On the classification of definable ccc forcing notions

Abstract

We show that for a Suslin ccc forcing notion $\mathbb{Q}$ adding a Hechler real, ``$\text{ZF}+{\text{DC}}_{{\omega }_1}+$all sets of reals are $I_{\mathbb{Q},{\aleph }_0}$-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.

This is publication 1097 of the third author