| Set
: The things correctly named by
some noun-like expression. As Cantor defined it: "By a 'set' we mean any
gathering into a whole ... of distinct perceptual or mental objecs" or
"A set is a many which allows itself to be thought of as one".
Set theory was first conceived of by Georg Cantor, in the second half of
the 19th Century, to solve problems and puzzles he faced when dealing
with infinities in mathematics.
Set theory is currently the standard
foundation of mathematics, especially in the form of the
Zermelo-Fraenkel set theory, often abbreviated as ZF. For this there are
three main reasons:
(1) Set theory is a lingua franca of
mathematics, that allows one to define many mathematical concepts
clearly and precisely, and to conduct many mathematical arguments
clearly and conclusively, while it is not difficult and indeed, as my
definition above suggests, intuitively based, in part at least, on how
one uses nouns and common names.
(2) Much of standard mathematics -
though not all of it - can be developed on the basis of definitions only
presuming the axioms of ZF. This constitutes at least theoretically and
notationally a great unification and clarification.
Furthermore, formally speaking set
theory only needs standard first-order logic and the primitive concept
"is an element of", usually written as "e".
(3) Set theory arose from the attempt to make the reasoning with
infinities in mathematics, especially in the calculus, that turns around
the concepts of infinite series like: 1, 1/2, 1/4, 1/8, .., 1/2n
.. and many others, precise and axiomatic, and succeeds in doing so, at
the price of or with the benefit of introducing an infinity of infinities
of sets. (See powerset.)
The terms "set", "class" and "collection" are often used as synonyms,
though it is noteworthy that one of the ways of avoiding paradoxes in
set theory, that was introduced by Von Neumann, is to distinguish
sets
and classes, or also classes and proper classes as
follows: What can be an element is always a set, but what cannot be an
element while still having elements is a class or proper class.
This neat distinction has the great merit of dissolving Russell's
paradox.
There are many introductions to set theory, of which "Naive Set
Theory" by Paul Halmos is one of the best, because it is
clear, fast-paced and not fussy about things that don't really
matter in a first introduction. Also it is a fairly thin book, so that
one doesn't need much time to decide whether this is something for one
or not.
A good introduction to and exposition of the many difficulties
involved in the concept of the infinite is A.W. Moore's "The
Infinite".
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