Abstract
The Weak Reflection Principle for ω 2 \omega _2 , or WRP ( ω 2 ) \textrm {WRP}(\omega _2) , is the statement that every stationary subset of P ω 1 ( ω 2 ) P_{ \omega _1}(\omega _2) reflects to an uncountable ordinal in ω 2 \omega _2 . The Reflection Principle for ω 2 \omega _2 , or RP ( ω 2 ) \textrm {RP}(\omega _2) , is the statement that every stationary subset of P ω 1 ( ω 2 ) P_{ \omega _1 } ( \omega _2 ) reflects to an ordinal in ω 2 \omega _2 with cofinality ω 1 \omega _1 . Let κ \kappa be a κ + \kappa ^+ -supercompact cardinal and assume 2 κ = κ + 2^{\kappa } = \kappa ^+ . Then there exists a forcing poset P \mathbb {P} which collapses κ \kappa to become ω 2 \omega _2 , and ⊩ P WRP ( ω 2 ) ∧ ¬ RP ( ω 2 ) \Vdash _{\mathbb {P}} \textrm {WRP}(\omega _2) \land \neg \textrm {RP}(\omega _2) .
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