Example of a free ultrafilter on natural numbers We know that free ultrafilter exist on natural numbers.
I would like to see an example of a free ultrafilter on natural numbers.
 A: Edit: The question originally asked for an example of a free filter on the natural numbers.
The filter $\{X\subseteq \mathbb{N}\mid \mathbb{N}\setminus X\text{ is finite}\}$ is free. This is known as the cofinite filter, or the Fréchet filter.
Maybe you meant to ask for an example of a free ultrafilter? No truly explicit example can be given - there is no constructive proof of the existence of a free ultrafilter on $\mathbb{N}$.
A: I'm assuming you meant "free ultrafilter."
This is a very reasonable question! Unfortunately, in a very real sense we can't exhibit a concrete example of a free ultrafilter on $\mathbb{N}$ - it is consistent with ZF (= set theory without the axiom of choice) that there are no free ultrafilters on $\mathbb{N}$. Any free ultrafilter has to be "hard to define" in various precise ways (this gets a bit technical - the relevant field is "descriptive set theory").
The situation is similar with regard to a number of other kinds of object whose existence relies on the axiom of choice, like non-measurable sets (as in the Banach-Tarski paradox).
