is a limit ordinal: set $M_{α} = ⋃_{i < α} M_{i}$ (then $| M_{α} | \leq | α | (| M | + | L |) = | M | + | L |$ as $α \leq | M |$ ).
Given $α$ (not a limit ordinal), look at $(a_{α}, b_{α}, c_{α})$ . Assume ${\bar{a}}_{α} \equiv^{M_{α}} {\bar{b}}_{α}$ so $f_{α} : {\bar{a}}_{α} \to {\bar{b}}_{α}$ partial elementary. Then we apply Proposition 13.4 (Note the elementary super structure construnted here is of size $\leq | M | + | L |$ ) to find $M_{α + 1} ≽ M_{α}$ with $| M_{α + 1} | \leq | M_{α} | + | L | \leq | M | + | L |$ with a $d \in M_{α + 1}$ realising $f_{α} (t p (c_{α} ∕ a_{α}))$ . Then by construction, $({\bar{a}}_{α} c) \equiv^{M_{α + 1}} ({\bar{b}}_{α} d)$ .

Now let $N = ⋃_{α < | M |} M_{α}$ . Note that we might have introduced new elements. Note $| N | \leq | M | (| M | + | L |) = | M | + | L |$ .

We build a new chain $M = N_{0} ≼ N_{1} ≼ N_{2} ≼ \dots$ of countably many steps with $| N_{1} | \leq | M | + | L |$ and such that for any $\bar{a}, \bar{b}, \bar{c} \in N$ , if $\bar{a} \equiv^{M} \bar{b}$ then there is $d \in N_{i + 1}$ such that $\bar{a} c \equiv^{N_{i + 1}} \bar{b} d$ . We do this by iterating the claim.

FInally let $N = ⋃_{i < ℵ_{0}} N_{i}$ . Then:
- $| N | \leq | M | + | L |$ .
- $N$ is $ℵ_{0}$ -homogeneous as any $\bar{a}, \bar{b}, c$ from $N$ lie in $N_{i}$ for some $i$ .

□

Definition 15.7 (Saturated). We say $M$ is saturated if it is $| M |$ -saturated.

Theorem 15.8. Assuming that:

$T$ a complete $L$ -theory with infinite models
$L$ is countable

Then

T

has a countable saturated model if and only if

S_{n} (T)

is countable for every

n \geq 1

Proof.

$\Rightarrow$ $M ⊨ T$ countable, saturated.
- $M^{n}$ is countable for all $n \in ℕ$ .
- We have a map $p \to \bar{a} ⊨ p$ ( $p \in S_{n} (T)$ , $\bar{a}$ some realisation). This is map since $M$ saturated, and injective (because complete types).
So $S_{n} (T)$ is countable.
$\Leftarrow$ Enumerate $⋃_{n \geq 1} S_{n} (T) = {p_{1}, p_{2}, p_{3}, \dots}$ . Fix a countable $M_{0} ⊨ T$ , and build a chain $M_{0} ≼ M_{1} ≼ M_{2} ≼ \dots$ such that $M_{i}$ realises $p_{i}$ and is countable, using Proposition 13.4.
Get $N = ⋃_{i \in ℕ} M_{i}$ . Then $N ⊨ T$ and is countable. $N$ realises all types over $\emptyset$ .

Apply Lemma 15.6 to get $M ≽ N$ countable and $ℵ_{0}$ -homogeneous structure.

So by Proposition 15.5 is is $ℵ_{0}$ -saturated. □

Example 15.9.

(i) Let $T = {ACF}_{p}$ and let
$F = {\begin{matrix} ℚ & if p = 0 \\ 𝔽_{p} & if p > \infty \end{matrix} .$

Then
$S_{n} (T) = Spec (F [x_{1}, \dots, x_{n}]) .$

Thus $S_{n} (T)$ is countable, since every ideal in $Spec (F [x_{1}, \dots, x_{n}])$ is finitely generated (Hilbert’s basis theorem). So by Theorem 15.8, ${ACF}_{p}$ has a countable saturated model which is the model of transcendence degree $ℵ_{0}$ .

Note: if $F ⊨ {ACF}_{p}$ has transcendence degree $n$ , the type determined by “ $x_{1}, \dots, x_{n + 1}$ ” is an algebraically independent set.
(ii) Let $T = T F D A G$ (torsion free, divisible abelian groups). This has a countable saturated model, which is the $ℚ$ -vector space of dimension $ℵ_{0}$ .
(iii) Let $T = T h (ℤ, +, 0)$ . For $n \geq 1$ , let $δ_{n} (x)$ be the $L$ -formula $\exists y (x = n y)$ , and let $ℙ$ be the set of primes. Given $X \subseteq ℙ$ finite,
$q_{x} = {δ_{n} (x) : n \in X} \cup {\neg δ_{n} (x) : n \in ℙ ∖ X} .$

$q_{X}$ is satisfiable in $ℤ$ , thus $\exists p_{X} \in S_{1} (T)$ with $q_{x} \leq p_{x}$ .

If $X \neq Y$ , then $p_{X} \neq p_{Y}$ , so $| S_{1} (T) | \geq 2^{ℵ_{0}}$ . By Theorem 15.8, $T$ doesn’t have a countable saturated model.

Example 15.10. Let $M ⊨ R G$ . We describe $S_{1}^{M} (M)$ . For $a \in M$ , let $p_{a} \in S_{1}^{M} (M)$ be the type containing “ $x = a$ ” (exercise: why is this unique).

FOr $V \leq M$ set

p_{V} = {x \neq a : a \in M} \cup {F (x, q) : a \in V} \cup {\neg E (x, a) : a \in M ∖ V} .

$p_{V}$ is a $1$ -type with respect to $M$ , not realised in $M$ , determines a complete $1$ -type, as we have determined all atomic formula, by $a \in p_{V}$ determines a complete type.

So $S_{1}^{M} (M) = {p_{a} : a \in M} \cup {p_{V} : V \subseteq M}$ .

$| S_{1}^{M} (M) | = 2^{| M |}$ .

Note: in general $| S_{1}^{M} (A) | \leq 2^{(| A | + | L | + ℵ_{0})}$ .

Proposition 15.11. Assuming that:

$T$ a complete $L$ -theory with infinite models
$M \equiv N$ are both countable and saturated

Then

M ≅ N

Proof. Exercise (use back and forth argument). □