A generalized theorem can be proved using basically the same ideas as those employed in the more special case discussed above.

If a theory has any infinite model, then, for any infinite cardinality α, that theory has a model of cardinality α. More explicitly, this theorem contains two parts: (1) If a theory has a model of infinite cardinality β, then, for each infinite cardinal α that is greater than β, the theory has a model of cardinality α. (2) If a theory has a model of infinite cardinality β, then, for each infinite cardinal α less than β, the theory has a model of cardinality α.

It follows immediately that any theory having an infinite model has two nonisomorphic models and is, therefore, not categorical. This applies, in particular, to the aforementioned theories T_a and T_b of arithmetic (based on the language of N), the natural models of which are countable, as well as to theories dealing with real numbers and arbitrary sets, the natural models of which are uncountable; both kinds of theory have both countable and uncountable models. There is much philosophical discussion about this phenomenon.

The possibility is not excluded that a theory may be categorical in some infinite cardinality. The theory T_d, for example, of dense linear ordering (such as that of the rational numbers) is categorical in the countable cardinality. One application of the Löwenheim-Skolem theorem is: If a theory has no finite models and is categorical in some infinite cardinality α, then the theory is complete; i.e., for every closed sentence in the language of the theory, either that sentence or its negation belongs to the theory. An immediate consequence of this application of the theorem is that the theory of dense linear ordering is complete.

A theorem that is generally regarded as one of the most difficult to prove in model theory is the theorem by Michael Morley, as follows:

A theory that is categorical in one uncountable cardinality is categorical in every uncountable cardinality.

Two-cardinal theorems deal with languages having some distinguished predicate U. A theory is said to admit the pair <α, β> of cardinals if it has a model (with its domain) of cardinality α wherein the value of U is a set of cardinality β. The central two-cardinal theorem says:

If a theory admits the pair <α, β> of infinite cardinals with β less than α, then for each regular cardinal γ the theory admits <γ⁺, γ>, in which γ⁺ is the next larger cardinal after γ.

The most interesting case is when γ is the least infinite cardinal, ℵ₀. (The general theorem can be established only when the “generalized continuum hypothesis” is assumed, according to which the next highest cardinality for an infinite set is that of its power set.)

Ultrafilters, ultraproducts, and ultrapowers

An ultrafilter on a nonempty set I is defined as a set D of subsets of I such that

• (1) the empty set does not belong to D,

• (2) if A, B are in D, so is their intersection, A ∩ B, the set of elements common to both,

• (3) if A is a subset of B, and A is in D, then B is in D, and

• (4) for every subset A of I, either A is in D or I minus A is in D.

Roughly stated, each ultrafilter of a set I conveys a notion of large subsets of I so that any property applying to a member of D applies to I “almost everywhere.”

The set {𝔄_i}, where 𝔄_i = < A_i, R_i> and the i are members of the set I, is taken to be a family of structures indexed by I, and D to be an ultrafilter on I. Consider now the Cartesian product B of {A_i} (for example, if I is {0, 1, 2, . . .}, then B is the set of all sequences f such that f(i) belong to A_i). The members of B are divided into equivalence classes with the help of D : f ≡ g if and only if {i|f(i) = g(i)} ∊ D—in other words, the set of indices i such that f(i) = g(i) belong to D [or f(i) and g(i) are equal “almost everywhere”]. Let W be the set of these equivalence classes—i.e., the set of all f* such that f* is the set of all members g of B with g ≡ f. Similarly, a relation S is introduced such that Sfg if and only if R_i holds between f(i) and g(i) for “almost all” i; i.e.,

{i|R_i [ f(i), g(i)]} ∊ D.

In this way, we arrive at a new structure U = <W,S>, which is called the ultraproduct of the original family {𝔄_i} over D. In the special case when all the 𝔄_i are the same, the resulting structure U is called the ultrapower of the original family over D.

The central theorems are the following:

1. If 𝔄_i (i ∊ I) are realizations of the same language, then a sentence p is true in the ultraproduct U if and only if the set of i such that p is true in 𝔄_i belongs to D. In particular, if each 𝔄_i is a model of a theory, then U is also a model of the theory.
2. Two realizations of the same language are said to be elementarily equivalent if they have the same set of true sentences. A necessary and sufficient condition for two realizations to be elementarily equivalent is that they admit ultrapowers that are isomorphic.

One application of these theorems is in the introduction of nonstandard analysis, which was originally instituted by other considerations. By using a suitable ultrapower of the structure of the field ℜ of real numbers, a real closed field that is elementarily equivalent to ℜ is obtained that is non-Archimedean—i.e., which permits numbers a and b such that no n can make na greater than b. This development supplies an unexpected exact foundation for the classical differential calculus using infinitesimals, which has considerable historical, pedagogical, and philosophical interest.

A widely known application to the area of algebra is that which deals with certain fields of rational numbers Q_p, called the p-adic completion of the rational numbers. The conjecture has been made that every form of degree d (in the same sense as degrees of ordinary polynomials) over Q_p, in which the number of variables exceeds d², has a nontrivial zero in Q_p. Using ultraproducts, it has been shown that the conjecture is true for arbitrary d with the possible exception of a finite set of primes p (depending on d). Subsequently, it was found that the original conjecture is not true when extended to full generality.

Other useful tools in model theory include the pigeonhole principles, of which the basic principle is that, if a set of large cardinality is partitioned into a small number of classes, some one class will have large cardinality. Those elements of the set that lie in the same class cannot be distinguished by the property defining that class.

A related concept is that of “indiscernibles,” which also has rather extensive applications in set theory. An ordered subset of the domain of a model 𝔄 of a theory is a homogeneous set, or a set of indiscernibles for 𝔄, if 𝔄 cannot distinguish the members of the subset from one another. More exactly, given any x₁ < . . . < x_n, y₁ < . . . <y_n in the subset, then for any sentence F(a₁, . . ., a_n) of the language of the theory, that sentence (with argument x) is satisfied by (symbolized ⊩) the structure—i.e.,

𝔄⊩F(x₁, . . ., x_n)

—if and only if that sentence (with argument y) is also satisfied by it—i.e.,

𝔄⊩F(y₁, . . ., y_n).

There is also a first theorem on this notion that says that, given a theory with an infinite model and a linearly ordered set X, there is then a model 𝔄 of the theory such that X is a set of indiscernibles for 𝔄.