Monster model

A monster model of a complete theory T is a model M which is $$\kappa$$saturated and strongly homogeneous for some relatively large cardinal $$\kappa$$. Generally speaking, one wants $$\kappa$$ to be bigger than any sets of parameters one expects to encounter.

Or one can construct a class-size monster in the BGC set theory, which is a conservative extension of ZFC. Which will then be $$\kappa$$saturated and strongly homogeneous for any cardinal $\kappa$.

Monster models are frequently denoted with symbols such as
 * $$\mathfrak{M}$$ or $$\mathbb{M}$$, "M" for "monster"
 * $$\mathbb{C}$$ and $$\mathfrak{C}$$, by analogy with the complex numbers, which are a saturated model of ACF.
 * $$\mathbb{U}$$ or $$\mathfrak{U}$$, "U" for "universe".

Working in a Monster Model
See the end of this section for a summary of the conventions and tools one has when working inside a monster.

Fix some complete theory T.

Let $$\kappa$$ be a cardinal with the following properties: The existence of such cardinals can be proven in ZFC (without much difficulty).
 * If $$\lambda < \kappa$$, then $$2^\lambda < \kappa$$, i.e., $$\kappa$$ is a strong limit cardinal.
 * The size of the language is strictly less than $$\kappa$$.
 * If $$\lambda < \kappa$$ and M is a model of T of size less than $$\kappa$$, then there is a $$\lambda$$-saturated and $$\lambda$$-strongly homogeneous elementary extension of M of size less than $$\kappa$$.
 * $$\kappa$$ has at least uncountable cofinality. (Ideally, $$\kappa$$ would have cofinality $$\kappa$$, but this would require $$\kappa$$ to be strongly inaccessible, and ZFC cannot guarantee the existence of such models.
 * Possibly additional properties, e.g., if we plan to use the Erdős-Rado theorem.

Henceforth, "small" will mean "of cardinality less than $$\kappa$$". Our assumptions imply for example that if $$A$$ is small, then so is the power set of $$A$$. Similarly, if M is a small model of T, then M has an |M|+-saturated and |M|+-strongly homogeneous elementary extension which is also small.

Now let $$\mathbb{M}$$ be a $$\kappa$$-saturated and $$\kappa$$-strongly homogeneous model of T. This will be the "monster."

Lemma: Let M and N be two structures. Let $$f : M \to N$$ be a partial elementary map whose domain happens to be all of M. Then f is an elementary embedding.

Proof: Let A be f(M). We can use the Tarski-Vaught criterion to show that A is an elementary substructure of N. We need to show for each formula $$\phi(x;y)$$, that if $$N \models \exists x : \phi(x;a)$$ for some a ∈ A, then $$N \models \exists x : \phi(a_0;a)$$ for some a0 ∈ A. Write a = f(m). Then because f is a partial elementary map,
 * $$N \models \exists x : \phi(x;f(m)) \iff M \models \exists x : \phi(x;m)$$

Thus there is some m0 ∈ M such that
 * $$M \models \phi(m_0;m)$$.

Then again, since f is a partial elementary map,
 * $$N \models \phi(f(m_0);f(m))$$,

and f(m0) ∈ A. So the Tarski-Vaught condition holds, and $$A \succeq N$$. And since f is a partial elementary map, it induces an isomorphism from M to A. So f is an elementary embedding. QED.

Lemma: Let M be a small model of T. Then M is isomorphic to a small elementary substructure of $$\mathbb{M}$$. More generally, if M is a small model of T, any partial elementary map from a subset of M to $$\mathbb{M}$$ can be extended to an elementary embedding of M into $$\mathbb{M}$$.

Proof: For the second claim, suppose $$f : A \to \mathbb{M}$$ is an elementary map, for some A ⊆ M. Let p be tp(M/A), and let f*p be the pushforward of p along f. So f*p is a complete type over $$f(A) \subset \mathbb{M}$$. Because M and f(A) have size less than $$\kappa$$, this type is realized in $$\mathbb{M}$$, by some N. But then by definition of the pushforward,
 * tp(MA/Ø) = tp(Nf(A)/Ø)

In particular, M -> N is an elementary map extending f. By the previous lemma, this map is an elementary embedding of M into $$\mathbb{M}$$.

