In the context of countable support iterations, there are two significant preservation theorems. The First Preservation Theorem, due to Shelah, preserves the dominating number of a \(F_{\sigma}\) relation small. The Second Preservation Theorem, due to Judah–Repický, preserves the bounding number of a \(F_{\sigma}\) relation small.
It is usual to use The First Preservation Theorem to preserve the right side of Cichoń’s diagram and to use The Second Preservation Theorem to preserve the left side of Cichoń’s diagram. But sometimes The Second Preservation Theorem is inconvenient since it does not help at successor steps. Then we develop methods using The First Preservation Theorem to preserve the left side of Cichoń’s diagram. This method also yields several new constellations of cardinal invariants, which we shall introduce.
This is joint work with Diego Mejía.