Jónsson function

Last updated April 13, 2019

In mathematical set theory, an ω-Jónsson function for a set x of ordinals is a function $f:[x]^{\omega }\to x$ with the property that, for any subset y of x with the same cardinality as x, the restriction of $f$ to $[y]^{\omega }$ is surjective on $x$ . Here $[x]^{\omega }$ denotes the set of strictly increasing sequences of members of $x$ , or equivalently the family of subsets of $x$ with order type $\omega$ , using a standard notation for the family of subsets with a given order type. Jónsson functions are named for Bjarni Jónsson.

Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used to define nearly all mathematical objects.

Ordinal number order type of a well-ordered set

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. Any finite collection of objects can be put in order just by the process of counting: labeling the objects with distinct natural numbers. Ordinal numbers are thus the "labels" needed to arrange collections of objects in order.

In mathematics, the cardinality of a set is a measure of the "number of elements of the set". For example, the set $contains 3 elements, and therefore has a cardinality of 3. There are two approaches to cardinality - one which compares sets directly using bijections and injections, and another which uses cardinal numbers. The cardinality of a set is also called its size, when no confusion with other notions of size is possible.$

ErdősandHajnal ( 1966 ) showed that for every ordinal λ there is an ω-Jónsson function for λ.

Kunen's proof of Kunen's inconsistency theorem uses a Jónsson function for cardinals λ such that 2^λ = λ^ℵ₀, and Kunen observed that for this special case there is a simpler proof of the existence of Jónsson functions. Galvin andPrikry ( 1976 ) gave a simple proof for the general case.

In set theory, a branch of mathematics, Kunen's inconsistency theorem, proved by Kenneth Kunen (1971), shows that several plausible large cardinal axioms are inconsistent with the axiom of choice.

Cardinal number unit of measure for the cardinality (size) of sets

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. The transfinite cardinal numbers describe the sizes of infinite sets.

Frederick William Galvin is a mathematician, currently a professor at the University of Kansas. His research interests include set theory and combinatorics.

The existence of Jónsson functions shows that for any cardinal there is an algebra with an infinitary operation that has no proper subalgebras of the same cardinality. In particular if infinitary operations are allowed then an analogue of Jónsson algebras exists in any cardinality, so there are no infinitary analogues of Jónsson cardinals.

In set theory, a Jónsson cardinal is a certain kind of large cardinal number.

Related Research Articles

In mathematics, especially in order theory, the cofinality cf(A) of a partially ordered set A is the least of the cardinalities of the cofinal subsets of A.

In mathematics, a Borel set is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel.

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, 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.

In mathematics, and in particular set theory, the aleph numbers are a sequence of numbers used to represent the cardinality of infinite sets that can be well-ordered. They are named after the symbol used to denote them, the Hebrew letter aleph.

In mathematics, an Erdős cardinal, also called a partition cardinal is a certain kind of large cardinal number introduced by Paul Erdős and András Hajnal (1958).

In mathematics, limit cardinals are certain cardinal numbers. A cardinal number λ is a weak limit cardinal if λ is neither a successor cardinal nor zero. This means that one cannot "reach" λ from another cardinal by repeated successor operations. These cardinals are sometimes called simply "limit cardinals" when the context is clear.

PCF theory is the name of a mathematical theory, introduced by Saharon Shelah (1978), that deals with the cofinality of the ultraproducts of ordered sets. It gives strong upper bounds on the cardinalities of power sets of singular cardinals, and has many more applications as well. The abbreviation "PCF" stands for "possible cofinalities".

In mathematical set theory and model theory, a stationary set is one that is not too small in the sense that it intersects all club sets, and is analogous to a set of non-zero measure in set theory. There are at least three closely related notions of stationary set, depending on whether one is looking at subsets of an ordinal, or subsets of something of given cardinality, or a powerset.

A Dynkin system, named after Eugene Dynkin, is a collection of subsets of another universal set $satisfying a set of axioms weaker than those of σ-algebra. Dynkin systems are sometimes referred to as λ-systems or d-system . These set families have applications in measure theory and probability.$

In mathematics, a $π$ -system on a set Ω is a collection P of certain subsets of Ω, such that

In set theory, the axiom of limitation of size was proposed by John von Neumann in his 1925 axiom system for sets and classes. This axiom formalizes the limitation of size principle, which avoids the paradoxes by recognizing that some classes are too big to be sets. Von Neumann realized that the paradoxes are caused by permitting these big classes to be members of a class. A class that is a member of a class is a set; a class that is not a set is a proper class. Every class is a subclass of V, the class of all sets. The axiom of limitation of size says that a class is a set if and only if it is smaller than V — that is, there is no function mapping it onto V. Usually, this axiom is stated in the equivalent form: A class is a proper class if and only if there is a function that maps it onto V.

In mathematics, infinitary combinatorics, or combinatorial set theory, is an extension of ideas in combinatorics to infinite sets. Some of the things studied include continuous graphs and trees, extensions of Ramsey's theorem, and Martin's axiom. Recent developments concern combinatorics of the continuum and combinatorics on successors of singular cardinals.

This is a glossary of set theory.

References

Erdős, P.; Hajnal, András (1966), "On a problem of B. Jónsson", Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, 14: 19–23, ISSN 0001-4117, MR 0209161
Galvin, Fred; Prikry, Karel (1976), "Infinitary Jonsson algebras and partition relations", Algebra Universalis, 6 (3): 367–376, doi:10.1007/BF02485843, ISSN 0002-5240, MR 0434822
Jónsson, Bjarni (1972), Topics in universal algebra, Lecture Notes in Mathematics, 250, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0058648, MR 0345895
Kanamori, Akihiro (2003), The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed.), Berlin, New York: Springer-Verlag, p. 319, ISBN 978-3-540-00384-7

Paul Erdős was a renowned Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. He was known both for his social practice of mathematics and for his eccentric lifestyle. He devoted his waking hours to mathematics, even into his later years—indeed, his death came only hours after he solved a geometry problem at a conference in Warsaw.

András Hajnal was a professor of mathematics at Rutgers University and a member of the Hungarian Academy of Sciences known for his work in set theory and combinatorics.

International Standard Serial Number unique eight-digit number used to identify a print or electronic periodical publication

An International Standard Serial Number (ISSN) is an eight-digit serial number used to uniquely identify a serial publication, such as a magazine. The ISSN is especially helpful in distinguishing between serials with the same title. ISSN are used in ordering, cataloging, interlibrary loans, and other practices in connection with serial literature.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.