The ultrafilter lemma

Well kiddo, imagine you have a bunch of different colored candies, like red, blue, yellow, and green. You want to pick out some of the candies but you can't pick them all because you have to go to bed soon. So you decide to use a special tool called an ultrafilter to help you choose the candies.

An ultrafilter is like a filter or sieve that can be used to sift through a bunch of things and only keep certain ones. For example, if you use an ultrafilter to sift through your candy pile, it might only let you keep the red and blue candies, and throw away the yellow and green ones.

Now imagine you have some numbers instead of candies. You have a big list of numbers, some are even and some are odd. You want to choose some of the numbers, but you don't know which ones. That's where the ultrafilter lemma comes in.

The ultrafilter lemma says that for any big list of numbers, there is at least one ultrafilter that can be used to choose a specific set of numbers in the list. This means that you can use an ultrafilter to sift through the list and only keep the even or odd numbers, or any other set of numbers you want.

So, the ultrafilter lemma helps us pick out specific sets of numbers from a larger list. It's like using a magnifying glass to focus on something very small in a big picture. And just like how you can use your ultrafilter to choose the red and blue candies, the ultrafilter lemma lets us choose the specific numbers we want from a big list.