Strongly increasing sequences

Abstract

Using a variation of Woodin's ${\mathbb{P}}_{\max }$ forcing, we force over a model of the Axiom of Determinacy to produce a model of ZFC containing a very strongly increasing sequence of length $\ddot{I}{‰}_2$ consisting of functions from $\ddot{I}‰$ to $\ddot{I}‰$. We also show that there can be no such sequence of length $\ddot{I}{‰}_4$.