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

Leibman, George

The Consistency Strength of

$\mathrm{MP_{CCC}}(\mathbb{R})$

Leibman, George

Notre Dame J. Formal Logic, Tome 51 (2010) no. 1, p. 181-193 / Harvested from Project Euclid

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Résumé

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

$R$ .

Publié le : 2010-04-15
Classification: forcing, forcing axioms, modal logic, 03E35, 03E40

@article{1276284781,
     author = {Leibman, George},
     title = {The Consistency Strength of $\mathrm{MP\_{CCC}}(\mathbb{R})$},
     journal = {Notre Dame J. Formal Logic},
     volume = {51},
     number = {1},
     year = {2010},
     pages = { 181-193},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1276284781}
}

Leibman, George. The Consistency Strength of $\mathrm{MP_{CCC}}(\mathbb{R})$. Notre Dame J. Formal Logic, Tome 51 (2010) no. 1, pp.  181-193. http://gdmltest.u-ga.fr/item/1276284781/