For we write if is finite, and we write if and .

A **tower** is a collection of co-infinite subsets of such that for all we have and either or . ( is co-infinite if is infinite.)

If are towers, we say that if and with jointly imply that . (In other words, this means that is a down-set or initial segment of with respect to ). It is easy to prove that is a partial order on the collection of all towers on .

The remainder of this post is about *maximal towers* with respect to . The proof of the following lemma is routine.

**Lemma 1.** If is a collection of towers, such that for all we have either or , then

1. is a tower, and

2. for all .

**Corollary 1.** Zorn’s Lemma and Lemma 1 imply that there is a tower in that is maximal with respect to .

**Lemma 2.** If is a countable tower then is not maximal.

*Proof.* Let be a sequence of co-infinite subsets of such that for all we have . We want to show there is co-infinite with for all .

*Step 1.* If , then is co-infinite.

*Step 2.* There is strictly increasing such that for all .

*Step 3.* Set

Then it follows that

1: for all since which is a finite set.

2: is co-infinite, since for all we have , so , and since is strictly increasing we have is infinite, so is co-infinite.

Letting we get that is a tower with but , so the countable tower is not maximal.