ON THE CLASSIFICATION OF DEFINABLE CCC FORCING NOTIONS

Abstract We show that for a Suslin ccc forcing notion Q $\mathbb Q$ double struck upper Q adding a Hechler real, “ ZF + DC ω 1 + $\mathrm {ZF}+\mathrm {DC}_{\omega _1}+$ upper Z upper F plus upper D upper C Subscript omega 1 plus all sets of reals are I Q , ℵ 0 $I_{\mathbb Q,\aleph _0}$ upper I Subscript double struck upper Q comma normal first transfinite cardinal 0 -measurable” implies the existence of an inner model with a measurable cardinal. We also introduce a wide class of Suslin ccc forcing notions which add a Hechler real, so that the above result applies to them.