# Naïve set theory

**Naïve set theory**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 name may be derived from the title of Paul Halmos' book

*Naive Set Theory*. Halmos later wrote that he thought he had heard or read the appelation before writing that book, but neither he nor anyone else has found the reference. Sets are of great importance in mathematics; in fact, in the modern formal treatment, the whole machinery of pure mathematics (numbers, relationss, functionss, etc.) is defined in terms of sets.

This article will give a brief introduction to naive set theory.

## Introduction

Naïve set theory was developed at the end of the 19th century (principally by Georg Cantor and Gottlob Frege) in order to allow mathematicians to work with infinite sets consistently.

As it turned out, assuming that one could perform any operations on sets without restriction led to paradoxes such as Russell's paradox. In response, axiomatic set theory was developed to determine precisely what operations were allowed and when. Today, when mathematicians talk about "set theory" as a field, they usually mean axiomatic set theory, but when they talk about set theory as a mere tool to be applied to other mathematical fields, they usual mean naïve set theory.

Axiomatic set theory can be quite abstruse and yet has little effect on ordinary mathematics. Thus, it is useful to study sets in the original naive sense in order to develop facility for working with them. Furthermore, a firm grasp of naïve set theory is important as a first stage in understanding the motivation for the axiomatic theory.

This article develops the naive theory. We begin by defining sets informally and investigating a few of their properties. Links in this article to specific axioms of set theory point out some of the relationships between the informal discussion here and the formal axiomatization of set theory, but we make no attempt to justify every statement on such a basis.

## Sets, membership and equality

If `x` is a member of `A`, then we also say that `x` is an **element** of `A`, or that `x` **belongs to** `A`, or that *x* is in `A`, or that `A` owns `x`. In this case, we write `x` ∈ `A`.
(The symbol "∈" is a derivation from the Greek letter epsilon, "ε", introduced by Peano in 1888.)

We define two sets to be **equal** when they have precisely the same elements. (See axiom of extensionality.) Thus a set is completely determined by its elements; the description is immaterial. For example, the set with elements 2, 3, and 5 is equal to the set of all prime numbers less than 6.
If `A` and `B` are equal, then this is denoted symbolically as `A` = `B` (as usual).

We also allow for an **empty set**, a set without any members at all.
Since a set is determined completely by its elements, there can only be one empty set. (See axiom of empty set.)

## Specifying sets

The simplest way to describe a set is to list its elements between curly braces. Thus {1,2} denotes the set whose only elements are 1 and 2. (See axiom of pairing.) Note the following points:

- Order of elements is immaterial; for example, {1,2} = {2,1}.
- Repetition of elements is irrelevant; for example, {1,2,2} = {1,1,1,2} = {1,2}.

We can also use the notation {`x` : `P`(`x`)} (or sometimes {`x` | `P`(`x`)}) to denote the set containing all objects for which the condition `P` holds.
For example, {`x` : `x` is a real number} denotes the set of real numbers, {`x` : `x` has blonde hair} denotes the set of everything with blonde hair, and {`x` : `x` is a dog} denotes the set {dogs} of all dogs.

This notation is called "**set builder** notation" (or "**set comprehension**", particularly in the context of Functional programming).
Some variants of set builder notation are:

- {
`x`∈`A`:`P`(`x`)} denotes the set of all`x`*that are already members of*such that the condition`A``P`holds for`x`. For example, if**Z**is the set of integers, then {`x`∈**Z**:`x`is even} is the set of all even integers. (See axiom of specification.) - {
`F`(`x`) :`x`∈`A`} denotes the set of all objects obtained by putting members of the set`A`into the formula`F`. For example, {2`x`:`x`∈**Z**} is again the set of all even integers. (See axiom of replacement.) - {
`F`(`x`) :`P`(`x`)} is the most general form of set builder notation. For example, {`x`'s owner :`x`is a dog} is the set of all dog owners.

## Subsets

Given two sets `A` and `B` we say that `A` is a **subset** of `B`, if every element of `A` is also an element of `B`.
Notice that in particular, `B` is a subset of itself; a subset of `B` that isn't equal to `B` is called **proper**.

If `A` is a subset of `B`, then one can also say that `B` is a **superset** of `A`, or that `A` is **contained in** `B`, or that `B` **contains** `A`.
In symbols, `A` ⊆ `B` means that `A` is a subset of `B`, and `B` ⊇ `A` means that `B` is a superset of `A`.
Some authors use the symbols "⊂" and "⊃" for subsets, and others use these symbols only for *proper* subsets.
In this encyclopedia, "⊆" and "⊇" are used for subsets while "⊂" and "⊃" are reserved for proper subsets.

As an illustration, let `A` be the set of real numbers, let `B` be the set of integers, let `C` be the set of odd integers, and let `D` be the set of current or former U.S. Presidents.
Then `C` is a subset of `B`, `B` is a subset of `A`, and `C` is a subset of `A`.
Note that not all sets are comparable in this way.
For example, it is not the case either that `A` is a subset of `D` nor that `D` is a subset of `A`.

## Universal sets and absolute complements

In certain contexts we may consider all of our sets as being subsets of some given universal set.
For instance, if we are investigating properties of real numbers (and sets of reals), then we may take **R**, the set of all reals, as our universal set. It is important to realise that a universal set is only temporarily defined by the context; there is no such thing as a "universal" universal set, "the set of everything" (see **Paradoxes** below).

