Usage of Filters and Ultrafilters

Question

I don't know why we need the concept of Filters and Ultrafilters.

they just seem nothing, and I don't know where to use them

can you tell me where do we use those concepts.

Answer 1

Filters are very useful in set theory, in logic and model theory and they have their applications to analysis and to topology as well. Not to mention that model theory has its applications to algebraic geometry and algebra as well.

All in all this makes filters quite very useful.

1. We can characterize continuity using filters and ultrafilters.
2. We can characterize compactness using ultrafilters.
3. We can construct mathematical structures and prove unprovability and inexpressibility using ultrafilters.
4. We can use ultrafilter to do some sort of amalgamation of structures. This way we can extract properties which happen "almost everywhere" in a uniform way.
5. Filters can be generated from [finitely-additive] measures. This means that in some aspect filters are measures, and sometimes you are dealing with sets which are too large to have real-valued measures in any effective way. There filters make a much much better measurement of sets. (E.g. the club filter on cardinals larger than the continuum.)

Generally filters allow us to say when a set is "large enough for our purposes". Then we can ask whether or not things happen on large sets. This is important because in mathematics we often wish to iron out the small pathologies, and in order to do so we need to know that they only occur on inconsequential sets and that they don't occur on large sets. Essentially this is what is common to all the above uses of filters, and probably all the uses of filters.

Answer 2

In model theory, one can use ultrafilters to build new models from old ones. I'll draw an example here, but you can find many others in model theory books.

Let us construct a non-standard model of Peano arithmetic, non-standard meaning that it doesn't "look like" the set $\mathbb N$ but still satisfies all the axioms of Peano arithmetic. Consider an ultrafilter $\mathcal U$ on $\mathbb N$ which contains all cofinite subsets of $\mathbb N$. Now consider the structure $\mathcal M$ on the base set $\mathbb{N}^{\mathbb N} / \mathcal U$, which is a quotient of the set of all sequences over natural integers, with $0^\mathcal M$ being the equivalence class of $(0,0,\ldots)$, $+^{\mathcal M}$ being addition component-wise, and $\cdot^{\mathcal M}$ is component-wise multiplication. You can check that this satisfies all Peano axioms. Now one can check that this isn't isomorphic to the usual $\mathbb N$, because here the sequence $(0,1,2,3,4,\ldots)$ (or more accurately it's equivalence class within the quotient) is not $0$ and is not some successor of $0$, whereas in $\mathbb N$ every integer is either $0$ or some successor of $0$.

Getting this kind of result proves for example that some formulae are not true or not expressible in first-order logic.