equivalence of forcing notionsEquivalenceOfForcingNotions2002-08-01 20:55:442003-01-11 18:15:41Definition% this is the default PlanetMath preamble. as your knowledge
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%\PMlinkescapeword{theory}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 $\mathfrak{M}[G]=\mathfrak{M}[H]$.