In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, and particularly in
set theory
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 concern ...
,
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
,
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 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
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
inside set-theoretical foundations. For instance, the
canonical motivating example of a category is
Set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
, 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.
In a specific context
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 number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, 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
Georg Ferdinand Ludwig Philipp Cantor ( , ; – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance o ...
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
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
s 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 ''As circle. Strictly speaking, this is the
relative complement
In set theory, the complement of a set , often denoted by (or ), is the set of elements not in .
When all sets in the universe, i.e. all sets under consideration, are considered to be members of a given set , the absolute complement of is ...
''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
Class or The Class may refer to:
Common uses not otherwise categorized
* Class (biology), a taxonomic rank
* Class (knowledge representation), a collection of individuals or objects
* Class (philosophy), an analytical concept used differently ...
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
join Join may refer to:
* Join (law), to include additional counts or additional defendants on an indictment
*In mathematics:
** Join (mathematics), a least upper bound of sets orders in lattice theory
** Join (topology), an operation combining two topo ...
s (which are
unions in set theory) apply, and apply even to the nullary meet and the nullary join (which is the
empty set).
In ordinary mathematics
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
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
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
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\t ...
may be considered, or
functions from ''X'' to itself, requiring universes like 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
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the leas ...
Then , the set whose only element is the empty set, will belong to S
2''X''; this is the von Neumann ordinal
Similarly, will belong to S
3''X'', and thus so will , as the union of and ; this is the von Neumann ordinal
Continuing this process, every
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
is represented in the superstructure by its von Neumann ordinal. Next, if ''x'' and ''y'' belong to the superstructure, then so does , which represents the
ordered pair
In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
(''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.
In set theory
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
Ernst Friedrich Ferdinand Zermelo (, ; 27 July 187121 May 1953) was a German logician and mathematician, whose work has major implications for the foundations of mathematics. He is known for his role in developing Zermelo–Fraenkel axiomatic ...
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
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the leas ...
. This can be extended to arbitrary
ordinal number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the leas ...
s:
:
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 predicate calculus
In an
interpretation
Interpretation may refer to:
Culture
* Aesthetic interpretation, an explanation of the meaning of a work of art
* Allegorical interpretation, an approach that assumes a text should not be interpreted literally
* Dramatic Interpretation, an event ...
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 is ambiguous, if no domain of discourse has been identified. In one interpretation, the domain of discourse could be the set of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s; in another interpretation, it could be the set of
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s. If the domain of discourse is the set of real numbers, the proposition is false, with as counterexample; if the domain is the set of naturals, the proposition is true, since 2 is not the square of any natural number.
In category theory
There is another approach to universes which is historically connected with
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
. 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:
#
implies
#
and
imply , (''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.
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 ''S'' ∈''U'', 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 it becomes possible to define other categories in terms of this 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, 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 type theory
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
In mathematical logic, System U and System U− are pure type systems, i.e. special forms of a typed lambda calculus with an arbitrary number of sorts, axioms and rules (or dependencies between the sorts). They were both proved inconsistent by Jea ...
(an analogue of
Russell's paradox for type theory), type theories are often equipped with a
countably infinite
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...
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
Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, a ...
) and Tarski-style universes (named after
Alfred Tarski).
["Universe in Homotopy Type Theory"]
in nLab A Russell-style universe is a type whose terms are types.
[ A Tarski-style universe is a type together with an interpretation operation allowing us to regard its terms as types.][
For example:]
See also
* Domain of discourse
* Grothendieck universe
* Herbrand universe
* Free object
* Open formula An open formula is a formula that contains at least one free variable.
An open formula does not have a truth value assigned to it, in contrast with a closed formula which constitutes a proposition and thus can have a truth value like ''true'' or ...
* Space (mathematics)
Notes
References
*Mac Lane, Saunders (1998). ''Categories for the Working Mathematician''. Springer-Verlag New York, Inc.
External links
*
*
{{Mathematical logic
Mathematical logic
Families of sets
Set theory