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In mathematical logic, model theory is the study of the relationship between formal theories, and their models. The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself. In particular, model theorists also investigate the sets that can be defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to Alfred Tarski, who first used the term "Theory of Models" in publication in 1954. Since the 1970s, the subject has been shaped decisively by Saharon Shelah's stability theory.
In set theory, a branch of mathematics, a Q-indescribable cardinal is a certain kind of large cardinal number that is hard to axiomatize in some language Q. There are many different types of indescribable cardinals corresponding to different choices of languages Q. They were introduced by Hanf & Scott (1961).
In mathematics, a measurable cardinal is a certain kind of large cardinal number. In order to define the concept, one introduces a two-valued measure on a cardinal κ, or more generally on any set. For a cardinal κ, it can be described as a subdivision of all of its subsets into large and small sets such that κ itself is large, ∅ and all singletons {α} are small, complements of small sets are large and vice versa. The intersection of fewer than κ large sets is again large.
In set theory, a supercompact cardinal is a type of large cardinal independently introduced by Solovay and Reinhardt. They display a variety of reflection properties.
In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by Erdős & Tarski (1961); weakly compact cardinals are large cardinals, meaning that their existence cannot be proven from the standard axioms of set theory.
Saharon Shelah is an Israeli mathematician. He is a professor of mathematics at the Hebrew University of Jerusalem and Rutgers University in New Jersey.
In mathematical logic, and particularly in its subfield model theory, a saturated modelM is one that realizes as many complete types as may be "reasonably expected" given its size. For example, an ultrapower model of the hyperreals is -saturated, meaning that every descending nested sequence of internal sets has a nonempty intersection.
In mathematical logic, a theory is categorical if it has exactly one model. Such a theory can be viewed as defining its model, uniquely characterizing the model's structure.
In model theory, a branch of mathematical logic, the spectrum of a theory is given by the number of isomorphism classes of models in various cardinalities. More precisely, for any complete theory T in a language we write I(T, κ) for the number of models of T (up to isomorphism) of cardinality κ. The spectrum problem is to describe the possible behaviors of I(T, κ) as a function of κ. It has been almost completely solved for the case of a countable theory T.
In the mathematical field of set theory, the proper forcing axiom (PFA) is a significant strengthening of Martin's axiom, where forcings with the countable chain condition (ccc) are replaced by proper forcings.
In model theory, a branch of mathematical logic, a complete first-order theory T is called stable in λ, if the Stone space of every model of T of size ≤ λ has itself size ≤ λ. T is called a stable theory if there is no upper bound for the cardinals κ such that T is stable in κ. The stability spectrum of T is the class of all cardinals κ such that T is stable in κ.
In the mathematical field of model theory, a theory is called stable if it satisfies certain combinatorial restrictions on its complexity. Stable theories are rooted in the proof of Morley's categoricity theorem and were extensively studied as part of Saharon Shelah's classification theory, which showed a dichotomy that either the models of a theory admit a nice classification or the models are too numerous to have any hope of a reasonable classification. A first step of this program was showing that if a theory is not stable then its models are too numerous to classify.
Richard Joseph Laver was an American mathematician, working in set theory.
In the mathematical field of set theory, the Solovay model is a model constructed by Robert M. Solovay in which all of the axioms of Zermelo–Fraenkel set theory (ZF) hold, exclusive of the axiom of choice, but in which all sets of real numbers are Lebesgue measurable. The construction relies on the existence of an inaccessible cardinal.
In model theory, a discipline within mathematical logic, an abstract elementary class, or AEC for short, is a class of models with a partial order similar to the relation of an elementary substructure of an elementary class in first-order model theory. They were introduced by Saharon Shelah.
This is a glossary of terms and definitions related to the topic of set theory.
In model theory, a discipline within the field of mathematical logic, a tame abstract elementary class is an abstract elementary class (AEC) which satisfies a locality property for types called tameness. Even though it appears implicitly in earlier work of Shelah, tameness as a property of AEC was first isolated by Grossberg and VanDieren, who observed that tame AECs were much easier to handle than general AECs.
References
- Harnik, Victor; Harrington, Leo (1984), "Fundamentals of forking", Ann. Pure Appl. Logic, 26 (3): 245–286, doi: 10.1016/0168-0072(84)90005-8 , MR 0747686
- Lascar, Daniel; Poizat, Bruno (1979), "An Introduction to Forking", The Journal of Symbolic Logic, 44 (3), Association for Symbolic Logic: 330–350, doi:10.2307/2273127, JSTOR 2273127
- Makkai, M. (1984), "A survey of basic stability theory, with particular emphasis on orthogonality and regular types", Israel Journal of Mathematics , 49 (1–3): 181–238, doi: 10.1007/BF02760649 , S2CID 121533246
- Marker, David (2002), Model Theory: An Introduction, Graduate Texts in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98760-6
- Ng, Siu-Ah (2001) [1994], "Forking", Encyclopedia of Mathematics , EMS Press
- Shelah, Saharon (1990) [1978], Classification theory and the number of nonisomorphic models , Studies in Logic and the Foundations of Mathematics (2nd ed.), Elsevier, ISBN 978-0-444-70260-9
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