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Encyclopedia > Universe (mathematics)

In mathematics, and particularly in applications to set theory and the foundations of mathematics, a universe or universal class (or if a set, universal set) is, roughly speaking, a class that is large enough to contain (in some sense) all of the sets that one may wish to use. Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... The term foundations of mathematics is sometimes used for certain fields of mathematics itself, namely for mathematical logic, axiomatic set theory, proof theory, model theory, and recursion theory. ... This article is about sets in mathematics. ...

Contents

In a specific context

There are several precise versions of this general idea. Perhaps the simplest is that any set can be a universe, so long as you are studying that particular set. So if you're studying the real numbers, then the real line R, which is the set of all real numbers, could be your universe. Implicitly, this is the universe that Georg Cantor was using when he first developed modern naive set theory and cardinality in the 1870s and 1880s in applications to real analysis. The only sets that Cantor was originally interested in were subsets of R. Please refer to Real vs. ... In mathematics, the real line is simply the set of real numbers. ... Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845 – January 6, 1918) was a mathematician who was born in Russia and lived in Germany for most of his life. ... Naive set theory1 is distinguished from axiomatic set theory by the fact that the former regards sets as collections of objects, called the elements or members of the set, whereas the latter regards sets only as that which satisfies certain axioms. ... The cardinality of a set is a property that describes the size of the set by describing it using a cardinal number. ... Events and Trends Franco-Prussian War (1870-1871) results in the collapse of the Second French Empire and in the formation of both the French Third Republic and the German Empire. ... Events and Trends Technology Development and commercial production of electric lighting Development and commercial production of gasoline-powered automobile by Karl Benz, Gottlieb Daimler and Maybach First commercial production and sales of phonographs and phonograph recordings. ... Real analysis is that branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ... A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y ⊇ X...


This concept of a universe is reflected in the use of Venn diagrams. In a Venn diagram, the action traditionally takes place inside a large rectangle that represents the universe U. One generally says that sets are represented by circles; but these sets can only be subsets of U. The complement of a set A is then given by that portion of the rectangle outside of A's circle. Strictly speaking, this is the relative complement U A of A relative to U; but in a context where U is the universe, we can regard this as this as the absolute complement AC of A. Similarly, we have a notion of the nullary intersection, that is the intersection of zero sets (meaning no sets, not null sets). Without a universe, the nullary intersection would be the set of absolutely everything, which is generally regarded as impossible; but with the universe in mind, we can treat the nullary intersection as the set of everything under consideration, which is simply U. Venn diagrams, Euler diagrams (pronounced oiler) and Johnston diagrams are similar-looking illustrations of set, mathematical or logical relationships. ... In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ... In mathematics, a relatively complemented lattice is a lattice L in which for all a, b, c in L with a ≤ b ≤ c there is some x in L such that x ∨ b = c and x ∧ b = a. ... In arithmetic, the empty product, or nullary product, is the result of multiplying no numbers. ... In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ... Zero redirects here. ... In measure theory, a null set is a set that it is negligible for the purposes of the measure in question. ...


These conventions are quite useful in the algebraic approach to basic set theory, based on Boolean lattices. Except in some non-standard forms of axiomatic set theory (such as New Foundations), the class of all sets is not a Boolean lattice (it is only a relatively complemented lattice). In contrast, the class of all subsets of U, called the power set of U, is a Boolean lattice. The absolute complement described above is the complement operation in the Boolean lattice; and U, as the nullary intersection, serves as the top element (or nullary meet) in the Boolean lattice. Then De Morgan's laws, which deal with complements of meets and joins (which are unions in set theory) apply, and apply even to the nullary meet and the nullary join (which is the empty set). In mathematics and computer science, Boolean algebras, or Boolean lattices, are algebraic structures which capture the essence of the logical operations AND, OR and NOT as well as the corresponding set theoretic operations intersection, union and complement. ... Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. ... In mathematical logic, the New Foundations (NF) of W. V. O. Quine is a candidate set theory, obtained from a streamlined version of the theory of types of Bertrand Russell. ... In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ... In mathematics, a relatively complemented lattice is a lattice L in which for all a, b, c in L with a ≤ b ≤ c there is some x in L such that x ∨ b = c and x ∧ b = a. ... In mathematics, given a set S, the power set of S, written P(S) or 2S, is the set of all subsets of S. In formal language, the existence of power set of any set is presupposed by the axiom of power set. ... In mathematics and computer science, Boolean algebras, or Boolean lattices, are algebraic structures which capture the essence of the logical operations AND, OR and NOT as well as the corresponding set theoretic operations intersection, union and complement. ... In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S. The term least element is defined dually. ... See lattice for other mathematical as well as non-mathematical meanings of the term. ... In logic, De Morgans laws (or De Morgans theorem) are the two rules of propositional logic, boolean algebra and set theory not (P and Q) = (not P) or (not Q) not (P or Q) = (not P) and (not Q) which allow us to move a negation over a... See: JOIN, join command in SQL, a relational database keyword. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... In mathematics, the empty set is the set with no elements. ...


