On Shelah 908; or, Strong chains of functions and sets

I will discuss the paper of Shelah from the title and results which can be proved using similar techniques. A sample result is that one cannot have a chain of subsets of Omega2 strongly increasing modulo finite. The negative results we discuss can be contrasted with positive results of Koszmider on strong chains and results of Baumgartner on strongly almost disjoint families. The paper of Shelah is very short and relies only on a basic understanding of set theory (regular cardinals, closed unbounded sets).