# Consistency of Strong reflection principle with the existence of a Suslin tree

In Woodin's book "The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal" Remark 2.55 (5), it states SRP by Todorcevic (defined below) is consistent with the existence of a Suslin tree (hence it does not imply MM). Is there any reference about this? My guess is we could probably get a restricted version of MM for a Suslin tree, like the one for PFA. More precisely, $PFA(S)$ is the forcing axioms for the class of forcings that preserve $S$. Then maybe we can derive SRP from $MM(S)$.

Definition (projectively stationary sets): Let $\lambda\geq \omega_1$, then $S\subset [H(\lambda)]^\omega$ is projectively stationary if for any stationary $T\subset \omega_1$, $\{X\in S: X\cap \omega_1\in T\}$ is stationary in $[\lambda]^\omega$.

Definition (SRP): SRP asserts for any projectively stationary $S\subset [H(\lambda)]^\omega$ for regular $\lambda\geq \omega_2$, there exists a continuous increasing $\in$-chain $\langle N_\alpha: \alpha<\omega_1\rangle$ such that $N_\alpha\in S$ for all $\alpha<\omega_1$.

## 1 Answer

This appears to be proven in the following paper of Miyamoto:

Miyamoto, Tadatoshi, On iterating semiproper preorders, J. Symb. Log. 67, No. 4, 1431-1468 (2002). ZBL1050.03034.

In section 5 of the paper, he introduces a forcing axiom, SPFA(Souslin), which is the forcing axiom for semi-proper posets which preserve every Souslin tree. He shows that SPFA(Souslin) is compatible with the existence of a Souslin tree and that SPFA(Souslin) implies SRP.