The Consistency Strength of M P C C C ( R )

Abstract

The Maximality Principle $\mathrm{M}{\mathrm{P}}_{\mathrm{C}\mathrm{C}\mathrm{C}}$ is a scheme which states that if a sentence of the language of ZFC is true in some CCC forcing extension $V^{\mathbb{P}}$, and remains true in any further CCC-forcing extension of $V^{\mathbb{P}}$, then it is true in all CCC-forcing extensions of V, including V itself. A parameterized form of this principle, $\mathrm{M}{\mathrm{P}}_{\mathrm{C}\mathrm{C}\mathrm{C}}(\mathbb{R})$, makes this assertion for formulas taking real parameters. In this paper, we show that $\mathrm{M}{\mathrm{P}}_{\mathrm{C}\mathrm{C}\mathrm{C}}(\mathbb{R})$ has the same consistency strength as ZFC, solving an open problem of Hamkins. We extend this result further to parameter sets larger than $\mathbb{R}$.