set theory

SET:
Set theory is a basis of modern mathematics, and notions of set theory are used in all
formal descriptions. The notion of set is taken as “undefined”, “primitive”, or “basic”, so
we don’t try to define what a set is, but we can give an informal description, describe
important properties of sets, and give examples. All other notions of mathematics can be
built up based on the notion of set.
Similar (but informal) words: collection, group, aggregate.
Description and terminology: a set is a collection of objects (entities) which are called
the members or elements of that set. If we have a set we say that some objects belong (or
do not belong) to this set, are (or are not) in the set. We say also that sets consist of their
elements.
Examples: the set of students in this room; the English alphabet may be viewed as the set
of letters of the English language; the set of even numbers; etc.
So sets can consist of elements of various natures: people, physical objects,
numbers, signs, other sets, etc. (We will use the words object or entity in a very broad
way to include all these different kinds of things.)
The membership criteria for a set must in principle be well-defined, and not
vague. If we have a set and an object, it is possible that we do not know whether this
object belongs to the set or not, because of our lack of information or knowledge. (E.g.
“The set of people in this room over the age of 28”, if we don’t know everyone’s age.)
But the answer should exist, at any rate in principle. It could be unknown, but it should
not be vague. If the answer is vague for some putative set-description, we can not
consider that a real description of a set. Another thing: If we have a set, then for every
two elements of it, x and y, it should not be vague whether x = y, or they are different.
Sometimes we simply assume for the sake of examples that a description is not
vague when perhaps for other purposes it would be vague – e.g., the set of all red objects.