On the classification of definable ccc forcing notions

We show that for a Suslin ccc forcing notion $\mathbb Q$ adding a Hechler real, ``$\text{ZF}+\text{DC}_{ω_1}+$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.

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