In mathematics, and particularly in set theory, category theory, type theory, and the foundations of mathematics, a **universe** is a collection that contains all the entities one wishes to consider in a given situation.

- In a specific context
- In ordinary mathematics
- In set theory
- In predicate calculus
- In category theory
- In type theory
- See also
- Notes
- References
- External links

In set theory, universes are often classes that contain (as elements) all sets for which one hopes to prove a particular theorem. These classes can serve as inner models for various axiomatic systems such as ZFC or Morse–Kelley set theory. Universes are of critical importance to formalizing concepts in category theory inside set-theoretical foundations. For instance, the canonical motivating example of a category is ** Set **, the category of all sets, which cannot be formalized in a set theory without some notion of a universe.

In type theory, a universe is a type whose elements are types.

Perhaps the simplest version is that *any* set can be a universe, so long as the object of study is confined to that particular set. If the object of study is formed by the real numbers, then the real line **R**, which is the real number set, could be the universe under consideration. 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**.

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, it can be regarded as the absolute complement *A*^{C} of *A*. Similarly, there is 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, the nullary intersection can be treated as the set of everything under consideration, which is simply *U*. 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).

However, once subsets of a given set *X* (in Cantor's case, *X* = **R**) are considered, the universe may need to be a set 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 **P***X*, the power set of *X*. This may be continued; the object of study may next consist of such sets of subsets of *X*, and so on, in which case the universe will be **P**(**P***X*). In another direction, the binary relations on *X* (subsets of the Cartesian product *X* × *X*) may be considered, or functions from *X* to itself, requiring universes like **P**(*X* × *X*) or *X*^{X}.

Thus, even if the primary interest is *X*, the universe may need to be considerably larger than *X*. Following the above ideas, one may want the **superstructure** over *X* as the universe. This can be defined by structural recursion as follows:

- Let
**S**_{0}*X*be*X*itself. - Let
**S**_{1}*X*be the union of*X*and**P***X*. - Let
**S**_{2}*X*be the union of**S**_{1}*X*and**P**(**S**_{1}*X*). - In general, let
**S**_{n+1}*X*be the union of**S**_{n}*X*and**P**(**S**_{n}*X*).

Then the superstructure over *X*, written **S***X*, is the union of **S**_{0}*X*, **S**_{1}*X*, **S**_{2}*X*, and so on; or

No matter what set *X* is the starting point, the empty set {} will belong to **S**_{1}*X*. The empty set is the von Neumann ordinal [0]. Then {[0]}, the set whose only element is the empty set, will belong to **S**_{2}*X*; this is the von Neumann ordinal [1]. Similarly, {[1]} will belong to **S**_{3}*X*, 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. The process also gives ordered *n*-tuples, represented as functions whose domain is the von Neumann ordinal [*n*], and so on.

So if the starting point is just *X* = {}, a great deal of the sets needed for mathematics appear as elements of the superstructure over {}. But each of the elements of **S**{} will be a finite set. Each of the natural numbers belongs to it, but the set **N** of *all* natural numbers does not (although it is a *subset* of **S**{}). In fact, the superstructure over {} consists of all of the hereditarily finite sets. As such, it can be considered the *universe of finitist mathematics *. Speaking anachronistically, one 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.

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 it may be necessary to start the process all over again and form **S**(**S**{}). However, to keep things simple, one can 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) belongs to **SN**. Even non-standard analysis can be done in the superstructure over a non-standard model of the natural numbers.

There is a slight shift in philosophy from the previous section, where the universe was any set *U* of interest. There, the sets being studied were *subset*s of the universe; now, they are *members* of the universe. Thus although **P**(**S***X*) is a Boolean lattice, what is relevant is that **S***X* 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 **P***A*, where *A* is any relevant set belonging to **S***X*; then **P***A* is a subset of **S***X* (and in fact belongs to **S***X*). In Cantor's case *X* = **R** in particular, arbitrary sets of real numbers are not available, so there it may indeed be necessary to start the process all over again.

It is possible to 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 an 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.

