Abstract

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