The Consistency Strength of $\mathrm{MP_{CCC}}(\mathbb{R})$

Notre Dame Journal of Formal Logic 51 (2):181-193 (2010)
Abstract
The Maximality Principle $\mathrm{MP_{CCC}}$ 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, $\mathrm{MP_{CCC}}(\mathbb{R})$ , makes this assertion for formulas taking real parameters. In this paper, we show that $\mathrm{MP_{CCC}}(\mathbb{R})$ has the same consistency strength as ZFC, solving an open problem of Hamkins. We extend this result further to parameter sets larger than R
Keywords forcing   forcing axioms   modal logic
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