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.