Saturated model
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mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...
, 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
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model of the hyperreals 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 relation ...
s has a nonempty intersection.


Definition

Let ''κ'' be a
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or
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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. T ...
and ''M'' a model in some
first-order language First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
. Then ''M'' is called ''κ''-saturated if for all subsets ''A'' ⊆ ''M'' of cardinality 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
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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 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 (e.g. ). The set of all rat ...
s with their usual ordering—is saturated. Intuitively, this is because any type consistent with the
theory A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be ...
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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
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 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 ...
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