TITLE: Non-Principal Ultrafilters Confused!!
QUESTION [8 upvotes]: I've just started learning about filters and non-principal ultrafilters. I'm getting confused on the requirement: $U$ contains no finite subsets of $J$; where $U$ is the ultrafilter and $J$ is a set.
I believe what this means is that $U$ can only contain sets that are infinite? (if it is non-principal). 
Furthermore I'm getting somewhat confused by the proof that non-principal ultrafilters can exist. By the following " Take the filter of all cofinite sets and extend to an ultrafilter". I can't seem to find a definition of a cofinte set? 
Sorry if this is incoherent, getting frustrated by this. 
Thanks in advance.

REPLY [2 votes]: You can also use the following construction:
Take X to be any infinite set, choose N a countable subset of X and enumerate N as a sequence, $N={\{x_n\}}_{n\in\mathcal{N}}$. 
Set $B_1=N$ and $B_n=N/\{x_1,\ldots,x_{n-1}\},n\in\mathcal{N} $
Then $\mathcal{B}=\{B_n : n\in\mathcal{N}\}$ is a filter base that can be extended to a non principal ultrafilter on X.