In ordinary mathematics

However, once you consider subsets of a given set X (in Cantor's case, X = R), you may become interested in sets of subsets of X. (For example, a topology on X is a set of subsets of X.) The various sets of subsets of X will not themselves be subsets of X but will instead be subsets of PX, the power set of X. Of course, it doesn't stop there; you might next be interested in sets of sets of subsets of X, and so on. In another direction, you may become interested in the Cartesian product X × X, or in functions from X to itself. Then you might want functions on the Cartesian product, or from X to X × PX, and so on. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, the Cartesian product (or direct product) X Y of two sets X and Y is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y. This concept is named after Ren Descartes. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...


Thus even if your primary interest is X, you may well want your universe to be quite a bit larger than X. Following the above ideas, you may want the superstructure over X. This can be defined by structural recursion as follows: Structural induction is a proof method that is used in mathematical logic (e. ...

  • Let S0X be X itself.
  • Let S1X be the union of X and PX.
  • Let S2X be the union of S1X and P(S1X).
  • In general, let Sn+1X be the union of SnX and P(SnX).

Then the superstructure over X, written SX, is the union of S0X, S1X, S2X, and so on; or In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ...

Note that no matter what set X you start with, the empty set {} will belong to S1X. Recall that the empty set is the von Neumann ordinal [0]. Then {[0]}, the set whose only element is the empty set, will belong to S2X; this is the von Neumann ordinal [1]. Similarly, {[1]} will belong to S3X, and thus so will {[0],[1]}, as the union of {[0]} and {[1]}; this is the von Neumann ordinal [2]. Continuing this process, every natural number is represented in the superstructure by its von Neumann ordinal. Next, if x and y belong to the superstructure, then so does {{x},{x,y}}, which represents the ordered pair (x,y). Thus the superstructure will contain the various desired Cartesian products. Then the superstructure also contains functions and relations, since these may be represented as subsets of Cartesian products. We also get ordered n-tuples, represented as functions whose domain is the von Neumann ordinal [n]. And so on. In mathematics, the empty set is the set with no elements. ... Ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... An ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... In mathematics, an n-ary relation (or often simply relation) is a generalization of binary relations such as = and < which occur in statements such as 5 < 6 or 2 + 2 = 4. It is the fundamental notion in the relational model for databases. ...


So if you start with just X = {}, then you can build up a great deal of the sets needed for mathematics as the elements of the superstructure over {}. But each of the elements of S{} will be finite sets! Each of the natural numbers belong to it, but the set N of all natural numbers does not (although it is a subset of S{}). In fact, the superstructure over X consists of all of the hereditarily finite sets. As such, it can be considered the universe of finitist mathematics. Speaking anachronistically, we could suggest that the 19th-century finitist Leopold Kronecker was working in this universe; he believed that each natural number existed but that the set N (a "completed infinity") did not. In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In mathematics, hereditarily finite sets are defined recursively as finite sets containing hereditarily finite sets (with the empty set as a base case). ... In the philosophy of mathematics, finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps. ... ( 18th century — 19th century — 20th century — more centuries) The 19th century lasted from 1801 to 1900 in the Gregorian calendar (using the Common Era system of year numbering). ... Leopold Kronecker (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on whole numbers, saying, God made the natural numbers; all else is the work of man (Bell 1986, p. ... Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ...


However, S{} is unsatisfactory for ordinary mathematicians (who are not finitists), because even though N may be available as a subset of S{}, still the power set of N is not. In particular, arbitrary sets of real numbers are not available. So we may have to start the process all over again and form S(S{}). However, to keep things simple, let's just take the set N of natural numbers as given and form SN, the superstructure over N. This is often considered the universe of ordinary mathematics. The idea is that all of the mathematics that is ordinarily studied refers to elements of this universe. For example, any of the usual constructions of the real numbers (say by Dedekind cuts) will belong to SN. Even nonstandard analysis can be done in the superstructure over a nonstandard model of the natural numbers. In mathematics, there are a number of ways of defining the real number system as an ordered field. ... In mathematics, a Dedekind cut in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that for any element x in S, if a is in A and x ≤ a, then x is in A as well) and B is... In the most restricted sense, nonstandard analysis or non-standard analysis is that branch of mathematics that formulates analysis using a rigorous notion of infinitesimal, where an element of an ordered field F is infinitesimal if and only if its absolute value is smaller than any element of F of... In the most restricted sense, nonstandard analysis or non-standard analysis is that branch of mathematics that formulates analysis using a rigorous notion of infinitesimal, where an element of an ordered field F is infinitesimal if and only if its absolute value is smaller than any element of F of...


One should note a slight shift in philosophy from the previous section, where the universe was any set U of interest. There, the sets being studied were subsets of the universe; now, they are members of the universe. Thus although P(SX) is a Boolean lattice, what is relevant is that SX itself is not. Consequently, it is rare to apply the notions of Boolean lattices and Venn diagrams directly to the superstructure universe as they were to the power-set universes of the previous section. Instead, one can work with the individual Boolean lattices PA, where A is any relevant set belonging to SX; then PA is a subset of SX (and in fact belongs to SX).


