Skip to main content
×
×
Home

A simple maximality principle

  • Joel David Hamkins (a1) (a2)
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 and all subsequent extensions Vℙ*ℚ 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 (◊ □ φ) ⇒ □ φ, 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 VδV 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 □ , which asserts that 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.

Copyright
References
Hide All
[1]Asperó, D., Bounded forcing axioms and the continuum. Ph.D. thesis, Universitat de Barcelona, 05 2000.
[2]Chalons, C., An axiom schemata, 1999, circulated email announcement.
[3]Chalons, C., Full set theory, 2000, electronic preprint.
[4]Hauser, K., The consistency strength of projective absoluteness. Annals of Pure and Applied Logic, vol. 74 (1995), no. 3, pp. 245295.
[5]Hughes, G. E. and Cresswell, M. J., An Introduction to Modal Logic, Methuan, London and New York. 1968.
[6]Jorgensen, M.. An equivalent form of LÉvy's Axiom Schema, Proceedings of the American Mathematical Society, vol. 26 (1970), no. 4, pp. 651654.
[7]Mitchell, B. and Schimmerling, E., Covering without countable closure, Mathematical Research Letters, vol. 2 (1995), pp. 595609.
[8]Mitchell, B. and Schindler, R., A universal extender model without large cardinals in V, this Journal, submitted.
[9]Stavi, J. and Väänänen, J., Reflection principles for the continuum. Logic and Algebra, American Mathematical Society Contemporary Mathematics Series, vol. 302, 07 2002.
[10]Steel, J., The core model iterability problem. Lecture Notes in Logic, vol. 8, Springer-Verlag, 1996.
[11]Steel, J., Core models with more Woodin cardinals, this Journal, vol. 67 (2002), pp. 11971226.
[12]Woodin, W. H., -Absoluteness andsupercompact cardinals. 05 1985, circulated notes.
[13]Woodin, W. H., Supercompact cardinals, sets of reals, and weakly homogeneous trees, Proceedings of the National Academy of Sciences, vol. 85 (1988), no. 18, pp. 65876591.
[14]Woodin, W. H., The Axiom of Determinacy, Forcing Axioms, and the Non-stationary Ideal, De Gruyter Series in Logic and its Applications, Walter de Gruyter, 1999.
[15]Woodin, W. H., On the consistency of Hamkins' axiom, 2001. lecture slides.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed