The fundamental theorem of ultraproducts

Imagine you have a bunch of groups, each with their own rules and structures. Say, one group might only allow you to use certain numbers, another might require all members to have a certain special skill.

Now imagine you take one member from each group and create a new group with those members. This new group doesn't have all the same rules as the original groups, but it's still a group. This process of creating a new group from members of other groups is called an "ultraproduct".

The fundamental theorem of ultraproducts says that under certain conditions, the ultraproduct group will have the same properties as the individual groups it was formed from. This means that even though the ultraproduct group may not follow all the specific rules of each individual group, it still has most of the same overall structure.

Think of it like making a fruit salad. Each fruit has its own unique flavor and texture, but when you combine them all in a salad, you still get a delicious overall taste and texture. The fundamental theorem of ultraproducts is like saying that even though the salad isn't the same as any individual fruit, it still has most of the characteristics of the fruits it's made from.