On the classification of definable ccc forcing notions

TL;DR

The paper investigates the classification of certain Suslin ccc forcing notions that add a Hechler real, showing that specific measurability assumptions imply the existence of an inner model with a measurable cardinal.

Contribution

It introduces a broad class of Suslin ccc forcing notions adding Hechler reals and establishes a link between measurability assumptions and large cardinal existence.

Findings

Measurability assumptions imply inner models with measurable cardinals.

A new class of Suslin ccc forcing notions adding Hechler reals is introduced.

The classification connects forcing notions with large cardinal hypotheses.

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.