For the first claim, observe that if M is a model of T, then M is elementarily equivalent to the monster. This is equivalent to the trivial map from Ø ⊆ M to Ø $$\subset \mathbb{M}$$ being a partial elementary map. By what we have just shown, this trivial map can be extended to an elementary embedding of M into $$\mathbb{M}$$. QED.

Corollary: Let M be a small elementary substructure of $$\mathbb{M}$$. Let N be a small elementary extension of M. Then we can assume that N is an elementary substructure of $$\mathbb{M}$$! More precisely, we can find $$N'$$ isomorphic to N over M, with $$M \preceq N' \preceq \mathbb{M}$$.

Proof: The identity map on M is a partial elementary map from N to $$\mathbb{M}$$. Extending it to an elementary embedding of N into $$\mathbb{M}$$ gives the desired result. QED.

A number of model-theoretic results say something like "under some hypotheses about M, such-and-such occurs in an elementary extension of M." The above machinery allows us to produce cleaner statements of these results in the monster. For example:

Corollary: Let M be a small elementary substructure of $$\mathbb{M}$$. Then we can find a small elementary substructure N of $$\mathbb{M}$$ which contains M and is |M|+-saturated and |M|+-strongly homogeneous.

Proof: It is known that M has a |M|+-saturated and |M|+-strongly homogeneous elementary extension N0. By assumption on $$\kappa$$, we may assume that N0 has cardinality less than $$\kappa$$, i.e., is small. Therefore we can move everything into the monster—we may assume that N0 is an elementary substructure of the monster. QED.

Corollary: Let M be a small elementary substructure of the monster. Let $$\{a_i\}_{i \in I}$$ be a sequence in M, and let J be a small ordered set. Then we can find an indiscernible sequence $$\{b_j\}_{j \in J}$$ inside the monster, realizing the Ehrenfeucht-Mostowski type of $$\{a_i\}_{i \in I}$$.

Proof: It is known (using Ramsey's theorem or Morley sequences) that there is some elementary extension N of M in which such an indiscernible sequence exists. By downwards Löwenheim-Skolem, we may assume N is small. Then we may assume that N sits inside the monster. QED.

We should also make the following observation:

Observation: Let A be a small subset of $$\mathbb{M}$$. Then there is a small elementary substructure of $$\mathbb{M}$$ containing A. This is immediate from downwards Löwenheim-Skolem.

Observation: Let A be a small subset. Then the definable and algebraic closures of A are also small. In fact, they lie inside any small elementary substructure of $$\mathbb{M}$$ containing A.

Lemma: The union of any countable increasing chain of small elementary substructures of $$\mathbb{M}$$ is a small elementary substructure of $$\mathbb{M}$$. More generally, we can replace "countable" with any cardinality less than the cofinality of $$\kappa$$.

Proof: The fact that the union is an elementary substructure of $$\mathbb{M}$$ is an exercise using the Tarski-Vaught Theorem and the first Lemma above (the one using the Tarski-Vaught criterion.) The fact that the union is small follows by the assumption on the cofinality of $$\kappa$$. QED.

In light of everything we have said, it is customary to assume that all "models" are elementary substructures of the monster. Certainly any small model of T is isomorphic to such a "model," and any theorem guaranteeing the existence of an elementary extension with a certain property can be rephrased in terms of these "models," if we assume that $$\kappa$$ has enough properties.

We should also point out what saturation and strong homogeneity give us:

Observation: Any type over a small set is realized. Any consistent small partial type is realized. If A is a small set, and a and b are two tuples of small length, then tp(a/A) = tp(b/A) if and only if some automorphism $$\sigma \in \text{Aut}(\mathbb{M}/A)$$ sends a to b componentwise.

Indeed, these all follow directly from the definitions of saturatedness and strong homogeneity.

As explained on the page "Definable and algebraic closure", we also have:

Fact: A tuple a is in the definable closure of a small set A if and only if a is fixed by $$\text{Aut}(\mathbb{M}/A)$$. Two small sets A and B are interdefinable (i.e., have dcl(A) = dcl(B)) if and only if they are fixed by the exact same automorphisms.

This gives rise to the notion of a code. If D is a definable set, or some other object in a class acted upon by $$\text{Aut}(\mathbb{M})$$, then we say that a (possibly infinite) tuple a is a code for D if and only if the stabilizer of D in $$\text{Aut}(\mathbb{M})$$ is exactly the stabilizer of a. For the case where D is actually a definable set, this is compatible with the notion of a "code" that comes up when discussing imaginaries. In the case where D is a global type, or an equivalence class of global types under some equivalence relation, this yields the notion of the canonical base of a type in a stable theory or simple theory.

