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mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
, and particularly in its subfield
model theory In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mat ...
, a saturated model ''M'' is one that realizes as many complete types as may be "reasonably expected" given its size. For example, an
ultrapower The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All fact ...
model of the
hyperreals In mathematics, hyperreal numbers are an extension of the real numbers to include certain classes of infinite and infinitesimal numbers. A hyperreal number x is said to be finite if, and only if, , x, for some integer n
is \aleph_1-saturated, meaning that every descending nested sequence of
internal set In mathematical logic, in particular in model theory and nonstandard analysis, an internal set is a set that is a member of a model. The concept of internal sets is a tool in formulating the transfer principle, which concerns the logical relati ...
s has a nonempty intersection.


Definition

Let ''κ'' be a finite or infinite
cardinal number In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the cas ...
and ''M'' a model in some
first-order language First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over ...
. Then ''M'' is called ''κ''-saturated if for all subsets ''A'' ⊆ ''M'' of
cardinality The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...
less than ''κ'', the model ''M'' realizes all complete types over ''A''. The model ''M'' is called saturated if it is , ''M'', -saturated where , ''M'', denotes the cardinality of ''M''. That is, it realizes all complete types over sets of parameters of size less than , ''M'', . According to some authors, a model ''M'' is called countably saturated if it is \aleph_1-saturated; that is, it realizes all complete types over countable sets of parameters. According to others, it is countably saturated if it is countable and saturated.Chang and Keisler 1990


Motivation

The seemingly more intuitive notion—that all complete types of the language are realized—turns out to be too weak (and is appropriately named weak saturation, which is the same as 1-saturation). The difference lies in the fact that many structures contain elements that are not definable (for example, any transcendental element of R is, by definition of the word, not definable in the language of fields). However, they still form a part of the structure, so we need types to describe relationships with them. Thus we allow sets of parameters from the structure in our definition of types. This argument allows us to discuss specific features of the model that we may otherwise miss—for example, a bound on a ''specific'' increasing sequence ''cn'' can be expressed as realizing the type which uses countably many parameters. If the sequence is not definable, this fact about the structure cannot be described using the base language, so a weakly saturated structure may not bound the sequence, while an ℵ1-saturated structure will. The reason we only require parameter sets that are strictly smaller than the model is trivial: without this restriction, no infinite model is saturated. Consider a model ''M'', and the type Each finite subset of this type is realized in the (infinite) model ''M'', so by compactness it is consistent with ''M'', but is trivially not realized. Any definition that is universally unsatisfied is useless; hence the restriction.


Examples

Saturated models exist for certain theories and cardinalities: * (Q, <)—the set of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...
s with their usual ordering—is saturated. Intuitively, this is because any type consistent with the
theory A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, ...
is implied by the order type; that is, the order the variables come in tells you everything there is to know about their role in the structure. * (R, <)—the set of
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s with their usual ordering—is ''not'' saturated. For example, take the type (in one variable ''x'') that contains the formula \textstyle for every natural number ''n'', as well as the formula \textstyle. This type uses ω different parameters from R. Every finite subset of the type is realized on R by some real ''x'', so by compactness the type is consistent with the structure, but it is not realized, as that would imply an upper bound to the sequence −1/''n'' that is less than 0 (its least upper bound). Thus (R,<) is ''not'' ω1-saturated, and not saturated. However, it ''is'' ω-saturated, for essentially the same reason as Q—every finite type is given by the order type, which if consistent, is always realized, because of the density of the order. *A dense totally ordered set without endpoints is a ηα set if and only if it is ℵα-saturated. * The countable random graph, with the only non-logical symbol being the edge existence relation, is also saturated, because any complete type is isolated (implied) by the finite subgraph consisting of the variables and parameters used to define the type. Both the theory of Q and the theory of the countable random graph can be shown to be ω-categorical through the back-and-forth method. This can be generalized as follows: the unique model of cardinality ''κ'' of a countable ''κ''-categorical theory is saturated. However, the statement that every model has a saturated elementary extension is not provable in ZFC. In fact, this statement is equivalent to the existence of a proper class of cardinals ''κ'' such that ''κ''<''κ'' = ''κ''. The latter identity is equivalent to for some ''λ'', or ''κ'' is strongly inaccessible.


Relationship to prime models

The notion of saturated model is dual to the notion of prime model in the following way: let ''T'' be a countable theory in a first-order language (that is, a set of mutually consistent sentences in that language) and let ''P'' be a prime model of ''T''. Then ''P'' admits an elementary embedding into any other model of ''T''. The equivalent notion for saturated models is that any "reasonably small" model of ''T'' is elementarily embedded in a saturated model, where "reasonably small" means cardinality no larger than that of the model in which it is to be embedded. Any saturated model is also
homogeneous Homogeneity and heterogeneity are concepts relating to the uniformity of a substance, process or image. A homogeneous feature is uniform in composition or character (i.e., color, shape, size, weight, height, distribution, texture, language, i ...
. However, while for countable theories there is a unique prime model, saturated models are necessarily specific to a particular cardinality. Given certain set-theoretic assumptions, saturated models (albeit of very large cardinality) exist for arbitrary theories. For ''λ''-
stable A stable is a building in which working animals are kept, especially horses or oxen. The building is usually divided into stalls, and may include storage for equipment and feed. Styles There are many different types of stables in use tod ...
theories, saturated models of cardinality ''λ'' exist.


Notes


References

* Chang, C. C.; Keisler, H. J. Model theory. Third edition. Studies in Logic and the Foundations of Mathematics, 73. North-Holland Publishing Co., Amsterdam, 1990. xvi+650 pp. * R. Goldblatt (1998). Lectures on the hyperreals. An introduction to nonstandard analysis. Springer. * Marker, David (2002). ''Model Theory: An Introduction''. New York: Springer-Verlag. * Poizat, Bruno; (translation: Klein, Moses) (2000), ''A Course in Model Theory'', New York: Springer-Verlag. * {{Mathematical logic Mathematical logic Model theory Nonstandard analysis