home | alphabetical index | |||||

## Naive set theoryNaive 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 the axioms. The name is perhaps derived from the title of Paul Halmos' book Naive Set Theory. Sets are of great importance in mathematics; in fact, in the modern formal treatment, the whole machinery of pure mathematics (numbers, relations, functions, etc.) is defined in terms of sets.This article will give a brief introduction to naive set theory. See also Simple theorems in set theory.
## IntroductionNaive set theory was developed at the end of the 19th century (principally by Georg Cantor and 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 naive 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 naive 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
We define two sets to be
We also allow for an ## Specifying setsThe 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 {
This notation is called " - {
`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
If
As an illustration, let ## 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
Given a universal set `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 { ## Intersections, unions, and relative complements
Given two sets `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
Now let ## Cartesian products
Given objects `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
Cartesian products were first developed by René Descartes in the context of analytic geometry.
If ## ParadoxesThus for every objectx, x belongs to Z if and only if x does not belong to x.
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. ## External link- Beginnings of set theory page at St. Andrews
| |||||

copyright © 2004 FactsAbout.com |