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Toward the Consistency Strength of Stationary Set Reflection on Small Cardinals


A set $S \subseteq \omega_3$ is stationary in $\omega_3$ if it intersects all closed and unbounded subsets in $\omega_3$. We say that $S$ reflects below $\omega_3$ if there is some $\alpha<\omega_3$ for which $S\cap \alpha$ is stationary in $\alpha$.

In my thesis I obtain a model $M$ for which every subset of $\omega_3$ focusing on cofinality $\omega$ reflects at a point of cofinality $\omega_1$. That is, $M\models$ Refl$(\omega_3, \omega, \omega_1)$. To do this, I adjust the methods of Gitik in order to get set reflection. This argument may help show the way to getting the consistency strength.

Due to existing bounds, our strategy begins with $\kappa$ of Mitchell order $\kappa$ which concentrates on $\alpha^+$-supercompact cardinals $\alpha$. We collapse $\kappa$ to $\omega_3$, change the cofinality of many measurables between $\omega_2$ and $\omega_3$, and shoot clubs through the non-reflecting stationary sets. For any non-reflecting stationary set $S$, we use the forcing to add a club $C$ for which $C\cap S=\emptyset$, thus rendering this $S$ no longer stationary. We take care that any more stationary sets we add in this process, have a club shot through them and so are no longer stationary in the final generic extension. Additionally, $S\subseteq S^{\omega_3}_\omega$ sets can only have a club shot through them under certain circumstances. In our situation, we use the fact that the complement of these non-reflecting stationary sets will be fat, i.e. for any club $C \subseteq \alpha$, $E\cap C$ contains closed sets of ordinals of arbitrarily large order-type below $\alpha$, to allow us to shoot a club without collapsing $\kappa$.

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