Introduction to Types - Model Theory

Definition ( $L$ -formula with parameters from $A$ ). Given a language $L$ , an $L$ -structure $M$ and a subset $A \subseteq M$ , we call an $L_{A}$ -formula an $L$ -formula with parameters from $A$ .

Write these as $φ (\bar{x}, \bar{a})$ for $φ (\bar{x}, \bar{y})$ an $L$ -formula, and $\bar{a} \in A$ (identify with ${\underset{̲}{a}}^{M}$ ).

SIngle formulas don’t give you much insight: suppose

a \in N

N ⊨ ϕ (a)

. Then there is some

a^{'} \in M

with

M ⊨ ϕ (a^{'})

Notation 13.1.

Let $p$ be a set of formulas in free variables $x_{1}, \dots, x_{n}$ . We often write $\underset{̲}{p (x_{1}, \dots, x_{n})}$ and $\underset{̲}{p}$ interchangeably.
Given $M$ and $a_{1}, \dots, a_{n} \in M$ , we write $M ⊨ p (a_{1}, \dots, a_{n})$ if $M ⊨ ϕ (a_{1}, \dots, a_{n})$ for every $ϕ \in p$ .
We say $p$ is consistent if it is realised in some $L$ -structure.

Exercise: Show

p

is consistent if and only if every finite subest of

p

is consistent ( Example Sheet 2, Q8).

Definition 13.2 ( $n$ -type). Let $M$ be an $L$ -structure and $A \subseteq M$ . An $n$ -type over $A$ with respect to $M$ is a set of $L$ -formulas with parameters from $A$ , in free variables $x_{1}, \dots, x_{n}$ such that $p \cup {T h}_{A} (M)$ is consistent.

An $n$ -type is complete if for every $L_{A}$ -formula with $n$ variables $ϕ$ , either $ϕ \in p$ or $\neg ϕ \in p$ .

Let $S_{n}^{M} (A)$ denote the set of all complete $n$ -types over $A$ with respect to $M$ .

Definition 13.3 ( $t p^{M}$ ). Given $a_{1}, \dots, a_{n} \in M$ , let ${t p}^{M} (a_{1}, \dots, a_{n} ∕ A)$ be the set of all $L_{A}$ -formulas $ϕ (x_{1}, \dots, x_{n})$ such that $M ⊨ ϕ (a_{1}, \dots, a_{n})$ (usually $a_{i} \notin A$ ).

${t p}^{M} (\bar{a} ∕ A) \in S_{n}^{M} (A)$ and $\bar{a} ⊨ {t p}^{M} (\bar{a} ∕ A)$ .

Proposition 13.4. Assuming that:

$p \in S_{n}^{M} (A)$

Then there is

N ≽ M

with

\bar{a} \in N^{n}

such that

p = {t p}^{N} (\bar{a} ∕ A)

Proof. By assumption $p \cup {T h}_{A} (M)$ is consistent.

Need to show $p \cup {T h}_{M} (M)$ is consistent.

Fix $Σ \subseteq p \cup {T h}_{M} (M)$ finite. $Σ \subseteq p \cup {φ_{1}, \dots, φ_{t}}$ , $φ_{i}$ an $L_{M}$ -sentence with $M ⊨ φ_{i}$ .

Let $φ^{*}$ be $\land_{i = 1}^{t} φ_{i}$ , then $φ^{*}$ can be written $φ^{*} (b_{1}, \dots, b_{m})$ where $b_{1}, \dots, b_{m} \in M ∖ A$ and $φ^{*} (x_{1}, \dots, x_{m})$ an $L_{A}$ -formula.

Since $M ⊨ φ^{*} (b_{1}, \dots, b_{m})$ we get $M ⊨ \exists \bar{v}, φ^{*} (v_{1}, \dots, v_{m})$ , so $\exists \bar{v} φ^{*} (\bar{v}) \in {T h}_{A} (M)$ .

So as ${T h}_{A} (M) \cup p$ is consistent, we have $N ⊨ {T h}_{A} (M) \cup p$ with

$\bar{c} \in N^{m}$ with $N ⊨ φ^{*} (\bar{c})$ .
$\bar{a} \in N^{m}$ with $N ⊨ p (\bar{a})$ .

Expand $N$ to an $L_{M}$ -structure, i.e. let

$b_{i}^{N} = c_{i}$ for $i \in {1, \dots, m}$ .
${\bar{b}}^{N}$ arbitary for $b \in M ∖ ⟨ (A \cup {b_{1}, \dots, b_{m}}) ⟩$ .

Then $N ⊨ φ (b_{1}, \dots, b_{m})$ . So $N ⊨ φ^{*}$ , so $N ⊨ Σ$ . □

Remark 13.6. $p$ is an $n$ -type over $A$ with respect to $M$ if and only if $\forall q \subseteq p$ finite, $\exists \bar{a} \in M^{n}$ such that $\bar{a} ⁄ ⊨ q$ .

Proof.

$\Rightarrow$ Clear.
$\Leftarrow$ Choose $N ≽ M$ realising $p$ . Fix $q \subseteq p$ finite, $φ (\bar{x})$ the conjunction of all $L_{A}$ -formulas in $q$ . Then $N ⊨ \exists \bar{x}, φ (\bar{x})$ . So $M ⊨ \exists \bar{x}, φ (\bar{x})$ , i.e. $q$ is realised in $M$ . □

Example 13.7. Suppose $K ⊨ ACF$ , $A \subseteq K$ . We want to describe $S_{n}^{K} (A)$ . Fix $p \in S_{n}^{K} (A)$ . By quantifier elimination we only need to consider quantifier free formulas.

Moreover,

\begin{array}{l} φ \land ψ \in p & ⟺ φ, ψ \in p \\ \neg ψ \in p & ⟺ φ \notin p \end{array}

So we can concentrate on atomic formulas $φ$ , polynomials in variables $x_{1}, \dots, x_{n}$ over the field generated by $A$ , say $F$ (i.e. $F [\bar{x}]$ ).

Let $I_{p} = {f (\bar{x}) \in F [\bar{x}] : f (\bar{x}) = 0 \in p}$ . Then $I_{p}$ is a prime ideal and $p \mapsto I_{p}$ is a bijection $S_{n}^{K} (A) \mapsto Spec F [\bar{x}]$ ( $Spec F [\bar{x}]$ is the set of prime ideals of $F [\bar{x}]$ ). So $S_{1}^{K} (A)$ consists of

{p_{a} : a \in A} \cup {q},

where $p_{a}$ contains (and thus is determined by) $x = a$ and $q = {x \neq a : a \in F}$ .

$| S_{1}^{K} (K) | = | K |$ .

13 Introduction to Types