In set theory

We can give a precise meaning to the claim that SN is the universe of ordinary mathematics; it is a model of Zermelo set theory, the axiomatic set theory originally developed by Ernst Zermelo in 1908. Zermelo set theory was successful precisely because it was capable of axiomatising "ordinary" mathematics, fulfilling the programme begun by Cantor over 30 years earlier. But Zermelo set theory proved insufficient for the further development of axiomatic set theory and other work in the foundations of mathematics, especially model theory. For a dramatic example, the description of the superstructure process above cannot itself be carried out in Zermelo set theory! The final step, forming S as a infinitary union, requires the axiom of replacement, which was added to Zermelo set theory in 1922 to form Zermelo-Fraenkel set theory, the set of axioms most widely accepted today. So while ordinary mathematics may be done in SN, discussion of SN goes beyond the "ordinary", into metamathematics. In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the models which underlie mathematical systems. ... Zermelo set theory, as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern set theory. ... Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. ... Ernst Friedrich Ferdinand Zermelo (July 27, 1871 – May 21, 1953) was a German mathematician and philosopher. ... 1908 is a leap year starting on Wednesday (link will take you to calendar). ... The term foundations of mathematics is sometimes used for certain fields of mathematics itself, namely for mathematical logic, axiomatic set theory, proof theory, model theory, and recursion theory. ... In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the models which underlie mathematical systems. ... In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of replacement is a schema of axioms in Zermelo-Fraenkel set theory. ... 1922 was a common year starting on Sunday (see link for calendar). ... The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ... Metamathematics is mathematics used to study mathematics. ...


But if we are going to bring in high-powered set theory, then we realise that the superstructure process above is merely the beginning of a transfinite recursion. Let us go back to X = {}, the empty set, and introduce the (standard) notation Vi for Si{}. Then we have V0 = {}, V1 = P{}, and so on as before. But what used to be called "superstructure" is now just the next item on our list: Vω, where ω is the first infinite ordinal number. If you're familiar with the ordinal numbers, then you know what happens next: Transfinite induction is the proof technique of mathematical induction when applied to (large) well-ordered sets, for instance to sets of ordinals or cardinals, or even to the class of all ordinals. ... Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... Ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ...

defines Vi for any ordinal number i. The union of all of the Vi is the Von Neumann universe V: In axiomatic set theory and related branches of mathematics, the Von Neumann universe, or Von Neumann hierarchy of sets is the class of all sets, divided into a transfinite hierarchy of individual sets. ...

.

Note that every individual Vi is a set, but their union V is a proper class. The axiom of foundation, which was added to ZF set theory at around the same time as the axiom of replacement, says that every set belongs to V. In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ... The axiom of regularity (also known as the axiom of foundation) is one of the axioms of Zermelo-Fraenkel set theory. ...

Kurt Gödel's constructible universe L and the axiom of constructibility
Inaccessible cardinals yield models of ZF and sometimes additional axioms

Kurt Gödel Kurt Gödel [kurt gøːdl], (April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher of mathematics. ... In mathematics, the constructible universe (or Gödels constructible universe) is a particular class of sets which can be described entirely in terms of simpler sets. ... The axiom of constructibility is a possible axiom for set theory in mathematics. ... In mathematics, a cardinal number k > (aleph-null) is called weakly inaccessible, or just inaccessible, if the following two conditions hold. ...

In category theory

There is another approach to universes which is historically connected with category theory. This is the idea of a Grothendieck universe. Conceptually, a Grothendieck universe is a set inside which all the usual operations of set theory can be performed. For example, the union of any two sets in a Grothendieck universe U is still in U. Similarly, intersections, unordered pairs, power sets, and so on are also in U. This is similar to the idea of a superstructure above. The advantage of a Grothendieck universe is that it is actually a set, and never a proper class; the disadvantage is that if one tries hard enough, one can leave a Grothendieck universe. Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, if κ is a strongly inaccessible cardinal, the corresponding Grothendieck universe is the set of all sets with rank less than κ. ...


The most common use of a Grothendieck universe U is to take U as a replacement for the category of all sets. One says that a set S is U-small if SU, and U-large otherwise. The category U-Set of all U-small sets has as objects all U-small sets and as morphisms all functions between these sets. Both the object set and the morphism set are sets, so it becomes possible to discuss the category of "all" sets without invoking proper classes. Then we can define other categories in terms of our new category. For example, the category of all U-small categories is the category of all categories whose object set and whose morphism set are in U. Then the usual arguments of set theory are applicable to the category of all categories, and one does not have to worry about accidentally talking about proper classes. Because Grothendieck universes are extremely large, this suffices in almost all applications.


Often when working with Grothendieck universes, there is an axiom hiding in the background: "For all sets x, there exists a universe U such that xU." The point of this axiom is that any set one encounters is then U-small for some U, so any argument done in a general Grothendieck universe can be applied. This axiom is closely related to the existence of strongly inaccessible cardinals. In mathematics, a strongly inaccessible cardinal is an uncountable cardinal number κ that is regular and a strong limit cardinal. ...

Set-like toposes

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