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In Model Theory by Chang & Keisler, saturated models are introduced on page 100.

A model $\mathfrak U$ is said to be $\omega$-saturated iff for every finite set $Y \subset A$, every set of formulas $\Gamma(x)$ of $\mathfrak L_Y$ consistent with $\operatorname{Th}(\mathfrak U_Y)$ is realized in $\mathfrak U_Y$.

For context, here's how the book is denoting things:

  • $\mathfrak U$ is a model with universe $A$
  • $\mathfrak L$ is a first-order language (of $\mathfrak U$).
  • $\operatorname{Th}(\mathfrak U)$ denotes the theory of $\ \mathfrak U$, the set of all formulas of $\mathfrak L$ which $\mathfrak U$ can satisfy. (Right? I don't think the book ever spelled this out, but this appears to be the meaning.)
  • $\mathfrak L_Y$ is the expansion of $\mathfrak L$ made by adding a new constant symbol $\bar y$ to $\mathfrak L$ for each $y \in Y$.
  • If $Y \subset A$, $\mathfrak U_Y$ denotes the expansion of $\mathfrak U$ to the language $\mathfrak L_Y$ built by interpreting each constant symbol $\bar y$ by the corresponding element of $Y$.

Now, I understand this definition. I can memorize it and make use of it in exercises. What I don't understand is why anyone would make this definition in the first place.

What is an $\omega$-saturated model? Why is it useful? What does this definition buy us?

It may help to give an answer in contrast to "atomic models", introduced just prior to saturated models in the book:

A formula $\phi(x_1\ldots x_n)$ is said to be complete (in $T$) iff for every formula $\psi(x_1 \ldots x_n)$ exactly one of $T \models \phi \rightarrow \psi$, $T \models \phi \rightarrow \lnot \psi$ holds.

A model $\mathfrak U$ is said to be an atomic model iff every $n$-tuple $a_1, \ldots, a_n \in A$ satisfies a complete formula in $\operatorname{Th}(\mathfrak U)$.

According to my understanding, a "complete formula" (free in at most $n$ variables) is one which nails down the interpretation of every other formula (free in at most $n$ variables). A model is "atomic" if any tuple drawn from the universe satisfies a complete formula.

In other words, "$\mathfrak U$ is atomic" means that any set of elements drawn from the universe of $\mathfrak U$ correspond to some formula that completely nails down the rest of the formulas (of similar freedom) in the language.

This gives us an "internal representation" of completeness. If you select $\emptyset$ as your element tuple, you get a complete sentence $\phi$ such that for every $\psi$ either $\mathfrak U \models \phi \rightarrow \psi$ or $\mathfrak U \models \phi \rightarrow \lnot \psi$.

This seems somewhat important, though I still don't have a solid grasp on the implications. (Am I on the right track?) I understand these definitions and can solve some simple problems, but I don't understand their purpose.

What's the motivation behind saturated models, especially compared to atomic models? Why should I care about the distinction?

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If $M$ is a saturated model (in its own cardinality) of a theory $T$, it has some other interesting properties that follow from saturation:

  1. $M$ is universal: for every $N\models T$ of cardinality at most equal to cardinality of $M$, there is an elementary embedding of $N$ into $M$.
  2. $M$ is strongly homogeneous: every partial elementary function defined on $M$ whose domain has cardinality less than that of $M$ extends to an automorphism.

This has some useful consequences: if we assume that we have some saturated model $\mathfrak C\models T$ of very large cardinality (where very large means much larger than any object we're really concerned with), we can assume without loss of generality that all models of $T$ are actually elementary substructures of $\mathfrak C$. We treat $\mathfrak C$ as a sort of universal domain (in fact, I've heard there is a notion of a universal domain which coincides with this in universal algebra).

Furthermore, any for any two elements $a,b$ in some model (or tuples, even infinite) and any (small) set $A$, under the previous assumption, the two have the same type over $A$ if and only if there is an automorphism of $\mathfrak C$ which takes $a$ to $b$ and fixes $A$ pointwise. This, in addition to the fact that every type over $A$ is realised in $\mathfrak C$, means that types over $A$ correspond exactly to the orbits of $\mathrm{Aut}(\mathfrak C/A)$ which often makes it simpler to imagine some things and prove them.

There is a minor gripe with the above: ZFC does not prove the existence of large saturated models for arbitrary theories (though it's easy to see that there's a saturated model in any strongly inaccessible cardinality). However, it does prove that for any cardinal $\kappa$, there is a model $\mathfrak C$ which is $\kappa$-saturated (which implies $\kappa$-universality, but not strong $\kappa$-homogeneity if $\kappa<\lvert \mathfrak C\rvert$) and $\kappa$-homogeneous, which is enough to have the things I mentioned above (where small means smaller than $\kappa$).

That said, many papers in model theory actually assume that there exists a large saturated model for brevity, though it's understood that its existence is not really necessary, and for most (if not all) purposes can be replaced by a model which is sufficiently saturated and homogeneous, however large its cardinality might be.

This is probably the most obvious "use" of saturated models, but there are others. For instance, it is true that any two saturated model of a given theory are isomorphic if and only if they have the same cardinality (more generally, any two homogeneous models of the same cardinality are isomorphic if and only if they realise the same $n$-types for each $n$).

Saturated models also arise in the context of nonstandard analysis. Though it usually just uses ultraproducts, the only purpose of those is to obtain $\aleph_1$-saturated models of analytic objects, which allows for naive expressions like "$f'(x)=\frac{f(x+h)-f(x)}{h}$ for infinitesimal $h$" or "$\int_0^1 f(x) dx=\sum_{k=0}^{n-1} f(k/n)/n$ for infinitely large natural number $n$" to become meaningful after minor modifications.

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