But if high-powered set theory is brought in, the superstructure process above reveals itself to be merely the beginning of a transfinite recursion. Going back to *X* = {}, the empty set, and introducing the (standard) notation *V*_{i} for **S**_{i}{}, *V*_{0} = {}, *V*_{1} = **P**{}, and so on as before. But what used to be called "superstructure" is now just the next item on the list: *V*_{ω}, where ω is the first infinite ordinal number. This can be extended to arbitrary ordinal numbers:

defines *V*_{i} for *any* ordinal number *i*. The union of all of the *V*_{i} is the von Neumann universe *V*:

- .

Every individual *V*_{i} 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*.

*Kurt Gödel's constructible universe*L*and the axiom of constructibility**Inaccessible cardinals yield models of ZF and sometimes additional axioms, and are equivalent to the existence of the Grothendieck universe set*

In an interpretation of first-order logic, the universe (or domain of discourse) is the set of individuals (individual constants) over which the quantifiers range. A proposition such as ∀ *x* (*x*^{2} ≠ 2) is ambiguous, if no domain of discourse has been identified. In one interpretation, the domain of discourse could be the set of real numbers; in another interpretation, it could be the set of natural numbers. If the domain of discourse is the set of real numbers, the proposition is false, with *x* = √2 as counterexample; if the domain is the set of naturals, the proposition is true, since 2 is not the square of any natural number.

There is another approach to universes which is historically connected with category theory. This is the idea of a Grothendieck universe. Roughly speaking, a Grothendieck universe is a set inside which all the usual operations of set theory can be performed. This version of a universe is defined to be any set for which the following axioms hold:^{ [1] }

- implies
- and imply {
*u*,*v*}, (*u*,*v*), and . - implies and
- (here is the set of all finite ordinals.)
- if is a surjective function with and , then .

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.^{[ citation needed ]}

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*-

Often when working with Grothendieck universes, mathematicians assume the Axiom of Universes: "For any set *x*, there exists a universe *U* such that *x* ∈*U*." 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 some type theories, especially in systems with dependent types, types themselves can be regarded as terms. There is a type called the universe (often denoted ) which has types as its elements. To avoid paradoxes such as Girard's paradox (an analogue of Russell's paradox for type theory), type theories are often equipped with a countably infinite hierarchy of such universes, with each universe being a term of the next one.

There are at least two kinds of universes that one can consider in type theory: **Russell-style universes** (named after Bertrand Russell) and **Tarski-style universes** (named after Alfred Tarski).^{ [2] }^{ [3] }^{ [4] } A Russell-style universe is a type whose terms are types.^{ [2] } A Tarski-style universe is a type together with an interpretation operation allowing us to regard its terms as types.^{ [2] }

- ↑ Mac Lane 1998, p. 22
- 1 2 3 "Universe in Homotopy Type Theory" in nLab
- ↑ Zhaohui Luo, "Notes on Universes in Type Theory", 2012.
- ↑ Per Martin-Löf,
*Intuitionistic Type Theory*, Bibliopolis, 1984, pp. 88 and 91.

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 make a selection of exactly 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 family of elements 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 **axiom of regularity** is an axiom of Zermelo–Fraenkel set theory that states that every non-empty set *A* contains an element that is disjoint from *A*. In first-order logic, the axiom reads:

In set theory and its applications throughout mathematics, a **class** is a collection of sets that can be unambiguously defined by a property that all its members share. Classes act as a way to have set-like collections while differing from sets as to avoid Russell's Paradox. The precise definition of "class" depends on foundational context. In work on Zermelo–Fraenkel set theory, the notion of class is informal, whereas other set theories, such as von Neumann–Bernays–Gödel set theory, axiomatize the notion of "proper class", e.g., as entities that are not members of another entity.

**Set theory** is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole.

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 set theory, an uncountable cardinal is **inaccessible** if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic. More precisely, a cardinal is **strongly inaccessible** if it is uncountable, it is not a sum of fewer than cardinals that are less than , and implies .

