Definition:Infinite Set
Definition
A set which is not finite is called infinite.
That is, it is a set for which there is no bijection between it and any $\N_n$, where $\N_n$ is the the set of all elements of $n$ less than $n$, no matter how big we make $n$.
Also known as
When Georg Cantor did his original work on his development of set theory, the concepts were considered alien and difficult to accept.
To sweeten the pill slightly, he coined the word transfinite, which he used instead of infinite, so as not to scare mathematicians who already had a conception of the meaning of infinite, and were having difficulty accepting the challenge to their notions.
Hence the concept of a transfinite set, which is exactly the same as an infinite set.
Contemporary mathematics does not bother much with the term transfinite, except inasmuch as it applies to the concept of a transfinite ordinal.
Also see
- Definition:Finite Set
- Definition:Countable Set
- Definition:Uncountable Set
- Definition:Dedekind-Infinite
- Results about infinite sets can be found here.
Historical Note
The idea of the actual existence of infinite sets as mathematical objects was pioneered by Georg Cantor.
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