Similarly, for algebraic closure, we have

Fact: A finite tuple a is in the algebraic closure of a small set A if and only if a has a finite orbit under $$\text{Aut}(\mathbb{M}/A)$$. (More generally, an infinite tuple a is in acl(A) if and only if a has a small orbit under $$\text{Aut}(\mathbb{M}/A)$$.)

Compactness or saturation can also be rephrased in the following way, which makes the connection with topological compactness more explicit:

Theorem: Let D be a definable set, or even a type-definable set over a small base. Let $$\{E_\alpha\}_{\alpha < \lambda}$$ be some small collection of definable sets, not necessarily fitting into any uniform family. Suppose that
 * $$ D \subset \bigcup_{\alpha < \lambda} E_\alpha$$.

Then D is covered by a finite union of the $$E_\alpha$$'s.

Proof: Consider the partial type p(x) which says that x ∈ D and $$x \notin E_\alpha$$ for each $$\alpha$$. Since D is type-definable over a small set, and since there are a small number of $$E_\alpha$$'s, this is a small partial type. By assumption, it is not realized in $$\mathbb{M}$$. But every consistent small partial type is realized in $$\mathbb{M}$$, so this type must not be consistent. Therefore some finite subset of it must be inconsistent. So there exist $$\alpha_1, \ldots, \alpha_n$$ such that
 * $$ D \setminus \bigcup_{i = 1}^n E_{\alpha_i}$$

is empty. But then
 * $$ D \subseteq E_{\alpha_1} \cup \cdots \cup E_{\alpha_n}$$.

QED.

In general, one only really discusses "type-definable" sets which are type-definable over small sets.

An important corollary of this compactness property is:

Corollary: Let D be an infinite type-definable set (over a small set, as always). Then D is not small.

Proof: Otherwise, D could be covered with a small number of singletons. By the compactness property, D could be covered with a finite number of singletons, contradicting the fact that it is infinite.

Consequently, the subsets of the monster we care about end up falling into two disjoint families:
 * Small subsets of the monster, including small models.
 * Definable and type-definable subsets of the monster, which are never small.

Various other kinds of compactness and uniformity results are more cleanly stated in a monster than not. For example:

Theorem: Let $$f : D \to E$$ be a definable surjective map from one definable set to another. Suppose that there is a small set A such that
 * $$ f^{-1}(\{e\}) \cap \text{dcl}(Ae) \ne \emptyset$$

for every e ∈ E. Then $$f$$ has a definable section.

Proof: For each e, there is an A-definable function g such that
 * $$ g(e) \in f^{-1}(\{e\})$$,

or equivalently, $$f(g(e)) = e$$. Now, for each A-definable function g, let
 * $$E_g = \{e \in E : g(e) \in D, ~ f(g(e)) = e\}$$.

Then the $$E_g$$'s cover E. Also, since A is small, there are only a small number of A-definable functions on g (essentially by our assumptions on $$\kappa$$). So we have covered the definable set E by a small family $$\{E_g\}$$. By compactness, E is covered by a finite family,
 * $$ E = E_{g_1} \cup \cdots \cup E_{g_n}$$.

Now for e ∈ E, define $$g(e)$$ to be $$g_i(e)$$ for the lowest $$i \in \{1,\ldots,n\}$$ such that $$e \in E_{g_i}$$. Then this function g is definable. And for any e, $$ g(e) = g_i(e)$$ where $$e \in E_{g_i}$$, i.e., $$f(g_i(e)) = e$$ and $$g_i(e) \in D$$. So for any e, $$f(g(e)) = e$$ and $$g(e) \in D$$. Then g is the desired definable section to f. QED.

The analogous statement without monsters would involve passing to elementary extensions.

