## Notre Dame Journal of Formal Logic

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

George Leibman

#### 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 $R$.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 51, Number 2 (2010), 181-193.

Dates
First available in Project Euclid: 11 June 2010

https://projecteuclid.org/euclid.ndjfl/1276284781

Digital Object Identifier
doi:10.1215/00294527-2010-011

Mathematical Reviews number (MathSciNet)
MR2667905

Zentralblatt MATH identifier
1205.03059

Keywords
forcing forcing axioms modal logic

#### Citation

Leibman, George. The Consistency Strength of $\mathrm{MP_{CCC}}(\mathbb{R})$. Notre Dame J. Formal Logic 51 (2010), no. 2, 181--193. doi:10.1215/00294527-2010-011. https://projecteuclid.org/euclid.ndjfl/1276284781

#### References

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