Infinite Set

AKA: Non-Finite Set.
Context:
- It can range from being an Infinite Unordered Set to being an Infinite Ordered Set.
- It can range from being a Countable Infinite Set to being an Uncountable Infinite Set.
- It can have a Cardinality of Infinity.
- …
Example(s):
- a Real Number Set.
- …
Counter-Example(s):
- a Finite Set.
See: Set Formal System, The Prime Number Sequence.

References

(Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Infinite_set
- In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. Some examples are:
  - the set of all integers, {..., -1, 0, 1, 2, ...}, is a countably infinite set; and
  - the set of all real numbers is an uncountably infinite set.
A set whose elements cannot be put in one-to-one correspondence with a finite set of positive integers.

http://primes.utm.edu/glossary/page.php?sort=Infinite
- An infinite set is a set which is equivalent to a proper subset of itself. For example, the set of integers is equivalent to the set of even integers--a proper subset (to see this, just note f(n)=2n is a one-to-one function from the integers to the even integers). This definition has some amusing consequences. For example, suppose we had a hotel with an infinite number of rooms numbered 1, 2, 3, 4, ..., and no vacancy (every room is filled). We still have room for another person! All we need to do is to have each person move over one room (1 goes to 2, 2 to 3, ..., n to n+1...). In fact, even if it is full, it has room for as many folks as are already in it (just send 1 to 2, 2 to 4, 3 to 6, ..., n to 2n, ..., this leaves 1, 3, 5, … empty). The moral is that "the number of" elements in an infinite set (its cardinality) does not behave like it does for a finite set.