Summary
In model theory, especially stability theory, it is common to "work in a monster model." This generally means the following:
 * 1) We assume that there is some $$\kappa$$-saturated and $$\kappa$$-strongly homogeneous model $$\mathbb{M}$$, called the "monster."
 * 2) A subset $$A \subset \mathbb{M}$$ is said to be "small" if $$|A| < \kappa$$. More generally, "small" means "cardinality less than $$\kappa$$."
 * 3) All "models" are assumed to be small elementary substructures of $$\mathbb{M}$$. This is relatively harmless, because any small model of the theory has an elementary embedding into the monster. And more generally, if M is a small elementary substructure of the monster, any small elementary extension of M may be moved into the monster. Finally, any small subset of monster is contained in a small "model" in this sense, i.e., in a small elementary substructure of the monster.
 * 4) With this convention, any inclusion of models will automatically be an elementary inclusion.
 * 5) The union of any increasing chain of small models will itself be a small model, provided the length of the chain is less than the cofinality of $$\kappa$$.
 * 6) Partial types are usually assumed to be small, i.e., to have fewer formulas than $$\kappa$$. Complete types are usually assumed to be over small sets. All such types are realized.
 * 7) Definable sets are never small (unless they are finite).
 * 8) Type-definable sets are usually type-definable over small sets. All such type-definable sets are never small (unless they are finite).
 * 9) We essentially care about two types of subsets of the monster: (a) small subsets and small models, and (b) definable sets and type-definable sets, which are (almost) never small.
 * 10) If A is a small subset of the monster, we can profitably look at the group $$\text{Aut}(\mathbb{M}/A)$$ of automorphisms of the monster which fix A pointwise, also known as the "group of automorphisms over A." Two (small) tuples have the same type over A if and only if they are in the same orbit of this group. So for example, the space $$S_n(A)$$ of n-types over A is exactly the space of orbits of $$\text{Aut}(\mathbb{M}/A)$$ on $$\mathbb{M}^n$$.
 * 11) For a small set A, a tuple is in dcl(A) if and only if it is fixed by the group of automorphisms over A. A finite tuple is in acl(A) if and only if it has a finite orbit under this group, and an infinite tuple is in acl(A) if and only if it has a small orbit. The definable closure and algebraic closure of any small set are small.
 * 12) If we assume enough about $$\kappa$$, then we have the following: any small set A is contained in an |A|+-saturated and |A|+-strongly homogeneous small model (elementary substructure of the monster). More generally, by assuming additional properties of $$\kappa$$ at the outset, we can replace |A|+ with practically any function of |A| in this statement.
 * 13) The symbol $$\models$$ is used by default to refer to truth in $$\mathbb{M}$$. So $$\models \phi(a)$$ by default means $$\mathbb{M}$$.
 * 14) The expression tp(a/B) is always interpreted within the model $$\mathbb{M}$$, rather than some other substructure.
 * 15) By assuming enough about $$\kappa$$, one has that there are a small number of types over any small subset.
 * 16) Indiscernible sequences of any small length can be extracted from sequences in the monster.
 * 17) When we do want to talk about types over the monster model, we call them "global types." Such types are usually not realized.

Obtaining a monster model
Ideally, one would have unlimited amounts of strongly inaccessible cardinals. Then one could simply define a monster model to be a saturated model of size $$\kappa$$, for $$\kappa$$ an inaccessible cardinal bigger than any cardinals we care about.

In some cases, saturated models have all the properties we hope for in a monster model. Some conditions guarantee the existence of saturated models, so this is one approach. For example, if the theory is stable, or if we assume the generalized continuum hypothesis, then saturated models can be created.

A more cautionary approach, the one we have been using above, is to require the monster model to be $$\kappa$$-saturated and $$\kappa$$-strongly homogeneous, and then impose numerous conditions on $$\kappa$$. In most (but not all) proofs using monster models, one can figure out what conditions on $$\kappa$$ are needed, impose these at the outset, and then carry out the proof.

Another approach is to have the monster model itself be a proper class. (This probably works best in Von Neumann-Bernays-Gödel Set Theory with global choice, a conservative extension of ZFC.) There is a technical difficulty: we can't define the truth of statements in class-sized models (or else we'd be able to break Tarski's undefinability theorem in ZFC). Presumably one avoids this problem by explicitly describing the class-sized model as a directed limit of small elementary substructures. This approach to monsters essentially works by pretending that our ambient universe of set theory is really the first $$\kappa$$ stages of the Von Neumann Hierarchy, for some inaccessible cardinal $$\kappa$$. The terminology "small" for "of size less than $$\kappa$$" is presumably motivated by this approach to monsters, as well as the paranoia regarding looking at types over big sets. (There are more than a class's worth of types over a class-sized model.)

Also, someone has proven some kind of meta-theorem that legitimizes the use of monster models. Does anyone know more details? Check the Ziegler and Tent book?