Given a universal set **U** and a subset `A` of **U**, we may define the **complement** of `A` (in **U**) as

`A`' := {`x`∈**U**: not (`x`∈`A`)},

**U**which are not members of

`A`. Thus with

`A`,

`B`and

`C`as in the section on subsets, if

`B`is the universal set, then

`C`' is the set of even integers, while if

`A`is the universal set, then

`C`' is the set of all real numbers that are either even integers or not integers at all.

The collection {`A` : `A` ⊆ **U**} of all subsets of a given universe **U** is called the **power set** of **U**.
(See axiom of power set.)
It is denoted `P`(**U**); the "`P`" is sometimes in a fancy font.

## Intersections, unions, and relative complements

`A`and

`B`, we may construct their

**union**. This is the set consisting of all objects which are elements of

`A`or of

`B`or of both (see axiom of union). It is denoted by

`A`∪

`B`. The

**intersection**of

`A`and

`B`is the set of all objects which are both in

`A`and in

`B`. It is denoted by

`A`∩

`B`. Finally, the

**relative complement**of

`B`relative to

`A`, also known as the

**set theoretic difference**of

`A`and

`B`, is the set of all objects that belong to

`A`but

*not*to

`B`. It is written as

`A`\\

`B`. Symbolically, these are respectively

`A`∪ B := {`x`: (`x`∈`A`) or (`x`∈`B`)};`A`∩`B`:= {`x`: (`x`∈`A`) and (`x`∈`B`)} = {`x`∈`A`:`x`∈`B`} = {`x`∈`B`:`x`∈`A`};`A`\\`B`:= {`x`: (`x`∈`A`) and not (`x`∈`B`) } = {`x`∈`A`: not (`x`∈`B`)}.

`A`doesn't have to be a subset of

`B`for

`B`\\

`A`to make sense; this is the difference between the relative complement and the absolute complement from the previous section.

To illustrate these ideas, let `A` be the set of left-handed people, and let `B` be the set of people with blond hair.
Then `A` ∩ `B` is the set of all left-handed blond-haired people, while `A` ∪ `B` is the set of all people who are left-handed or blond-haired or both.
`A` \\ `B`, on the other hand, is the set of all people that are left-handed but not blond-haired, while `B` \\ `A` is the set of all people that have blond hair but aren't left-handed.

Now let `E` be the set of all human beings, and let `F` be the set of all living things over 1000 years old.
What is `E` ∩ `F` in this case?
No human being is over 1000 years old, so `E` ∩ `F` must be the empty set {}.

## Cartesian products

Given objects `a` and `b` the **ordered pair** containing `a` and `b` is denoted (`a`,`b`).
For the time being we shall take this as a primitive notion (but see also Ordered pair).
That is, we shall *assume* that (`a`,`b`) has the property that if (`a`,`b`) = (`x`,`y`), then `a` = `x` and `b` = `y`.
The objects `a` and `b` are called respectively the first and second **components** of (`a`,`b`).
Now, given two sets `A` and `B`, we define their **Cartesian product** to be

`A`×`B`= {(`a`,`b`) :`a`is in`A`and`b`is in`B`}.

`A`×

`B`is the set of all ordered pairs whose first component is an element of

`A`and whose second component is an element of

`B`.

We can extend this definition to a set `A` × `B` × `C` of ordered triples, and more generally to sets of ordered `n`-tuples for any positive integer `n`.
It is even possible to define infinite Cartesian products, but to do this we need a more recondite definition of the product.

Cartesian products were first developed by René Descartes; in the context of analytic geometry.
If **R** denotes the set of all real numbers, then **R**^{2} := **R** × **R** represents the Euclidean plane and **R**^{3} := **R** × **R** × **R** represents three-dimensional Euclidean space.

## Paradoxes

Now for the problem: is`Z`a member of

`Z`? If yes, then by the defining quality of

`Z`,

`Z`is not a member of itself, i.e.,

`Z`is not a member of

`Z`. This forces us to declare that

`Z`is not a member of

`Z`. Then

`Z`is not a member of itself and so, again by definition of

`Z`,

`Z`is a member of

`Z`. Thus both options lead us to a contradiction and we have an inconsistent theory. Axiomatic developments place restrictions on the sort of sets we are allowed to form and thus prevent problems like our set

`Z`from arising. (This particular paradox is Russell's paradox.)

The penalty is a much more difficult development. In particular, it is problematic to speak of a set of everything, or to be (possibly) a bit less ambitious, even a set of all sets. In fact, in the standard axiomatisation of set theory, there is no set of all sets. In areas of mathematics that seem to require a set of all sets (such as category theory), one can sometimes make do with a universal set so large that all of ordinary mathematics can be done within it (see universe (mathematics)). Alternatively, one can make use of proper classeses. Or, one can use a different axiomatisation of set theory, such as W. V. Quine's New Foundations, which allows for a set of all sets and avoids Russell's paradox in another way. The exact resolution employed rarely makes an ultimate difference.

## See also

simple theorems in the algebra of sets, internal set theory

## References

- Paul Halmos,
*Naive set theory*, Springer-Verlag, New York. 1974. ISBN 0-387-90092-6.

## External link

- Beginnings of set theory page at St. Andrews