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 mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory, as it provides a useful method for constructing models of any set of sentences that is finitely consistent.
In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named after the symbol he used to denote them, the Hebrew letter aleph (ℵ).
In mathematical logic, the Löwenheim–Skolem theorem is a theorem on the existence and cardinality of models, named after Leopold Löwenheim and Thoralf Skolem.
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.
Rami Grossberg is a full professor of mathematics at Carnegie Mellon University and works in model theory.
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 mathematical logic, a non-standard model of arithmetic is a model of first-order Peano arithmetic that contains non-standard numbers. The term standard model of arithmetic refers to the standard natural numbers 0, 1, 2, …. The elements of any model of Peano arithmetic are linearly ordered and possess an initial segment isomorphic to the standard natural numbers. A non-standard model is one that has additional elements outside this initial segment. The construction of such models is due to Thoralf Skolem (1934).
In the mathematical field of graph theory, the Rado graph, Erdős–Rényi graph, or random graph is a countably infinite graph that can be constructed by choosing independently at random for each pair of its vertices whether to connect the vertices by an edge. The names of this graph honor Richard Rado, Paul Erdős, and Alfréd Rényi, mathematicians who studied it in the early 1960s; it appears even earlier in the work of Wilhelm Ackermann. The Rado graph can also be constructed non-randomly, by symmetrizing the membership relation of the hereditarily finite sets, by applying the BIT predicate to the binary representations of the natural numbers, or as an infinite Paley graph that has edges connecting pairs of prime numbers congruent to 1 mod 4 that are quadratic residues modulo each other.
In set theory, an Aronszajn tree is a tree of uncountable height with no uncountable branches and no uncountable levels. For example, every Suslin tree is an Aronszajn tree. More generally, for a cardinal κ, a κ-Aronszajn tree is a tree of height κ in which all levels have size less than κ and all branches have height less than κ. They are named for Nachman Aronszajn, who constructed an Aronszajn tree in 1934; his construction was described by Kurepa (1935).
The Vaught conjecture is a conjecture in the mathematical field of model theory originally proposed by Robert Lawson Vaught in 1961. It states that the number of countable models of a first-order complete theory in a countable language is finite or ℵ0 or 2ℵ0. Morley showed that the number of countable models is finite or ℵ0 or ℵ1 or 2ℵ0, which solves the conjecture except for the case of ℵ1 models when the continuum hypothesis fails. For this remaining case, Robin Knight has announced a counterexample to the Vaught conjecture and the topological Vaught conjecture. As of 2021, the counterexample has not been verified.
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.
A timeline of mathematical logic; see also history of logic.
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.
In order theory and model theory, branches of mathematics, Cantor's isomorphism theorem states that every two countable dense unbounded linear orders are order-isomorphic. For instance, Minkowski's question-mark function produces an isomorphism between the numerical ordering of the rational numbers and the numerical ordering of the dyadic rationals.
