# equivalence of forcing notions

Let $P$ and $Q$ be two forcing notions such that given any generic subset $G$ of $P$ there is a generic subset $H$ of $Q$ with $\mathfrak{M}[G]=\mathfrak{M}[H]$ and vice-versa. Then $P$ and $Q$ are equivalent.

Since if $G\in\mathfrak{M}[H]$, $\tau[G]\in\mathfrak{M}$ for any $P$-name $\tau$, it follows that if $G\in\mathfrak{M}[H]$ and $H\in\mathfrak{M}[G]$ then $\mathfrak{M}[G]=\mathfrak{M}[H]$.

Title equivalence of forcing notions EquivalenceOfForcingNotions 2013-03-22 12:54:24 2013-03-22 12:54:24 Henry (455) Henry (455) 5 Henry (455) Definition msc 03E35 msc 03E40 equivalent Forcing ProofThatForcingNotionsAreEquivalentToTheirComposition