Everything about Sets totally explained
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This article gives an introduction to what mathematicians call "intuitive" or "naive" set theory; for a more detailed account see Naive set theory. For a rigorous modern axiomatic treatment of sets, see Axiomatic set theory.
A
set is a collection of distinct objects considered as a whole. Sets are one of the most fundamental
concepts in
mathematics. The study of the structure of sets,
set theory, is rich and ongoing. Having only been invented at the end of the
19th century, set theory is now a ubiquitous part of
mathematics education, being introduced from
primary school in many countries. Set theory can be viewed as a foundation from which nearly all of mathematics can be derived.
In
philosophy, sets are ordinarily considered to be
abstract objects
the physical
tokens of which are, for instance; three cups on a table when spoken of together as "
the cups", or the chalk lines on a board in the form of the opening and closing curly bracket
symbols along with any other symbols in between the two bracket symbols. However, proponents of
mathematical realism including
Penelope Maddy have argued that sets are
concrete objects.
Definition
At the beginning of his
Beiträge zur Begründung der transfiniten Mengenlehre,
Georg Cantor, the principal creator of set theory, gave the following definition of a set:
The
elements of a set, also called its
members, can be anything: numbers, people, letters of the alphabet, other sets, and so on. Sets are conventionally denoted with
capital letters. The statement that sets
A and
B are equal means that they've precisely the same members (for example, every member of
A is also a member of
B and vice versa).
Unlike a
multiset, every element of a set must be unique; no two members may be identical. All set operations preserve the property that each element of the set is unique. The order in which the elements of a set are listed is irrelevant, unlike a
sequence or
tuple.
Describing sets
There are two ways of describing, or specifying the members of, a set. One way is by
intensional definition, using a rule or
semantic description. See this example:
» A is the set whose members are the first four positive
integers.
B is the set of colors of the
French flag.
The second way is by
extension, that is, listing each member of the set. An
extensional definition is notated by enclosing the list of members in
braces:
» C = doesn't exist.
Cantor's paradox - It shows that "the set of all sets" can't exist.
The reason is that the phrase well-defined isn't very well-defined. It was important to free set theory of these paradoxes because nearly all of mathematics was being redefined in terms of set theory. In an attempt to avoid these paradoxes, set theory was axiomatized based on first-order logic, and thus the axiomatic set theory was born.
For most purposes however, the naive set theory is still useful.
Mathematical realism
Penelope Maddy has suggested that sets can be causally efficacious, and in fact share all the causal and spatiotemporal properties of their elements. Thus, when I see the three cups on the table in front of me, I also see the set as well. She used recent work in cognitive science and psychology to support this position, pointing out that just as at a certain age we begin to see objects rather than mere sense perceptions, there's also a certain age at which we begin to see sets rather than just objects.
Further Information
Get more info on 'Sets'.
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