February 8
Sean Cox,
Virginia Commonwealth University
Martin's Maximum and the Diagonal Reflection Principle
Several years ago I introduced the Diagonal Reflection Principle (DRP), a maximal form of simultaneous stationary reflection. Roughly, DRP asserts that for all regular $\theta \ge \omega_2$, there are stationarily many sets $W$ of size $\omega_1$ such that every stationary element of $W$ reflects to $W$ (i.e. if $S \subset [\theta]^\omega$ is stationary and $S \in W$, then $S \cap [W \cap \theta]^\omega$ is stationary). In that paper I showed that that $\text{MM}^{+\omega_1}$--even just the '$+\omega_1$' version of the forcing axiom for $\sigma$-closed forcings--implies DRP. In this talk I will prove that MM, even its technical strengthening $\text{MM}^{+\omega}$, does NOT imply DRP, contrary to my initial expectations. This is joint work with Hiroshi Sakai.