Algebra Confusion

Recently, I've been working with friends of mine to try to understand various topics in mathematics. Two concepts that have come up flirt closely with the concepts of the infinite but seem to evade some of my understanding regarding such topics. The particular topics here are the notions of algebras of a set and sigma-algebras of a set. An algebra of a set X is a set or list of subsets of X. For any two subsets you find in your algebra, the subset of X that contains all the elements of both must also be in your algebra (which is just a set, remember). Similarly, whenever you pick a subset in your algebra, the complement of that subset with regards to X must also be in the algebra. That means that the set containing all the things in X that are NOT in your selected set (which then makes another subset) is also in your algebra. Finally, for any two subsets in your algebra, the intersection of those subsets (the set containing all the members of BOTH) must also be in the algebra.

So, when you take a set X, like {1, 2, 3, 4} for example, you can write a set of subsets of X by pulling out subsets of X and putting them in your algebra. I'll take the subset {1, 2} of my set X, and put that in my algebra. Because I put that in, the compliment of it must also go in, which is {3, 4}. Now I have two things in my algebra, {1, 2} and {3, 4}. If I take the union of those two, {1, 2, 3, 4}, that ALSO has to go into the algebra! Finally, if I take the intersection of {1, 2} and {3, 4}, which contains nothing (i.e. it's the empty set, {}), that also has to go into the algebra. By making my algebra from the subset {1, 2}, I've been forced to add in more subsets, to get the final structure being { {}, {1, 2}, {3, 4}, and {1, 2, 3, 4} }. All of that had to go in just to make sure that the properties of being an algebra were satisfied. Now, when I take any pair of those and take the union or intersection, I won't leave that list. The same is true if I take any one and take the compliment.

The sigma algebra is not defined by pairwise unions (i.e. taking two subsets from the algebra and taking all the elements from both), but rather from countably many unions. That means that I can grab as many of the subsets in the algebra as I want so long as I pull them out one at a time, in order. Then, I can take all the elements from those selections and the subset containing all of those must be in the algebra. The same is true for intersections.

What I don't get is this: how are those two definitions describing different things? If you can take any number of pairwise combinations and take the union of them one at a time, then you should be able to take the countable combination of the same sets where the order in which you pick them is the order in which you did the pairwise combinations previously. Clearly I'm not understanding some part of this. I'll share the resolution here if I should successfully comprehend what's going on.