On the Weak Reflection Principle

Krueger, John

doi:10.1090/S0002-9947-2011-05310-9

On the Weak Reflection Principle
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by John Krueger PDF

Trans. Amer. Math. Soc. 363 (2011), 5537-5576 Request permission

Abstract:

The Weak Reflection Principle for $\omega _2$, or $\textrm {WRP}(\omega _2)$, is the statement that every stationary subset of $P_{ \omega _1}(\omega _2)$ reflects to an uncountable ordinal in $\omega _2$. The Reflection Principle for $\omega _2$, or $\textrm {RP}(\omega _2)$, is the statement that every stationary subset of $P_{ \omega _1 } ( \omega _2 )$ reflects to an ordinal in $\omega _2$ with cofinality $\omega _1$. Let $\kappa$ be a $\kappa ^+$-supercompact cardinal and assume $2^{\kappa } = \kappa ^+$. Then there exists a forcing poset $\mathbb {P}$ which collapses $\kappa$ to become $\omega _2$, and $\Vdash _{\mathbb {P}} \textrm {WRP}(\omega _2) \land \neg \textrm {RP}(\omega _2)$.

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