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.