set theory

Set theory is special among mathematical theories, in two ways: It playsa central role in putting mathematics on a reliable axiomatic foundation,and it provides the basic language and apparatus in which most ofmathematics is expressed.

1 Axiomatic set theory

I will informally list the undefined notions, the axioms, and two of the“schemes” of set theory, along the lines of Bourbaki’s account. Theaxioms are closer to the von Neumann-Bernays-Gödel model than to theequivalent ZFC model. (But some of the axioms are identical to some in ZFC;see the entry ZermeloFraenkelAxioms (http://planetmath.org/ZermeloFraenkelAxioms).) The intention here is just togive an idea of the level and scope of these fundamental things.

There are three undefined notions:

1. the relation of equality of two sets

2. the relation of membership of one set in another ($x\in y$)

3. the notion of an ordered pair, which is a set comprised from two othersets, in a specific order.

Most of the eight schemes belong more properly to logic than toset theory, but they, or something on the same level, areneeded in the work of formalizing any theory that uses the notion ofequality, or uses quantifiers such as $\exists$.Because of their formal nature, let me just (informally)state two of the schemes:

S6. If $A$ and $B$ are sets, and $A=B$, then anything true of $A$ is true of$B$, and conversely.

S7. If two properties $F(x)$ and $G(x)$ of a set $x$ are equivalent,then the “generic” set having the property $F$, is the same as thegeneric set having the property $G$.

(The notion of a generic set having a given property, is formalizedwith the help of the Hilbert $\tau$ symbol; this is one way,but not the only way, to incorporate what is called the Axiom of Choice.)

Finally come the five axioms in this axiomatization of set theory. (Someare identical to axioms in ZFC, q.v.)

A1. Two sets $A$ and $B$ are equal iff they have the same elements, i.e.iff the relation $x\in A$ implies $x\in B$ and vice versa.

A2. For any two sets $A$ and $B$, there is a set $C$ such that the$x\in C$ is equivalent to $x=A$ or $x=B$.

A3. Two ordered pairs $(A,B)$ and $(C,D)$ are equal iff $A=C$ and $B=D$.

A4. For any set $A$, there exists a set $B$ such that $x\in B$ isequivalent to $x\subset A$; in other words, there is a set of allsubsets of $A$, for any given set $A$.

A5. There exists an infinite set.

The word “infinite” is defined in terms of Axioms A1-A4. But to formulatethe definition, one must first build up some definitions and results aboutfunctions and ordered sets, which we haven’t done here.

2 Product sets, relations, functions, etc.

Moving away from foundations and toward applications, all the more complexstructures and relations of set theory are built upout of the three undefined notions. (See the entry “Set”.) For instance,the relation $A\subset B$ between two sets, means simply“if $x\in A$ then $x\in B$”.

Using the notion of ordered pair, we soon get the very important structurecalled the product $A\times B$ of two sets $A$ and $B$. Next, we can get suchthings as equivalence relations and order relations on a set $A$, for theyare subsets of $A\times A$. And we get the critical notion of a function$A\to B$, as a subset of $A\times B$. Using functions, we get such thingsas the product ${\prod }_{i\in I}{A}_i$ of a family of sets. (“Family” is avariation of the notion of function.)

To be strictly formal, we should distinguish between a function and thegraph of that function, and between a relation and its graph, but thedistinction is rarely necessary in practice.

3 Some structures defined in terms of sets

The natural numbers provide the first example. Peano, Zermelo and Fraenkel,and others have given axiom-lists for the set $\mathbb{N}$, with itsaddition, multiplication, and order relation; butnowadays the custom is to define even the natural numbers in terms ofsets. In more detail, a natural number is the order-type of a finitewell-ordered set.The relation $m\le n$ between $m,n\in \mathbb{N}$is defined with the aid of a certain theorem which says, roughly, thatfor any two well-ordered sets, one is a segment of the other.The sum or product of two natural numbers is defined as the cardinalof the sum or product, respectively, of two sets. (For an extension ofthis idea, see surreal numbers.)

(The term “cardinal” takes some work to define.The “type” of an ordered set, or any other kind of structure, is the“generic” structure of that kind, which is defined using $\tau$.)

Groups provide another simple example of a structure defined in terms of setsand ordered pairs. A group is a pair $(G,f)$ in which $G$ is just a set, and$f$ is a mapping $G\times G\to G$ satisfying certain axioms; the axioms(associativity etc.) can all be spelled out in terms of sets and orderedpairs, although in practice one uses algebraic notation to do it. When wespeak of (e.g.) “the” group $S_3$ of permutations ofa 3-element set, we mean the “type” of such a group.

Topological spaces provide another example of how mathematical structurescan be defined in terms of, ultimately, the sets and ordered pairs in settheory. A topological space is a pair $(S,U)$, where the set $S$ isarbitrary, but $U$ has these properties:

– any element of $U$ is a subset of $S$

– the union of any family (or set) of elements of $U$ is also an element of $U$

– the intersection of any finite family of elements of $U$ is an element of $U$.

Many special kinds of topological spaces are defined by enlarging this listof restrictions on $U$.

Finally, many kinds of structure are based on more than one set. E.g. aleft module is a commutative group $M$ together with a ring $R$,plus a mapping $R\times M\to M$ which satisfies a specific set ofrestrictions.

4 Categories, homological algebra

Although set theory provides some of the language and apparatus usedin mathematics generally, that language and apparatus have expandedover time, and now include what are called “categories” and “functors”.A category is not a set, and a functor is not a mapping, despitesimilarities in both cases. A category comprises all the structuredsets of the same kind, e.g. the groups, and contains also adefinition of the notion of a morphism from one such structuredset to another of the same kind. A functor is similar to a morphism butcompares one category to another, not one structured set to another.The classic examples are certain functors from the category of topologicalspaces to the category of groups.

“Homological algebra” is concerned with sequences of morphismswithin a category, plus functors from one category to another.One of its aims is to get structure theories for specific categories;the homology of groups and the cohomology of Lie algebras are examples.For more details on the categories and functors of homological algebra, Irecommend a search for “Eilenberg-Steenrod axioms”.

Title	set theory
Canonical name	SetTheory
Date of creation	2013-03-22 13:20:53
Last modified on	2013-03-22 13:20:53
Owner	mathwizard (128)
Last modified by	mathwizard (128)
Numerical id	12
Author	mathwizard (128)
Entry type	Topic
Classification	msc 03E30
Synonym	theory of sets
Related topic	Set
Related topic	ZermeloFraenkelAxioms
Related topic	Superset
Related topic	AbstractRelationalBiology
Related topic	Definition