Journal of Symbolic Logic

A simple maximality principle

Joel David Hamkins

Abstract

In this paper, following an idea of Christophe Chalons, I propose a new kind of forcing axiom, the Maximality Principle, which asserts that any sentence φ holding in some forcing extension $V\P$ and all subsequent extensions V\P*\Qdot holds already in V. It follows, in fact, that such sentences must also hold in all forcing extensions of V. In modal terms, therefore, the Maximality Principle is expressed by the scheme $(\possible\necessaryφ)\implies\necessaryφ$, and is equivalent to the modal theory S5. In this article, I prove that the Maximality Principle is relatively consistent with \ZFC. A boldface version of the Maximality Principle, obtained by allowing real parameters to appear in φ, is equiconsistent with the scheme asserting that $Vδ\elesub V$ for an inaccessible cardinal δ, which in turn is equiconsistent with the scheme asserting that $\ORD$ is Mahlo. The strongest principle along these lines is $\necessary\MPtilde$, which asserts that $\MPtilde$ holds in V and all forcing extensions. From this, it follows that 0# exists, that x# exists for every set x, that projective truth is invariant by forcing, that Woodin cardinals are consistent and much more. Many open questions remain.

Article information

Source
J. Symbolic Logic, Volume 68, Issue 2 (2003), 527- 550.

Dates
First available in Project Euclid: 11 May 2003

https://projecteuclid.org/euclid.jsl/1052669062

Digital Object Identifier
doi:10.2178/jsl/1052669062

Mathematical Reviews number (MathSciNet)
MR1976589

Zentralblatt MATH identifier
1056.03028

Citation

Hamkins, Joel David. A simple maximality principle. J. Symbolic Logic 68 (2003), no. 2, 527-- 550. doi:10.2178/jsl/1052669062. https://projecteuclid.org/euclid.jsl/1052669062

References

• D. Asperó Bounded forcing axioms and the continuum, Ph.D. thesis, Universitat de Barcelona, May 2000.
• C. Chalons An axiom schemata,1999, circulated email announcement.
• K. Hauser The consistency strength of projective absoluteness, Annals of Pure and Applied Logic, vol. 74 (1995), no. 3, pp. 245--295.
• G. E. Hughes and M. J. Cresswell An Introduction to Modal Logic, Methuan, London and New York,1968.
• M. Jorgensen An equivalent form of Lévy's Axiom Schema, Proceedings of the American Mathematical Society, vol. 26 (1970), no. 4, pp. 651--654.
• B. Mitchell and E. Schimmerling Covering without countable closure, Mathematical Research Letters, vol. 2 (1995), pp. 595--609.
• B. Mitchell and R. Schindler A universal extender model without large cardinals in $V$, Journal of Symbolic Logic, submitted.
• J. Stavi and J. Väänänen Reflection principles for the continuum, Logic and Algebra, American Mathematical Society Contemporary Mathematics Series, vol. 302, July 2002.
• J. Steel The core model iterability problem, Lecture Notes in Logic, vol. 8, Springer-Verlag,1996.
• W. H. Woodin $\Undertilde\Sigma^2_1$-Absoluteness and supercompact cardinals, May 1985, circulated notes.