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In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that a Cartesian product of a collection of non-empty sets is non-empty. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to construct a set by arbitrarily choosing one object from each bin, even if the collection is infinite. Formally, it states that for every indexed family of nonempty sets, there exists an indexed set such that for every . The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem.
In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. It states:
There is no set whose cardinality is strictly between that of the integers and the real numbers.
In mathematics, an integer sequence is a sequence of integers.
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 set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated ZFC, where C stands for "choice", and ZF refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded.
In group theory, a branch of abstract algebra, the Whitehead problem is the following question:
Is every abelian group A with Ext1(A, Z) = 0 a free abelian group?
In mathematics, Suslin's problem is a question about totally ordered sets posed by Mikhail Yakovlevich Suslin (1920) and published posthumously. It has been shown to be independent of the standard axiomatic system of set theory known as ZFC: Solovay & Tennenbaum (1971) showed that the statement can neither be proven nor disproven from those axioms, assuming ZF is consistent.
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 paper, 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 the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large". The proposition that such cardinals exist cannot be proved in the most common axiomatization of set theory, namely ZFC, and such propositions can be viewed as ways of measuring how "much", beyond ZFC, one needs to assume to be able to prove certain desired results. In other words, they can be seen, in Dana Scott's phrase, as quantifying the fact "that if you want more you have to assume more".
In mathematics, and particularly in axiomatic set theory, the diamond principle◊ is a combinatorial principle introduced by Ronald Jensen in Jensen (1972) that holds in the constructible universe and that implies the continuum hypothesis. Jensen extracted the diamond principle from his proof that the axiom of constructibility implies the existence of a Suslin tree.
The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible. The axiom is usually written as V = L, where V and L denote the von Neumann universe and the constructible universe, respectively. The axiom, first investigated by Kurt Gödel, is inconsistent with the proposition that zero sharp exists and stronger large cardinal axioms. Generalizations of this axiom are explored in inner model theory.
In set theory, a branch of mathematical logic, an inner model for a theory T is a substructure of a model M of a set theory that is both a model for T and contains all the ordinals of M.
In mathematical logic, independence is the unprovability of a sentence from other sentences.
In set theory, inner model theory is the study of certain models of ZFC or some fragment or strengthening thereof. Ordinarily these models are transitive subsets or subclasses of the von Neumann universe V, or sometimes of a generic extension of V. Inner model theory studies the relationships of these models to determinacy, large cardinals, and descriptive set theory. Despite the name, it is considered more a branch of set theory than of model 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 set theory, a branch of mathematics, the minimal model is the minimal standard model of ZFC. The minimal model was introduced by Shepherdson and rediscovered by Cohen (1963).
This is a glossary of set theory.
Joel David Hamkins is an American mathematician and philosopher who is O'Hara Professor of Philosophy and Mathematics at the University of Notre Dame. He has made contributions in mathematical and philosophical logic, set theory and philosophy of set theory, in computability theory, and in group theory.
In mathematics, Wetzel's problem concerns bounds on the cardinality of a set of analytic functions that, for each of their arguments, take on few distinct values. It is named after John Wetzel, a mathematician at the University of Illinois at Urbana–Champaign.
References
- Antos, Carolin; Friedman, Sy-David; Honzik, Radek; Ternullo, Claudio (2015), "Multiverse conceptions in set theory", Synthese, 192 (8): 2463–2488, doi:10.1007/s11229-015-0819-9, MR 3400617
- Hamkins, J. D. (2012), "The set-theoretic multiverse", Rev. Symb. Log., 5 (3): 416–449, arXiv: 1108.4223 , Bibcode:2011arXiv1108.4223H, MR 2970696
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