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In mathematical logic, and particularly in its subfield model theory, a saturated model ''M'' is one that realizes as many complete types as may be "reasonably expected" given its size. For example, an ultrapower model of the
hyperreals In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains number ...
is \aleph_1-saturated, meaning that every descending nested sequence of internal sets has a nonempty intersection.


Definition

Let ''κ'' be a
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb Traditionally, a finite verb (from la, fīnītus, past particip ...
or infinite
cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. Th ...
and ''M'' a model in some first-order language. Then ''M'' is called ''κ''-saturated if for all subsets ''A'' ⊆ ''M'' of
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
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
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
s). 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 numbers with their usual ordering—is saturated. Intuitively, this is because any type consistent with the theory 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 numbers 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 In mathematical logic, especially set theory and model theory, the back-and-forth method is a method for showing isomorphism between countably infinite structures satisfying specified conditions. In particular it can be used to prove that * any two ...
. 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 In model theory, a branch of mathematical logic, two structures ''M'' and ''N'' of the same signature ''σ'' are called elementarily equivalent if they satisfy the same first-order ''σ''-sentences. If ''N'' is a substructure of ''M'', one often ...
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 In set theory, an uncountable cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic. More precisely, a cardinal is strongly inaccessible if it is uncountable, it is not a sum of f ...
.


Relationship to prime models

The notion of saturated model is dual to the notion of
prime model In mathematics, and in particular model theory, a prime model is a model that is as simple as possible. Specifically, a model P is prime if it admits an elementary embedding into any model M to which it is elementarily equivalent (that is, into ...
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. 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 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; Trans: Klein, Moses (2000), ''A Course in Model Theory'', New York: Springer-Verlag. * {{Mathematical logic Mathematical logic Model theory Nonstandard analysis