Stability Theory

The Notion of Types in Model Theory

Suppose $\mathcal{M}$ is an $\mathcal{L}$-structure and $A\subseteq M$. Let ${\mathcal{L}}_A$ be the language obtained by adding $\mathcal{L}$ constant symbols for each $a\in A$, $\mathcal{M}$ can be seen as a ${\mathcal{L}}_A$-structure by interpreting the new symbols. Let $T{h}_A(\mathcal{M})$ be the set of all ${\mathcal{L}}_A$-sentences true in $\mathcal{M}$.

Types (Model Theory): Let $p$ be the set of ${\mathcal{L}}_A$-formulas in free variable $v_1,\dots ,v_n$. We call $p$ an $n$-type if $p\cup T{h}_A(\mathcal{M})$ is satisfiable. We say that $p$ is a complete $n$-type if $\varphi \in p$ or $\neg \varphi \in p$ for all ${\mathcal{L}}_A$-formulas with free variable from $v_1,\dots ,v_n$. Let $S_n^{\mathcal{M}}(A)$ be the set of all complete $n$-types. $\to$ The intuition behind a type is that is a set of formulas which describe a possible element (or tuple of elements) in a model

Incomplete types are also referred to as ‘partial’ types

Realizing and Omitting Types For $p$ is an $n$-type over a subset $A\subseteq M$, we say that $\overline{a}\in M^n$ realizes p if $\mathcal{M}\models \varphi (\overline{a})$ for all $\varphi \in p$. If $p$ is not realized in $\mathcal{M}$ ($\mathcal{M}/\models \varphi (\overline{a})$ for $\overline{a}\in M^n$), then we say that $\mathcal{M}$ omits $p$.

Witnesses: A witness is an instance of $t$ where when substituted into an existential formula $\exists x\phi (x)$ then $\phi (t)$ is true.

E.g., An (arithmetic) theory $T$ is inconsistent if there is a proof of $0=1$ Let $I(T)$ denote ”$T$ is inconsistent”

$I(T)$ is thus an existential formula, and so a witness for this formula is a proof of $0=1$

Types of Models in Stability Theory

Homogeneous models: Let $\kappa$ be an infinite cardinal. We say that $\mathcal{M}\models T$ is $\kappa$-homogeneous if whenever $A\subset M$ with $|A|<\kappa ,f:A\to M$ is a partial elementary map, and $a\in M$, there is $f^{\ast }\supseteq f$ such that $f^{\ast }:A\cup \{a\}\to M$ is partial elementary. We say that $\mathcal{M}$ is homogeneous if it is $|M|$-homogeneous.

Prime models: We say that $\mathcal{M}\models T$ is a prime model of $T$ if whenever $\mathcal{N}\models T$ there is an elementary embedding $f:\mathcal{M}\to \mathcal{N}$ $\to$ A prime model is a model of a (complete) theory $T$ which elementary embeds into every other model of $T$

If $\mathcal{M}$ and $\mathcal{N}$ are prime models, then $\mathcal{M}\cong \mathcal{N}$

Atomic models: We say that $\mathcal{M}\models T$ is atomic if tp${}^{\mathcal{M}}(\overline{a})$ is isolated for all $\overline{a}\in {\mathcal{M}}^n$.

Note: prime models are atomic.

Theorem: Let $\mathcal{L}$ be a countable language and let $T$ be a complete $\mathcal{L}$-theory with infinite models. Then, $\mathcal{M}\models T$ is prime if and only if it is countable and atomic

Theorem: Let $\mathcal{L}$ be a countable language and $T$ a complete $\mathcal{L}$-theory with infinite models. Then the following are equivalent: i) $T$ has a prime model; ii) $T$ has an atomic model; iii) The isolated types in $S_n(T)$ are dense for all $n$

Saturated models: Let $\kappa$ be an infinite cardinal, we say that a model $\mathcal{M}\models T$ (for $T$ is some theory) is $\kappa$-saturated if, for all $A\subseteq M$, if $|A|<\kappa$ and $p\in S_n^{\mathcal{M}}(A)$, then $p$ is realized in $\mathcal{M}$. We say that $\mathcal{M}$ is saturated if it is $|M|$-saturated.

Stability: Let $T$ be a complete theory in a countable language and let $\kappa$ be an infinite cardinal. We say that $T$ is $\kappa$-stable if whenever $\mathcal{M}\models T,A\subseteq M,$ and $|A|=\kappa$, then $|S_n^{\mathcal{M}}(A)|=\kappa$. We say that $\mathcal{M}$ is $\kappa$-stable if Th($\mathcal{M}$) is $\kappa$-stable.

Theorem: Let $T$ be a complete theory in a countable language. If $T$ is $\omega$-stable, then $T$ is $\kappa$-stable for all infinite cardinals $\kappa$.

References

Model Theory D.Marker.pdf