# uniform and free ultrafilter

I saw two definition for uniform ultrafilter:

For $‎\kappa‎‎$ an infinite cardinal number, an ultrafilter $\mathscr{F}$ over $‎\kappa‎‎$ is called uniform if |F| = $‎\kappa‎‎$ for all $F \in \mathscr{F}$.

A ultrafilter $\mathscr{F}$ on $X$is called uniform if $|F|= |X|$ for all $F \in \mathscr{F}$.

so,

Is defined the uniform ultrafilter on only infinite set?

Are equal uniform and free ultrafilters on infinite set?

## 1 Answer

An ultrafilter could be defined on a finite set. However, it is of no interest, since it is of necessity principal. Uniform ultrafilters on a finite set are even less interesting, for the underlying set of such an ultrafilter would have to be a $1$-element set.

A free ultrafilter need not be uniform. For example, let $U$ be a free ultrafilter on $\mathbb{N}$. Define $U'$ on $\mathbb{R}$ by saying that $X\in U'$ if and only if $X\cap \mathbb{N}\in U$. It is not hard to verify that $U'$ is a free ultrafilter on $\mathbb{R}$. But both $\mathbb{N}$ and $\mathbb{R}$ are in $U'$, so $U'$ is not uniform.

The two definitions you quote are roughly equivalent. The first is on a cardinal number, the second on a general set. But (with AC) for any set there is a bijection between the set and some cardinal. In certain situations, we may be interested in, say, the interaction between an ultrafilter on $\mathbb{N}$ and arithmetic progressions. In that case, structure on the underlying set, and not just cardinality, may be relevant.