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In mathematics, a set is countable if it has the same cardinality as some subset of the set of natural numbers N = {0, 1, 2, 3, ...}. Equivalently, a set S is countable if there exists an injective function f : S → N from S to N; it simply means that every element in S has the correspondence to a different element in N.
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 mathematics, an uncountable set is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers.
In mathematics, a well-order on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. The set S together with the well-order relation is then called a well-ordered set. In some academic articles and textbooks these terms are instead written as wellorder, wellordered, and wellordering or well order, well ordered, and well ordering.
In the mathematical discipline of set theory, forcing is a technique for proving consistency and independence results. It was first used by Paul Cohen in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo–Fraenkel set theory.
In the mathematical discipline of set theory, 0# is the set of true formulae about indiscernibles and order-indiscernibles in the Gödel constructible universe. It is often encoded as a subset of the integers, or as a subset of the hereditarily finite sets, or as a real number. Its existence is unprovable in ZFC, the standard form of axiomatic set theory, but follows from a suitable large cardinal axiom. It was first introduced as a set of formulae in Silver's 1966 thesis, later published as Silver (1971), where it was denoted by Σ, and rediscovered by Solovay, who considered it as a subset of the natural numbers and introduced the notation O#.
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.
Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in contrast to the ordinary mathematical practice of deriving theorems from axioms. It can be conceptualized as sculpting out necessary conditions from sufficient ones.
In mathematics, in set theory, the constructible universe, denoted by L, is a particular class of sets that can be described entirely in terms of simpler sets. L is the union of the constructible hierarchyLα . It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis". In this, he proved that the constructible universe is an inner model of ZF set theory, and also that the axiom of choice and the generalized continuum hypothesis are true in the constructible universe. This shows that both propositions are consistent with the basic axioms of set theory, if ZF itself is consistent. Since many other theorems only hold in systems in which one or both of the propositions is true, their consistency is an important result.
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 set theory, an ordinal number α is an admissible ordinal if Lα is an admissible set ; in other words, α is admissible when α is a limit ordinal and Lα ⊧ Σ0-collection.
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).
This article contains a discussion of paradoxes of set theory. As with most mathematical paradoxes, they generally reveal surprising and counter-intuitive mathematical results, rather than actual logical contradictions within modern axiomatic set theory.
In mathematical logic, a formula is said to be absolute if it has the same truth value in each of some class of structures. Theorems about absoluteness typically establish relationships between the absoluteness of formulas and their syntactic form.
In the mathematical field of model theory, a complete theory is called stable if it does not have too many types. One goal of classification theory is to divide all complete theories into those whose models can be classified and those whose models are too complicated to classify, and to classify all models in the cases where this can be done. Roughly speaking, if a theory is not stable then its models are too complicated and numerous to classify, while if a theory is stable there might be some hope of classifying its models, especially if the theory is superstable or totally transcendental.
In mathematics, the first uncountable ordinal, traditionally denoted by ω1 or sometimes by Ω, is the smallest ordinal number that, considered as a set, is uncountable. It is the supremum of all countable ordinals. The elements of ω1 are the countable ordinals, of which there are uncountably many.
In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a collection of objects in order, one after another.
This is a glossary of set theory.
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. It is named after Georg Cantor, and can be proved by the back-and-forth method sometimes attributed to Cantor, but Cantor's original proof only used the "going forth" half of this method.
References
- Jech, Thomas (2003). Set Theory. Springer Monographs in Mathematics (Third Millennium ed.). Berlin, New York: Springer-Verlag. ISBN 978-3-540-44085-7. Zbl 1007.03002.