**Zermelo set theory**, as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern Zermelo-Fraenkel set theory (ZF) and its extensions, such as von Neumann–Bernays–Gödel set theory (NBG). It bears certain differences from its descendants, which are not always understood, and are frequently misquoted. This article sets out the original axioms, with the original text and original numbering.

In set theory and related branches of mathematics, the **von Neumann universe**, or **von Neumann hierarchy of sets**, denoted by * V*, is the class of hereditary well-founded sets. This collection, which is formalized by Zermelo–Fraenkel set theory (ZFC), is often used to provide an interpretation or motivation of the axioms of ZFC. The concept is named after John von Neumann, although it was first published by Ernst Zermelo in 1930.

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

In mathematical logic, **New Foundations** (**NF**) is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of *Principia Mathematica*. Quine first proposed NF in a 1937 article titled "**New Foundations for Mathematical Logic**"; hence the name. Much of this entry discusses **NFU**, an important variant of NF due to Jensen (1969) and exposited in Holmes (1998). In 1940 and in a revision of 1951, Quine introduced an extension of NF sometimes called "Mathematical Logic" or "ML", that included proper classes as well as sets.

In mathematics, a **Grothendieck universe** is a set *U* with the following properties:

- If
*x*is an element of*U*and if*y*is an element of*x*, then*y*is also an element of*U*. - If
*x*and*y*are both elements of*U*, then is an element of*U*. - If
*x*is an element of*U*, then*P*(*x*), the power set of*x*, is also an element of*U*. - If is a family of elements of
*U*, and if*I*is an element of*U*, then the union is an element of*U*.

In mathematics, a set *A* is **Dedekind-infinite** if some proper subset *B* of *A* is equinumerous to *A*. Explicitly, this means that there exists a bijective function from *A* onto some proper subset *B* of *A*. A set is **Dedekind-finite** if it is not Dedekind-infinite. Proposed by Dedekind in 1888, Dedekind-infiniteness was the first definition of "infinite" that did not rely on the definition of the natural numbers.

In mathematics, specifically in axiomatic set theory, a **Hartogs number** is an ordinal number associated with a set. In particular, if *X* is any set, then the Hartogs number of *X* is the least ordinal α such that there is no injection from α into *X*. If *X* can be well-ordered then the cardinal number of α is a minimal cardinal greater than that of *X*. If *X* cannot be well-ordered then there cannot be an injection from *X* to α. However, the cardinal number of α is still a minimal cardinal *not less than or equal to* the cardinality of *X*. The map taking *X* to α is sometimes called **Hartogs's function**. This mapping is used to construct the aleph numbers, which are all the cardinal numbers of infinite well-orderable sets.

Constructive set theory is an approach to mathematical constructivism following the program of axiomatic set theory. The same first-order language with "" and "" of classical set theory is usually used, so this is not to be confused with a constructive types approach. On the other hand, some constructive theories are indeed motivated by their interpretability in type theories.

In set theory, the **axiom of limitation of size** was proposed by John von Neumann in his 1925 axiom system for sets and classes. It formalizes the limitation of size principle, which avoids the paradoxes encountered in earlier formulations of set theory 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*.

**Tarski–Grothendieck set theory** is an axiomatic set theory. It is a non-conservative extension of Zermelo–Fraenkel set theory (ZFC) and is distinguished from other axiomatic set theories by the inclusion of **Tarski's axiom**, which states that for each set there is a Grothendieck universe it belongs to. Tarski's axiom implies the existence of inaccessible cardinals, providing a richer ontology than that of conventional set theories such as ZFC. For example, adding this axiom supports category theory.

This is a **glossary of set theory**.

In mathematics, in the framework of one-universe foundation for category theory, the term **"conglomerate"** is applied to arbitrary sets as a contraposition to the distinguished sets that are elements of a Grothendieck universe.

- Mac Lane, Saunders (1998).
*Categories for the Working Mathematician*. Springer-Verlag New York, Inc.

- "Universe",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994] - Weisstein, Eric W. "Universal Set".
*MathWorld*.

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